THE RISK-SPREAD OPTION IN A POTENTIAL THEORETIC FRAMEWORK

By

MICHAEL C. SWEARINGEN

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2000

Copyright 2000

by

Michael C. Swearingen

To My Late Father Thank You for a Lifetime of Encouragement

Wish You Were Here

iv

PREFACE

Participants in fixed-income markets trade in various securities whose value

ultimately depends upon a particular rate of interest at which a financial entity is willing

to issue debt. The financial entities involved may be governments or corporations, which

determines whether the interest rate is riskless or risky, respectively. The risk involved in

this context is the credit risk associated with the possibility of default on a corporate

bond. Since there is no credit risk with government bonds, it is reasonable to assume that

the risky interest rate should always be greater than the riskless interest rate. But, what

happens when there is inflation? Does the difference between these interest rates, or risk

spread, remain the same as the government rate rises? This raises the issue of hedging

against the inflation risk associated with corporate bonds.

The main result of this dissertation is the development and pricing of an originally

designed interest rate derivative which shall be known as the risk-spread option. This

option may be used by investors to hedge away the risk associated with the difference

between the government riskless interest rate and that of a corporate bond. The potential

theoretic approach to this pricing problem is general enough to generate various

stochastic models of both the riskless and risky interest rates. Moreover, it provides a

model that is analogous to physical systems that employ potential theory; thus, the

physics of fixed-income finance is illuminated revealing a mathematical structure behind

economic intuitions.

v

The first section of Chapter 1 establishes the basic definitions and assumptions in

fixed-income finance and arbitrage pricing theory that will be used throughout this work.

Sections 2 and 3 lay down a general potential theoretic framework in which to evaluate

interest rate derivatives. As an example of the procedure developed in these sections,

Section 4 uses an Ornstein-Uhlenbeck process to derive models for the risky and riskless

interest rates as well as bond prices. In Section 5, the risk-spread option is introduced by

means of a discrete example.

Chapter 2 contains the main theoretical work necessary to represent the price of the

risk-spread option as the solution to a Cauchy problem. In Chapter 3, the Fourier and

Laplace transforms are used to represent the solution in a more tractable form. Also, it is

shown how to hedge the risk-spread option using a portfolio of riskless and risky bonds.

In the final section of Chapter 3, the graphs of the riskless and risky yield curves are

displayed for various parameters. A summary of the results together with some remarks

on advantages, disadvantages, and proposed future improvements is found in the

concluding chapter.

vi

TABLE OF CONTENTS

page

PREFACE .......................................................................................................................... iv

ABSTRACT.....................................................................................................................viii

CHAPTERS

1 A POTENTIAL THEORETIC FRAMEWORK FOR FIXED-INCOME MARKETS....1

1.1. The Essentials of Mathema tical Finance ................................................................ 1 1.1.1 The Fundamentals of Fixed-Income Finance................................................ 2

1.1.2 A Review of Arbitrage Pricing Theory......................................................... 6 1.2 Potential Approach I: Riskless Bonds and the Martingale Measure....................... 8 1.3 Potential Approach II: Risky Bonds and the Forward Martingale Measure......... 12

1.3.1 Risky Bonds ................................................................................................ 13 1.3.2 The Forward Martingale Measure............................................................... 21

1.4 A Simple Example of the Potential Theoretic Approach...................................... 26 1.5 The Risk-Spread Option........................................................................................ 28

2 A CAUCHY PROBLEM FOR THE RISK-SPREAD OPTION ....................................31

2.1 Derivation of the Cauchy Problem........................................................................ 31 2.2 The Potential Theoretic Parametrix Method......................................................... 37

2.2.1 The Gaussian Semigroup............................................................................ 37 2.2.2 The Fundamental Solution ......................................................................... 39

2.3 Preliminary Technical Results .............................................................................. 43 2.3.1 Differentiability of the Gaussian Semigroup.............................................. 43 2.3.2 Basic Potential Theory ............................................................................... 48

2.4 The Derivatives of the Gaussian Potential............................................................ 54 2.5 A Series Representation of the Fundamental Solution ......................................... 61

2.5.1 Convergence and Continuity...................................................................... 62 2.5.2 Hölder Continuity....................................................................................... 65

2.6 The Solution to the Cauchy Problem.................................................................... 71 3 NUMERICAL RESULTS AND APPLICATIONS .......................................................73

3.1 The Fourier and Laplace Transforms .................................................................... 73 3.2 Delta Hedging with the Risk-Spread Option ........................................................ 79 3.3 Numerical Properties of the Yield Curve.............................................................. 82

vii

4 SUMMARY AND CONCLUSIONS ............................................................................ 91 4.1 Summary of Results .............................................................................................. 91 4.2 Future Projects and Model Extensions ................................................................. 92

REFERENCES ..................................................................................................................94

BIOGRAPHICAL SKETCH .............................................................................................96

viii

Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

THE RISK-SPREAD OPTION IN A POTENTIAL THEORETIC FRAMEWORK

By

Michael C. Swearingen

August 2000

Chairman: Joseph Glover Major Department: Mathematics

A fixed-income economy, which includes defaultable securities, is developed through

a potential theoretic approach to modeling the spot rate of interest. Under the assumption

of an arbitrage free market, the riskless and risky state-price densities are used as inputs

to generate the respective spot rates in a Markovian setting. The riskless state-price

density is simply the discounted conditional expectation of the derivative of the

martingale measure Q with respect to the reference probability P associated with the

underlying Markov process tX . The risky state-price density is an original modification

of its riskless counterpart. If the time to default is modeled as the first jump in a

generalized Poisson process with intensity ( )t tXλ = λ , then the risky state-price density

is defined as the discounted conditional expectation of the derivative of the forward

martingale measure F with respect to P. However, the discounting is done with respect to

the default intensity λ rather than the riskless spot rate. Furthermore, it is revealed

ix

through the resulting expression for the risky bond price, that the default intensity λ is the

risk spread between the riskless and risky spot rates.

The main example used to illustrate this procedure is the well-known Ornstein-

Uhlenbeck process from which a Cox-Ingersoll-Ross model of both spot rates is derived.

In addition to computing bond prices with this example, a Cauchy problem for an

originally designed option on the risk spread is derived through the Feynmann-Kac

Theorem. A series solution is then developed using a modern potential theoretic version

of the classical parametrix method for parabolic partial differential equations.

1

CHAPTER 1 A POTENTIAL THEORETIC FRAMEWORK FOR FIXED-INCOME MARKETS

1.1 The Essentials of Mathematical Finance

The two main prerequisites of mathematical finance that are imperative to an

understanding of this dissertation are fixed-income finance and arbitrage pricing theory.

This section begins by establishing the probabilistic setting in which these concepts

will be reviewed. According to Musiela and Rutkowski (1998), an economy is a family

of filtered probability space ( ) , , :Ω µ µ ∈F P , where the filtration t t [0,T]∈=F F

satisfies the usual conditions, and P is a collection of mutually equivalent probability

measures on the measurable space( )T,Ω F . We model the subjective market uncertainty

of each investor by associating to each investor a probability measure from P. Investors

with more risk tolerance will be represented by probability measures that weight

unfavorable events relatively lower, whereas conservative investors are characterized by

probability measures that weight unfavorable events relatively higher. Moreover, it is

assumed that investment information is revealed to each investor simultaneously as

events in the filtration F. Since the measures in P are mutually equivalent, the investors

agree on the events that have and have not occurred. It is convenient to further assume

that investors initially have no other information, i.e. 0F is trivial with respect to each

probability measure in P. This assumption asserts that the initial information available to

investors is objective.

2

1.1.1 The Fundamentals of Fixed-Income Finance The foundation of a working knowledge of fixed-income finance rests on an

understanding of the inherent relationship between the various interest rates and bonds.

Consider the economy ( ) , , :Ω µ µ ∈F P on the interval [ ]0,T and a Markov process

tX with ( )t sX :0 s t≡ σ ≤ ≤F . Implicit in this statement is the assumption that the state-

variable probability xP P≡ associated with tX belongs to P for some fixed element x of

the state space of tX . A zero-coupon bond, or discount bond, of maturity T is a security

that pays the holder one unit of currency at time T. The prices of government and

corporate discount bonds at time t ≤ T are denoted by B(t,T) and B(t,T)% , respectively.

The local expectations hypothesis (L-EH) relates the discount bond to the instantaneous

interest rate, or spot rate, for borrowing and lending over the time interval [ ]t , t dt+ .

Denote the riskless spot rate by ( )t tr r X= and assume that it is a nonnegative, adapted

process with almost all sample paths integrable on [0,T] with respect to Lebesgue

measure. The L-EH asserts that

( )( )T

P s ttB(t,T) E exp r X ds = −

∫ F (1.1)

According to Musiela and Rutkowski (1998), the economic interpretation of this

hypothesis is that “...the current bond price equals the expected value ... of the bond price

in the next (infinitesimal) period, discounted at the current short-term rate” (p. 283). This

statement is better understood in a discrete-time setting. In fact, using a left sum

approximation to the integral in (1.1) with the partition n

i i 0t

= of [ ]0,T yields

3

( )i 1

n

P t ii 1

B(0,T) E exp r X t−

=

= − ∆ ∑ (1.2)

( )( ) ( )0 i 1

n

P t 1 t ii 2

E exp r X t exp r X t−

=

= − ∆ − ∆ ∑

( )( ) ( )i 1 1

n

1 P P t i ti 2

exp r x t E E exp r X t−

=

= − − ∆ ∑ F

( )( ) ( )( )x1 P 1exp r x t E B t ,T= − .

Under the assumption of no arbitrage, it will be shown that (1.1) holds under the risk-

neutral measure in Section 2. Naturally, a similar relationship holds between the risky

bond B% and the risky spot rate tr% , which will be derived in Section 3.

The savings account tB is a process that represents the price of a riskless security that

continuously compounds at the spot rate. More precisely, it is the amount of cash at time

t that accumulates by investing one dollar initially, and continually rolling over a bond

with an infinitesimal time to maturity. Hence, we have

( )( )t

t s0B exp r X ds≡ ∫ . (1.3)

When a security tS is divided by the savings account, the resultant process is the price

process of the security discounted at the riskless rate.

Another bond of importance is known as the coupon bond, which pays the holder

fixed coupon payments 1 nc ,...,c at fixed times 1 nT,...,T with nT T= . The price of the

coupon bond is simply the present value of the sum of these cash flows. Denoting the

price of a riskless coupon bond at time t by cB (t,T) , we have

i

c i iT t

B (t,T) c B(t,T)>

= ∑ . (1.4)

A similar relationship holds for the risky coupon bond cB% .

4

In practice, the coupons are typically structured by setting ic c= for i 1,...,n 1= − ,

and nc N c= + , where N is the principal, or face value, and c is a fixed amount that is

generally quoted as a percentage of N called the coupon rate.

A problem that arises in comparing coupon bonds is that the uncertainty of the rate at

which the coupons will be reinvested causes uncertainty in the total return of the coupon

bond. Hence, coupon bonds of different coupon rates and payment dates are not directly

comparable. The continuously compounded riskless yield-to-maturity (YTM) Y(t,T) is

the unique solution to the equation

( )( )( )i

c i iT t

B (t,T) c exp Y t,T T t>

= − −∑ , (1.5)

and represents the total return on the coupon bond under the assumption that each of the

coupon payments occurring after t are reinvested at the rate Y(t,T) . The risky YTM

Y(t,T)% is defined in a similar fashion.

The interested reader should verify that there exists a unique, adapted, nonnegative

process 0 t TY(t,T)

≤ ≤ given the adapted coupon bond process, coupon payments, and

payment dates. In fact, this follows by noting that the LHS of (1.5) is a decreasing

function of Y, and that the price of a coupon bond will never exceed the sum of the

coupon payments.

The yield-to-maturity expectations hypothesis (YTM-EH) relates the riskless YTM

and the riskless spot rate. Musiela and Rutkowski (1998) state this hypothesis as the

assertion that “...the [continuously compounded] yield from holding any [discount] bond

is equal to the [continuously compounded] yield expected from rolling over a series of

single-period [discount] bonds” (p.284). To gain a better understanding of this statement,

5

we first observe that the YTM of a discount bond is simply the continuously compounded

interest rate. Hence, in a discrete-time setting with the partition n

i i 0t

= of [ ]t,T , we have

that the yield of a discount bond ( )i 1 iB t , t− is given by

( )i 1i 1 i tY(t , t ) r X−− = , (1.6)

from which we deduce that the bond price is given by

( ) ( )( )i 1i 1 i t iB t , t exp r X t−− = − ∆ . (1.7)

Since the YTM-EH asserts that the yield of ( )B t,T is equal to the yield expected

from rolling over a series of discount bonds ( )i 1 iB t , t− , it follows that

Y(t,T) ( )( )n

P i 1 i ti 1

1 1ln B t,T E ln B(t , t )

T t T t −=

≡ − = − − −

∏ F (1.8)

( )i 1

n

P t i ti 1

1E r X t

T t −=

= ∆ −

∑ F .

Taking the limit as the mesh of the partition tends to zero, we obtain the continuous-

time discount bond price and YTM under the YTM-EH:

( )T

P s ttB(t,T) exp E r X ds = − ∫ F (1.9)

and

( )T

P s tt

1Y(t,T) E r X ds

T t = − ∫ F . (1.10)

The last interest rate that we will consider is the instantaneous forward interest rate,

or forward rate for borrowing or lending over the time interval [ ]s,s ds+ as seen from

time t s≤ . This will be denoted by f(t, s) in the riskless case and f(t,s)% in the risky case.

If the dynamics of the process t s Tf(t,s)

≤ ≤ are specified, then the price of the

discount bond is defined by

6

( )T

tB(t,T) exp f(t,s)ds≡ −∫ . (1.11)

Alternatively, if the dynamics of the discount bond are known, then we have

f(t,T) lnB(t,T)T

∂≡ −

∂, (1.12)

provided that this derivative exists. By combining (1.9) and (1.12) we obtain

( )( )P T tf(t,T) E r X= F . (1.13)

Therefore, the YTM-EH asserts that the forward rate is an unbiased estimate of the

spot rate under the state-variable probability measure P. Under the assumption of no

arbitrage, it is shown in Section 3 that this holds under the forward martingale measure.

1.1.2 A Review of Arbitrage Pricing Theory

The terminology presented in this review may be found in Musiela and Rutkowski

(1998). Consider the economy ( ) , , :Ω µ µ ∈F P on the interval [ ]0,T . A trading

strategy, or portfolio, tφ is a vector of locally bounded, adapted processes of tradable

asset holdings. Moreover, it is assumed that every sample path is right continuous with

left limits. A trading strategy tφ is called self-financing, if the wealth process ( )tV φ of

the trading strategy neither receives nor pays out cash flows external to the assets that

comprise the strategy. More precisely, let iφ denote the holding of asset iS . Then, a

self-financing trading strategy ( )1 n,...,φ = φ φ is defined by asserting that ( )n

i it t t

i 1

V S=

φ ≡ φ∑

satisfies

( ) 1 1 n nt t t t tdV dS ... dSφ = φ + + φ . (1.14)

7

A self-financing strategy φ is called an arbitrage portfolio, if its associated wealth

process satisfies all of the following conditions for some (thus for all) P ∈ P:

• ( )0V 0φ = (Zero Investment)

• ( )( )TP V 0 1φ ≥ = (Zero Risk)

• ( )( )TP V 0 0φ > > (Possible Gain).

Hence, an investor taking advantage of an arbitrage opportunity may become infinitely

wealthy without risk. Under the assumption that arbitrage portfolios do not exist, it has

been shown that there exists a risk-neutral, or martingale measure, Q in our economy

under which the discounted asset process 1t t tZ B S−≡ follows a martingale. This result,

known as the Fundamental Theorem of Asset Pricing, is proven in a quite general setting

by Delbaen and Schachermayer (1994, 1998).

The next topic for review is the arbitrage pricing of financial derivatives. A self-

financing trading strategy φ is Q-admissible if the discounted wealth process

( ) ( )1t t tV B V−φ ≡ φ is a Q-martingale and uniformly bounded below with respect to

[ ]t 0,T∈ . The uniform boundedness condition is included to disallow trading strategies

in which the investor’s debt may become arbitrarily large. A contingent claim, or TF -

measurable random variable, C is Q-attainable if there exists a Q-admissible trading

strategy φ that replicates the value of C at time T (i.e. ( )TV Cφ = ). The market is

defined by M(Q) ≡ (S,Φ), where Φ consists of the Q-admissible trading strategies. A

market is said to be complete if every contingent claim is attainable.

Under the assumption of no arbitrage, an attainable claim C is uniquely replicated for

each martingale measure Q. In fact, we define the arbitrage price process ( )t C Qπ of C

8

to be the wealth process of the uniquely replicating trading strategy. Since this strategy is

Q-admissible, it follows that

( ) ( )1t t Q T tC Q B E B C−π = F . (1.15)

It is shown in Musiela and Rutkowski (1998) that ( ) ( )t 1 t 2C Q C Qπ = π for distinct

martingale measures 1Q and 2Q , if C is attainable with respect to both measures. Hence,

the definition of arbitrage price is independent of the choice of martingale measure and

will be denoted by ( )t Cπ . Therefore, if we assume that the market is complete, then the

pricing of contingent claims does not depend on the choice of martingale measure.

Alternatively, we may assume that the martingale measure is unique from which it

follows that the market is complete in the restricted sense that every contingent claim C

with ( )1T 1B C , ,Q− ∈ ΩL F is attainable (Björk, 1996). Either assumption will suffice for

the contingent claims considered in this dissertation. Furthermore, it will be shown in the

next section that the expression for the arbitrage price (1.15) can be rewritten with respect

to the state-variable probability P associated with the Markov process tX used to model

market uncertainty.

1.2 Potential Approach I: Riskless Bonds and the Martingale Measure

Equipped with the notions from our review of mathematical finance, we will now

present the potential approach to developing models of the riskless spot rate. The

fundamentals of this approach will carry over to the next section where a framework in

which credit derivatives such as risky bonds and the risk-spread option can be priced.

9

Consider the economy ( ) , , :Ω µ µ ∈G P on the interval [ ]0,T and a Markov

process tX with ( )t sX : 0 s t≡ σ ≤ ≤G . Combining the concepts of fixed-income finance

with those of arbitrage pricing theory, we see that a discount bond is simply a contingent

claim with the constant value one. Under the assumption of no arbitrage, it follows from

(1.15) that there exists a risk-neutral measure Q such that

( ) ( ) ( )T1

t t Q T t Q s ttB(t,T) 1 B E B E exp r(X )ds− = π = = −

∫G G . (2.1)

This proves the previously stated assertion that the L-EH given in (1.1) is satisfied under

the risk-neutral measure Q.

Rogers (1997) has shown that the expectation in (2.1) can be rewritten with respect to

the state-variable probability P using the state-price density

( )t 1t s P t t t0

dQexp r(X )ds E B N

dP−

ζ ≡ − =

∫ G , (2.2)

where t P t

dQN E

dP

≡

G . Before proving this result, we recall the following abstract

version of Bayes Rule:

Lemma 1 – Let Q and P be probabilities on the measurable space ( ),Ω J , H be a sub-

σ-algebra of J , ( )1f , ,Q∈ ΩL J , and dQ

NdP

≡ . Then

( ) ( )( )

PQ

P

E f NE f

E N=

HH H . (2.3)

proof : See pg. 458 of Musiela and Rutkowski (1998). <

10

Theorem 2 – For any contingent claim C, we have ( ) P T tt

t

E CC

ζ π =ζ

G. (2.4)

proof : From (1.15) and Lemma 1, it follows that

( ) ( ) ( )( )

1P T T t P T t1

t t Q T t ttP T t

E CB N E CC B E B C B

E N

−− ζ π = = =

ζ

G GG G . < (2.5)

An immediate consequence of Theorem 2 is the desired expression:

( ) P T t

t

EB t,T

ζ =ζ

G. (2.6)

This is the fundamental result of the potential approach. Since r is nonnegative, it is

easily verified that tζ is a positive supermartingale with respect to P. In fact, we have

( ) ( )( )t s

t s

P t s t P s t0

dQE E exp r X ds

dP+

+

+

ζ = −

∫

GG G (2.7)

( )( )t s

t

s P t0

dQexp r X ds E

dP+

≤ −

∫

GG

( )( )t

s P t t0

dQexp r X ds E

dP

= − = ζ

∫ G .

Furthermore, if we assume that ( )P tE 0ζ → as t → ∞, then tζ is a potential. From

(2.6) it follows that this assumption translates into the reasonable financial assumption

that the price of the riskless bond B(0,t) tends to zero as the time until maturity increases

to infinity.

The general potential approach to fixed-income finance outlined in Rogers (1997) is

to generate models of the spot rate through a judicious choice of the state-price density.

The only mathematical restriction is that tζ must be a positive supermartingale with

11

respect to P. A specific procedure is to choose a positive function g defined on the state

space of tX and use this to define tζ by

t tt

0

U g(X )e

U g(X )

α−α

αζ ≡ , (2.8)

where ( )0

U gα

α > is the α-potential of g defined by

( ) ( )( )x sP s0

U g x E e g X ds∞α −α= ∫ . (2.9)

Since this tζ is clearly nonnegative, we must only verify that it is a supermartingale

with respect to P. In fact, consider the martingale

( )st P s t0

M E e g X ds∞

−α = ∫ G . (2.10)

Applying the Markov property of tX , we deduce

( )tt t tM A e U g X−α α= + , (2.11)

where ( )t s

t s0A e g X ds−α≡ ∫ is an increasing process. It follows from (2.8) that

( )t t t0

1M A

U g(X )αζ = − (2.12)

from which we deduce that tζ is a supermartingale.

Given the above model for the state-price density, (2.6) may be employed to derive

the price of a riskless bond. We will now derive an expression for the riskless spot rate

by comparing (2.2) with (2.12). It follows from (2.2) that

1t t t t td B dN r(X ) dt−ζ = − ζ , (2.13)

On the other hand, we deduce from (2.12) that

( ) ( )( )t

t t t

0

1d dM e g X dt

U g X−α

αζ = − . (2.14)

12

Since tN is a (local) martingale, a comparison of (2.13) and (2.14) reveals the desired

expression for the riskless spot rate

( ) ( ) ( )tt t

t0 t t

e g X g Xr X

U g(X ) U g(X )

−α

α α= =ζ

, (2.15)

where the model of the state-price density (2.8) was used in the final equality.

Rather than specify g directly, it is convenient to model its α-potential by a

nonnegative function f, defined on the state space of tX , which lies in the domain of the

infinitesimal generator G of tX . With ( ) ( )t tf X U g Xα≡ we rewrite (2.8) as

( )( )

ttt

0

f Xe

f X−αζ = . (2.16)

Since Uα is the inverse of α – G, we have g = (α – G)f. Hence, the expression for the

spot rate (2.15) may be rewritten as

( ) ( )

( )t

tt

G f Xr

f X

α −= . (2.17)

1.3 Potential Approach II: Risky Bonds and the Forward Martingale Measure

The main goal of this section is to extend the results of the previous section to the

risky setting by developing a model in which the risky bond price is expressed in a

similar fashion to the riskless bond in (2.6). The main difference is that the riskless state-

price density tζ is replaced with the risky state-price density tϕ , which is expressed in

terms of the forward martingale measure instead of the risk-neutral measure.

Consider the economy ( ) , , :Ω µ µ ∈F P on the interval [ ]0,T with F defined

below. Let the Markov process tX denote the state-variable process, and ν be a random

13

time denoting the time of default. Following Lando (1998), we define F so as to make ν

a stopping time and adapted to tX as follows:

• tG ( )sX :0 s t≡ σ ≤ ≤

• tH [ ]( )s : 0 s t≡ σ ν > ≤ ≤

• tF t t≡ ∨G H . In the first of the following two sub-sections, an expression for the risky bond price is

derived under the assumption that ν may be represented by the first jump of a generalized

Poisson process. The forward martingale measure is introduced in the second sub-

section to derive the risky bond analog of (2.6).

1.3.1 Risky Bonds

A risky bond is a contingent claim that pays the holder one unit of currency at

maturity in the event that there is no default. Hence, under the assumption of no

arbitrage, it follows from (1.15) that there exists a risk-neutral measure Q such that

[ ]( ) [ ]( ) ( ) [ ]T

1t t Q T t Q s tT T Tt

B(t,T) 1 B E B 1 E exp r(X )ds 1−ν > ν > ν >

= π = = − ∫% F F . (3.1)

In applying the potential approach to risky bonds, it will be convenient to rewrite the

conditional expectation in (3.1) with respect to tG . Before this can be achieved, we will

recall the Monotone Class Theorem (MCT) and use it to prove some preliminary results.

Theorem 1 (The Monotone Class Theorem) – Let A and D be collections of subsets

of a set C . Then ( )σ ⊆A D if the following conditions are satisfied:

(i) A B∈∩ A for every A and B in A

(ii) ⊆A D (iii) ∈C D (iv) B \ A ∈D for every A and B in D with A ⊂ D

(v) ii 1

A∞

=

∈∪ D for every increasing sequence i i 1A

∞

= of sets from D .

14

A proof of this result may be found in Blumenthal and Getoor (1968) among other places.

The first result that we will prove using the MCT is

Lemma 2 – For every t T≤ and tA ∈ H , we have that either

[ ]( ) [ ]( )T TQ t A Q tν > = ν >∩ G G (3.2)

or

[ ]( )TQ t A 0ν > =∩ G . (3.3)

proof: It suffices to show that for every t T≤ , tA ∈ H , and TC∈G we have that either

[ ]( ) [ ]( )Q t A C Q t Aν > = ν >∩ ∩ ∩ (3.4)

or

[ ]( )Q t A C 0ν > =∩ ∩ . (3.5)

Let t T≤ . We will apply the MCT with ≡ ΩC ,

[ ] t : 0 s t≡ ν > ≤ ≤A , (3.6)

and

[ ]( ) [ ]( ) t TA :Q t A C Q t C or 0 for every C≡ ∈ ν > = ν > ∈∩ ∩ ∩D H G . (3.7)

Let 1 20 s s t≤ < ≤ . Since [ ] [ ] [ ]1 2 2s s sν > ν > = ν >∩ , it follows that A is closed

under intersections and ⊆A D . Hence, A satisfies the hypotheses of the MCT. We

proceed to verify the hypotheses on D .

Since it is clear that Ω ∈D , we begin by showing that D is closed under proper

differences. Let A ∈ D , B∈D , and TC∈G with A B⊂ . We have

[ ]( )( ) [ ]( ) [ ]( )( )Q B \ A t C Q B t C \ A t Cν > = ν > ν >∩ ∩ ∩ ∩ ∩ ∩ (3.8)

[ ]( ) [ ]( )Q B t C Q A t C= ν > − ν >∩ ∩ ∩ ∩ .

If [ ]( )Q B t C 0ν > =∩ ∩ , then the RHS of (3.8) is also equal to zero since A B⊂ .

So, assume that [ ]( )Q B t C 0ν > >∩ ∩ . Since B∈D we have that

15

[ ]( ) [ ]( )Q B t C Q t Cν > = ν >∩ ∩ ∩ . (3.9)

Hence, if [ ]( )Q A t C 0ν > =∩ ∩ , then the RHS of (3.8) is equal to [ ]( )Q t Cν > ∩ . On

the other hand, if [ ]( )Q A t C 0ν > >∩ ∩ , then

[ ]( ) [ ]( )Q A t C Q t Cν > = ν >∩ ∩ ∩ (3.10)

since A ∈ D . From (3.9) and (3.10) we deduce that the RHS of (3.8) is zero. It follows

that B \ A ∈D .

The final hypothesis of the MCT that must be verified is that D is closed under

increasing sequences. Let i i 1A

∞

= be an increasing sequence of sets from D and define

ii 1

A A∞

=

≡ ∪ . We will show that A ∈ D . We begin by defining the pairwise disjoint

sequence of sets i i 1B

∞

= by 1 1B A≡ and i 1 i 1 iB A \ A+ +≡ . Since D is closed under

proper differences, we have that iB ∈D for every i 1≥ . Furthermore, it is clear that

ii 1

A B∞

=

= ∪ . It follows that

[ ]( ) [ ] [ ]( )i ii 1i 1

1 Q A t C Q B t C Q B t C∞ ∞

==

≥ ν > = ν > = ν >

∑∩ ∩ ∩ ∩ ∩ ∩∪ . (3.11)

Since this series is finite, we have that

[ ]( )iQ B t C 0ν > =∩ ∩ (3.12)

for all but finitely many i. Define

[ ]( ) iI i 1 :Q B t C 0≡ ≥ ν > >∩ ∩ . (3.13)

Since iB ∈D for every i 1≥ , it follows that

[ ]( ) [ ]( ) [ ]i ii I i I

Q A t C Q B t C Q B t C∈ ∈

ν > = ν > = ν >

∑∩ ∩ ∩ ∩ ∩ ∩∪ , (3.14)

[ ]( ) [ ]( )maxiQ A t C Q t C≤ ν > ≤ ν >∩ ∩ ∩ ,

16

where maxi max i I= ∈ and the last inequality holds since maxiA ∈D . Furthermore,

since iB ∈D for every i I∈ , we also have that

[ ]( ) [ ]( ) [ ]( )ii I

Q A t C Q B t C nQ t C∈

ν > = ν > = ν >∑∩ ∩ ∩ ∩ ∩ , (3.15)

where n denotes the cardinality of I. Comparing (3.14) with (3.15), we see that n must be

zero or one from which we deduce that A ∈ D .

Therefore, we conclude from the MCT that ( )t = σ ⊆H A D . This implies that

either (3.2) or (3.3) holds for every t T≤ and tA ∈ H . <

Theorem 3 – For every t T≤ we have

( ) [ ]( )( )

TT t t

T

Q T |Q T | 1

Q t |ν>

ν >ν > ∨ =

ν >GG H G . (3.16)

proof : Define ( )t TY Q t≡ ν > G for every t T≤ and rewrite (3.16) as

[ ]( ) [ ]Q t T t TT tE 1 Y 1 Yν> ν >∨ =G H . (3.17)

In the following proof, be aware that tY is TG -measurable for each t T≤ and is not

necessarily tG -measurable.

Since the RHS of (3.17) is measurable with respect to T t∨G H , it suffices to show

that for every t T≤ and T tD ∈ ∨G H we have

[ ]( ) [ ]( )Q t Q TT D t DE 1 Y E 1 Yν> ν >=∩ ∩ . (3.18)

Let t T≤ . We will prove (3.18) using the MCT with ≡ ΩC ,

T tA B : A ,B≡ ∈ ∈∩A G H , (3.19)

and

[ ]( ) [ ]( ) T t Q t Q TT D t DD : E 1 Y E 1 Yν> ν>≡ ∈ ∨ =∩ ∩D G H . (3.20)

17

We begin by checking that A satisfies the hypotheses of the MCT. Clearly, A is

closed under intersections. Let A B∈∩ A for some TA ∈G and tB∈H . We will show

that ⊆A D by considering the following two cases:

Case 1 – [ ]( )TQ T B 0ν > >∩ G

Since t TB ∈ ⊆H H , it follows from Lemma 2 that

[ ]( ) ( )T TQ T B Q Tν > = ν >∩ G G . (3.21)

Furthermore, since [ ] [ ]T tν > ⊆ ν > we have that

[ ]( ) [ ]( )T TQ t B Q T B 0ν > ≥ ν > >∩ ∩G G . (3.22)

Thus, another application of Lemma 2 yields

[ ]( ) ( )T TQ t B Q tν > = ν >∩ G G . (3.23)

From (3.21) and (3.23) we deduce that

[ ] ( )( )Q tT A BE 1 Yν> ∩ ∩ [ ]( )( ) [ ]( )( )Q A t T Q A t TE 1 Y Q T B E 1 Y Q T= ν > = ν >∩ G G (3.24)

( ) ( )( )Q A t T Q A T TE 1 Y Y E 1 Y Q t= = ν > G

[ ]( )( ) [ ] ( )( )Q A T T Q Tt A BE 1 Y Q t B E 1 Yν>= ν > = ∩ ∩∩ G .

Hence, A B ∈∩ D which implies that ⊆A D in this case.

Case 2 – [ ]( )TQ T B 0ν > =∩ G

From the assumption of this case, we have that

[ ] ( )( ) [ ]( )( )Q t Q A t TT A BE 1 Y E 1 Y Q T B 0ν> = ν > =∩ ∩ ∩ G . (3.25)

So, it suffices to show that

[ ] ( )( )Q Tt A BE 1 Y 0ν> =∩ ∩ . (3.26)

18

Hence, we may assume without loss of generality that TY 0> . This implies that

[ ]( ) [ ]( )C CT TQ t B Q T Bν > ≥ ν >∩ ∩G G (3.27)

[ ]( ) [ ]( )CT T TQ T B Q T B Y 0= ν > + ν > = >∩ ∩G G .

It follows from Lemma 2 that

[ ]( ) ( )CT TQ t B Q tν > = ν >∩ G G (3.28)

since CtB ∈H . On the other hand, we have that

( ) [ ]( ) [ ]( )CT T TQ t Q t B Q t Bν > = ν > + ν >∩ ∩G G G . (3.29)

Combining (3.28) and (3.29) we obtain

[ ]( )TQ t B 0ν > =∩ G (3.30)

from which we deduce that

[ ] ( )( ) [ ]( )( )Q T Q A T Tt A BE 1 Y E 1 Y Q t B 0ν> = ν > =∩ ∩ ∩ G . (3.31)

Therefore, we deduce from (3.25) and (3.31) that ⊆A D in this case as well.

We proceed to verify that D satisfies the hypotheses of the MCT. Clearly, Ω ∈D .

In fact, we have that

[ ]( ) ( )( ) ( )Q t Q t T Q t TTE 1 Y E Y Q T E Y Yν> = ν > =G (3.32)

( )( ) [ ]( )Q T T Q TtE Y Q t E 1 Yν>= ν > =G .

The next step is to show that D is closed under proper differences. Let A and B be

sets in D with A B⊂ . We deduce that B \ A ∈D from the following result:

[ ] ( )( )Q tT B\AE 1 Yν> ∩ [ ]( ) [ ]( )( ) [ ]( )( ) [ ]( )( )Q t Q t Q tT B \ T A T B T AE 1 Y E 1 Y E 1 Y

ν> ν > ν> ν >= = −∩ ∩ ∩ ∩ (3.33)

[ ]( )( ) [ ]( )( )Q T Q Tt B t AE 1 Y E 1 Y

ν> ν>= −∩ ∩

[ ]( ) [ ]( )( ) [ ] ( )( )Q T Q Tt B\At B \ t AE 1 Y E 1 Yν>ν> ν>

= = ∩∩ ∩ .

19

The final hypothesis of the MCT that must be verified is that D is closed under

increasing sequences. Let i i 1A

∞

= be an increasing sequence of sets from D and define

ii 1

A A∞

=

≡ ∪ . We will show that A ∈ D . As in Lemma 2, we begin by defining the

pairwise disjoint sequence of sets i i 1B

∞

= by 1 1B A≡ and i 1 i 1 iB A \ A+ +≡ . Since D is

closed under proper differences, we have that iB ∈D for every i 1≥ . Furthermore, it is

clear that ii 1

A B∞

=

= ∪ . Thus, it follows that A ∈ D from the following equality:

[ ]( ) [ ]( ) [ ]( ) [ ]( )i iQ t Q t Q T Q TT A T B t B t A

i 1 i 1

E Y 1 E Y 1 E Y 1 E Y 1∞ ∞

ν> ν> ν> ν >= =

= = =∑ ∑∩ ∩ ∩ ∩ . (3.34)

Hence, we conclude from the MCT that ( )T t∨ = σ ⊆G H A D . Thus, (3.18) holds

for every t T≤ and T tA ∈ ∨G H from which we obtain the desired result (3.16). <

We now return to the problem of rewriting the conditional expectation in (3.1) with

respect to tG . Conditioning with respect to T t∨G H first yields

( )T

Q Q s [ T] T t ttB(t,T) E E exp r(X )ds 1 ν>

= − ∨ ∫% G H F (3.35)

( ) ( )T

Q s T t ttE exp r(X )ds Q T | = − ν > ∨ ∫ G H F

[ ] ( ) ( )( )

T TQ s tt t

T

Q T1 E exp r(X )ds

Q tν>

ν >= −

ν > ∫

G FG ,

where the last inequality follows from Theorem 3. This expression for the risky bond is

more attractive than (3.1) since the argument of the conditional expectation is now TG -

measurable. However, we are still unable to replace the conditioning on tF with

conditioning on tG . This will require an independence assumption and the following

lemma:

20

Lemma 4 – Let 1F , 2F , and 3F be sub σ-algebras of the σ-algebra F such that 1 2∨F F

is independent of 3F . Then for every integrable, 1F -measurable function Y we have

( ) ( )2 3 2E Y | E Y |∨ =F F F .

proof : See Section 9.2 of Chung (1974). <

Assuming that TG is independent of tH for each t T≤ , we may apply Lemma 4 with

1 T≡F G , 2 t≡F G , and 3 t≡F H to (3.35) to obtain

( ) [ ] ( ) ( )( )

T TQ s tt t

T

Q TB t,T 1 E exp r(X )ds

Q tν>

ν >= −

ν > ∫ % G GG . (3.36)

The financial interpretation of this independence assumption is that the riskless spot

rate up until the time horizon T is independent of the default status of the bond prior to

maturity. We will now present a model of the default time in which this independence

assumption is somewhat relaxed.

As in Lando (1998), we model the default time by t

s0inf t 0 : (X )dsν ≡ > λ ≥∫ E ,

where E is distributed under Q as a unit exponential random variable that is independent

of TG , and λ is a nonnegative, continuous function on the state space. The default time

may be regarded as the first jump in a generalized Poisson process with intensity

( )t

t s0X dsΛ ≡ λ∫ , and we deduce that ( ) ( )T tQ t | expν > = −ΛG . In fact, we clearly have

that tΛ is measurable with respect to tG . So, it suffices to show that for every tA ∈G we

have [ ]( ) ( )( )Q A tQ t A E 1 expν > = −Λ∩ . It follows that

[ ]( ) [ ]( ) ( ) ( )( )t

ut Q A Q A tQ t A Q A E 1 e du E 1 exp

∞ −

Λν > = Λ < = = −Λ∫∩ ∩E . (3.37)

21

Hence, (3.35) becomes

[ ] ( )( )T

Q s tt tB(t,T) 1 E exp r X dsν>

= − ∫% % F , (3.38)

where r r≡ + λ% . Now, since E is independent of TG , we may apply Lemma 4 with

1 T≡F G , 2 t≡F G , and ( )3 ≡ σF E to obtain

( )( ) ( ) ( )( )T T

Q s t Q s tt tE exp r X ds E exp r X ds − ∨ σ = − ∫ ∫% %G E G . (3.39)

Since ( )t t t t t⊆ ≡ ∨ ⊆ ∨ σG F G H G E , we may condition with respect to tF on both

sides of (3.39) to obtain

( )( ) ( )( )T T

Q s t Q s tt tE exp r X ds E exp r X ds − = − ∫ ∫% %F G . (3.40)

Combining this with (3.38) we deduce the desired expression for the risky bond:

[ ] ( )( )T

Q s tt tB(t,T) 1 E exp r X dsν>

= − ∫% % G . (3.41)

We conclude that the process tλ represents the risk spread between the riskless spot

rate tr and risky spot rate tr% . Intuitively, this makes sense since the probability of default

increases with tλ .

1.3.2 The Forward Martingale Measure

The next step before applying the potential approach to risky bonds is to develop an

understanding of the forward martingale measure (FMM) which is denoted by F. We

define this probability measure through the derivative

t

tt

dF B(t,T)Y

dQ B B(0,T)≡ ≡

G. (3.42)

22

It should be obvious from (2.1) that (3.42) defines a Q-martingale, since it is shown there

that the discounted bond price follows a martingale under Q.

The FMM arises when pricing forward contracts in a market without arbitrage.

Formally, a forward contract is an agreement established at time 0t T< to exchange an

asset for a prearranged delivery price at time T. More precisely, we have

Definition 3 – A forward contract written at time 0t T< on an attainable contingent

claim C for settlement at time T with delivery price K is an attainable contingent claim

H C K≡ − , where the delivery price is a fixed amount of cash determined at time 0t .

Since there is no initial exchange of money between the participants of a forward

contract, the delivery price must be set such that ( )0t H 0π = . In other words, the delivery

price K is set equal to the arbitrage price of C. If this is not the case, then it can be shown

that an arbitrage portfolio exists.

Example 4 – Let H be a forward contract as in the previous definition with 0t 0= , and

suppose that ( )0 H 0π > . Let Ctφ and H

tφ denote the replicating trading strategies of the

attainable claims C and H, respectively. Then an arbitrage portfolio can be constructed

by taking a long position in the forward contract, short selling Ctφ , and purchasing

( )( )0 C

B 0,T

π riskless bonds of maturity T. Denoting this portfolio by tψ we see that

( )

( )0 C H

t t t

C,0,...,0

B 0,T

πψ = − φ + φ

, (3.43)

where the first coordinate denotes riskless bond holdings. It follows that

( ) ( )( ) ( ) ( ) ( ) ( ) ( )0 C H C

0 0 0 0 0

CV B 0,T V V C V 0

B 0,T

πψ = − φ + φ = π − φ ≡ . (3.44)

This implies that the trading strategy ψ requires no initial investment. Furthermore,

23

( )TV ψ ( )

( ) ( ) ( ) ( ) ( )( ) ( )0 0C H

T T

C CB T,T V V C C K

B 0,T B 0,T

π π= − φ + φ = − + − (3.45)

( ) ( )( )

( )( )

0 0C KB 0,T H0

B 0,T B 0,T

π − π= = > a.s.

Hence, ψ also satisfies the zero risk and possible gain conditions of an arbitrage

portfolio. In fact, ψ represents extreme arbitrage in the sense that a positive profit will

almost surely be realized.

On the other hand, if ( )0 H 0π < then an arbitrage portfolio can be constructed by

taking a short position in the forward contract, short selling ( )

( )0 C

B 0,T

π riskless bonds of

maturity T, and purchasing Ctφ . That is, we construct an arbitrage portfolio by negating

the holdings of the previous case. Therefore, ( )0 H 0π = in an arbitrage free market. <

Although the forward contract has an initial price of zero, its arbitrage price may

fluctuate before it matures. We define the forward price of C at time t as the delivery

price for which ( )t H 0π = and form the adapted process ( ) 0

C t t TF t,T

≤ ≤. It follows from

(3.45) that the forward price is given by

( ) ( )( )t

C

CF t,T

B t,T

π= . (3.46)

Hence, the forward price C is simply the arbitrage price of C discounted by the

riskless bond price. This is not surprising since the FMM is a special case of a change of

numeraire with the riskless bond chosen as the new numeraire (Björk, 1996).

The next result justifies the name “forward martingale measure” for F.

Theorem 5 – Let C be an attainable contingent claim that is integrable with respect to F.

Then, the forward price process ( ) C 0 t TF t,T

≤ ≤ is a martingale under F.

24

proof : Since C is integrable with respect to F and ( )CF T,T C= , it suffices to show that

for every t ∈ [0, T] we have

( ) ( )C F tF t,T E C= G . (3.47)

We apply Bayes rule (Lemma 2.1) to obtain

( ) ( )( ) ( )Q T t 1

F t Q t T tQ T t

E Y CE C E Y Y C

E Y−= =

GG GG , (3.48)

where the martingale tY is defined in (3.42). From (3.42) and (3.46) we deduce

( ) ( )( )

( )( ) ( )

1t Q T t t

F t C

B E B C CE C F t,T

B t,T B t,T

−π

= = =G

G . < (3.49)

Theorem 6 – [ ] ( )( )T

F s tt tB(t,T) 1 B(t,T)E exp X dsν>

= − λ ∫% G . (3.50)

proof : An immediate consequence of (3.49) is

( ) ( ) ( )t F tC B t,T E Cπ = G . (3.51)

A comparison of (1.15) (with tF replaced by tG ) and (3.51) yields

( )( ) ( ) ( ) ( )T

Q s t t F ttE exp r X ds C C B t,T E C − = π =

∫ G G . (3.52)

Applying (3.52) to the attainable contingent claim ( )( )T

stC exp X ds= − λ∫ , we obtain the

desired result (3.50) from the risky bond equation (3.38). <

It follows from this theorem that the YTM-EH (1.13) holds under the forward

martingale measure. In fact, it follows from (3.52) with TC r= that

( ) ( ) ( )( )T

F T t Q T s tt

1E r E r exp r X ds

B t,T = − ∫G G (3.53)

( ) ( )( )T

Q s tt

1E exp r X ds

B t,T T

∂ = − − ∂ ∫ G

( ) ( ) ( ) ( )1

B t,T lnB t,T f t,TB t,T T T

∂ ∂= − = − =∂ ∂

.

25

We proceed to extend the potential approach to the risky bond. A comparison of the

expectation in (2.1) with that in (3.50) provides motivation for the following:

Definition 7 – The risky state-price density is given by ( )t

t t

dFexp

dPϕ ≡ −Λ

G.

It follows from Theorem 6 that we may proceed as in Theorem 2.2 to obtain the risky

analog of (2.6):

[ ] ( ) P T t

tt

EB(t,T) 1 B t,Tν >

ϕ =ϕ

% G. (3.54)

In fact, it follows from Bayes Rule (Lemma 2.1) that

( )( )( )( )T

P s ttT P T tF s tt

tP t

dFE exp X ds

EdPE exp X ds

dFE

dP

− λ ϕ − λ = = ϕ

∫∫

G GGG

. (3.55)

Hence, (3.54) follows from (3.50) and (3.55). Therefore, the risky bond price may be

determined by specifying the risky state-price density in a similar fashion to the

procedure outlined in Section 2.

We finish this section by noting that the risky state-price density may be expressed in

terms of its riskless counterpart and the risk spread. From the relation

( ) ( )

t t tt

tt

dF dF dQ B(t,T) dQ B(t,T)

dP dQ dP B B 0,T dP B 0,T= = = ζ

G G GG (3.56)

it follows that

( ) ( )t t t

B(t,T)= exp

B 0,Tϕ −Λ ζ . (3.57)

Inserting this into (3.54) yields

[ ]

( )( )T

P s T tt

tt

E exp X dsB(t,T) 1 ν >

− λ ζ =ζ

∫%G

. (3.58)

26

This expression for the risky bonds resembles (2.5); however, it does not follow

directly from Theorem 2.2 since [ ]t1 ν> is not tG -measurable. This result illustrates the

importance of the independence assumption used to derive (3.41).

1.4 A Simple Example of the Potential Theoretic Approach

Consider the economy ( ) , , :Ω µ µ ∈F P on the interval [ ]0,T with F defined as in

the beginning of the previous section. Let the state-variable process tX denote the

Gaussian diffusion with state space dR satisfying

t t tdX dW X dt= − θ (4.1) for some positive parameter θ, where tW is a d-dimensional Brownian motion. Hence,

tX is the well-known stationary Ornstein-Uhlenbeck process given by

( )tt st 0 s0

X e X e dW−θ θ= + ∫ . (4.2)

It follows that tX has distribution ( )t0 tN e X , V−θ , where ( )2 t

t

1V 1 e

2− θ≡ −

θ. Also, the

generator of this process is given by

d

2i

i 1 i

1 fGf f x

2 x=

∂≡ ∇ − θ

∂∑ (4.3)

for every f ∈ ( )2 dC R . Define the function df : +→R R by

2

f(x) exp xd

α =

(4.4)

for some positive parameter α. It follows from (2.17) that

( ) ( )

( )2t

t tt

G f X 1r X

f X 2

α −= = σ , (4.5)

where ˆ ˆ4 ( )σ ≡ α θ − α and ˆd

αα ≡ . Since the riskless spot rate must be positive, this

27

induces the condition ˆθ > α . We will show in Chapter 3 that this is the well-known

mean-reverting Cox-Ingersoll-Ross (CIR) process.

Next, the riskless state-price density given by (2.16) is

( )( ) ( )t

tt 0 t

0

e f(X )exp t r exp r

f(X )

−α

ζ = = − α + κ κ , (4.6)

where2

d

ακ ≡

σ. Hence, the price of the riskless bond may be calculated from (2.6):

( )B t,T ( ) ( )( ) ( )( )P T t

t P T tt

Eexp r E exp r

ζ= = − α τ + κ κ

ζ

G G (4.7)

( )( ) ( )( )tX

t Pexp r E exp rτ= − α τ + κ κ

( ) ( )( )d

2t

ˆˆ1 2 V exp V r−τ τ= − α − +ατ ,

where V

Vˆ1 2 Vτ

ττ

≡− α

and T tτ ≡ − . We deduce that the riskless YTM is given by

( ) ( ) ( )t

1 1 dˆ ˆY t,T lnB t,T V r ln 1 2 V2τ τ

≡ − = +ατ+ − α τ τ . (4.8)

Finally, we use (1.12) to derive an expression for the riskless forward rate:

( ) ( ) ( )( )

tlnB t,T exp 2 r ˆf t,T 2 V

ˆ ˆT 1 2 V 1 2 V ττ τ

∂ − θτ = − = − αα ∂ − α − α

. (4.9)

Similarly, in the risky setting we use a function dh : +→R R defined by

2

h(x) exp xd

β =

(4.10)

to model the risky state-price density, where β is a positive parameter. We deduce that

( ) ( )

( )2t

t tt

G h X 1X

h X 2

β −λ = = σ% , (4.11)

where ˆ ˆ4 ( )σ ≡ β θ − β% and ˆ

ˆd

ββ ≡ . Since the risk spread must be positive, this induces the

28

condition ˆθ > β . It follows that the risky state-price density is given by

( )( ) ( )t

tt 0 t

0

e h(X )exp t exp

h(X )

−β

ϕ = = − β + κλ κλ% % , (4.12)

where2

d

βκ ≡

σ% % . From (3.54), (4.11), and (4.12) we obtain the risky analog of (4.7):

B(t,T)% [ ] ( ) P T t

tt

E1 B t,Tν >

ϕ =ϕ

G (4.13)

[ ] ( )( ) ( )( )d2

ttˆ1 B t,T 1 2 V exp V

−

τ τν >= − β − λ +βτ% ,

where V

V ˆ1 2 Vτ

τ

τ

≡− β

% . We deduce that the risky YTM is given by

( ) ( ) [ ] ( ) ( )tt

1 1 d ˆY t,T lnB t,T 1 Y t,T V ln 1 2 V2

τ τν >

≡ − = + λ +βτ+ − β τ τ % % % . (4.14)

We conclude this example with an expression for the risky forward rate:

( ) ( )[ ] ( ) ( )

( )t

t

lnB t,T exp 2 ˆf t,T 1 f t,T 2 VˆˆT 1 2 V1 2 V

τν >ττ

∂ − θτ λ = − = + − ββ ∂ − β− β

%% . (4.15)

1.5 The Risk-Spread Option

Continuing with the economy of the previous section, we will conclude this chapter

by presenting the payoff of the risk-spread option. Consider an investor who purchases a

risky bond at t 0= that matures at time T. We would like to construct an option that will

guarantee that he would receive a minimum return of γ above the riskless spot rate. We

define the risk-spread option by its payoff of

( )( )T

s0C exp ds 1

+= γ − λ −∫ , (5.1)

and we refer to γ as the risk-spread insurance level.

29

We begin with a simple deterministic, discrete-time example under the assumption

that the risky bond does not default. Suppose that the riskless spot rate is given by

1 0t

2 0

p if t tr

p if t t

<= ≥

, (5.2)

and that the risk spread is given by

1 0t

2 0

q if t t

q if t t

<λ = ≥

, (5.3)

where 1 2q q> γ > . From (5.1) and (5.3) we have that

( )( )( )2 0C exp q T t 1= γ − − − . (5.4)

Since the risky rate t t tr r= + λ% is deterministic, it follows from (3.41), (5.2) and (5.3) that

( ) ( ) ( )( ) ( )( )( )T

s 0 1 1 0 2 20B 0,T exp r ds exp t p q exp T t p q= − = − + − − +∫% % . (5.5)

Because we are assuming that the risky bond does not default, the investor receives one

dollar at maturity. Hence, the total (continuously compounded) return the investor

receives from the option and the risky bond at maturity is

R( ) ( )0

1 1 Cln

T B 0,T C

+= + π % (5.6)

Assuming that ( )0 C 0π = , we have that

( )0

0 00 t

t T tR r r

T T

−= + + γ% .

It can be seen from this result that the investor is compensated when the risk spread

drops below the insurance level γ after time 0t . If 0t 0= , then the investor is guaranteed

30

to receive a return of γ above the riskless spot rate. On the other hand, if 0t T= , then the

option expires out-of-the-money and the investor receives the risky spot rate of return.

However, from (1.15) we see that ( ) 10 TC B C 0−π = > . This implies that the return on

the risk-spread option is the riskless spot rate. In fact, the rate of return of holding any

deterministic derivative is the riskless spot rate. Of course, there is no need for the risk-

spread option in a deterministic world.

In the general stochastic, continuous-time case we deduce from (5.4) and (5.6) that

R ( )( )

( ) ( )

T

s0

0

exp ds1ln

T B 0,T C

+ γ − λ = + π

∫% (5.7)

( ) ( ) ( )( )( )T

s 00

1ds ln B 0,T C

T

+= γ − λ − + π∫ % .

We would like to find 0γ >% such that

( )( )T

s s0

1R r ds

T

+= + γ − λ∫ % % . (5.8)

If there exists a constant γ% such that (5.7) and (5.8) are equal, we define the effective

risk-spread insurance level γ% as the solution to

( )( ) ( )( ) ( ) ( )( )P T P T T 0E E r ln B 0,T CT

+ + ∂γ − λ = γ − λ − − + π

∂%% . (5.9)

In the next chapter, the general stochastic, continuous-time case for the risk-spread

option is studied in detail. In particular, a representation for the arbitrage price of this

option is derived as the solution to a Cauchy problem.

31

CHAPTER 2 A CAUCHY PROBLEM FOR THE RISK-SPREAD OPTION

2.1 Derivation of the Cauchy Problem

Consider the economy ( ) , ,P : PΩ ∈F P on the interval [ ]0,T with F defined as in

Section 1.3. Let ( )T

t0C exp ( (X )) dt 1+= γ − λ −∫ denote the payout at time T of a risk-

spread option with fixed risk-spread insurance level γ, where t 0 t TX ≤ ≤ is the state-

variable process with natural filtration tG and probability P ∈P defined in Section 1.4.

According to (1.2.6), the price of this option at time t is given by

( )t Cπ ( )( )( )( )T

P s T t0P T t

t t

E exp X ds 1E C+ γ − λ − ζ ζ = =

ζ ζ

∫ GG (1.1)

( )T

P s T t0

t

E exp dsB(t,T)

Ψ ζ = −ζ

∫ G,

where

( ) ( )( )s s sX X+

Ψ ≡ Ψ ≡ γ − λ (1.2)

and ( )B t,T denotes the riskless bond price.

Using the model of the state-price density developed in the simple example of Section

1.4, it follows from (1.4.6) that

( )T

P s T t0E exp ds Ψ ζ ∫ G ( )( ) ( ) ( )T

0 P T s t0exp T r E exp r exp ds = − α + κ κ Ψ ∫ G (1.3)

( )( ) ( ) ( ) ( )tt X

0 s P s0 0exp T r exp ds E exp r exp ds

τ

τ = − α + κ Ψ κ Ψ ∫ ∫ .

32

Combining (1.1) and (1.3) yields

( ) ( )( ) ( ) ( ) ( )t

t t s t0C exp r exp ds u X , B t,Tπ = − α τ + κ Ψ τ −∫ . (1.4)

where

( ) ( ) ( ) ( ) ( )x xP s P s0 0

u x, E exp r exp ds E g X exp dsτ τ

τ τ τ ≡ κ Ψ = Ψ ∫ ∫ , (1.5)

( )2ˆg(x) exp x≡ α , d

αα ≡ , and T tτ ≡ − .

The goal of this section is to prove that (1.5) is the unique solution of class

( )2,1 du C [0,T]∈ ×R to the following Cauchy problem:

Lu(x, ) 0τ = (1.6)

for every ( ) dx, (0,T]τ ∈ ×x R , and

u(x,0) g(x)= (1.7)

for every dx ∈R , where

L G∂

≡ + Ψ −∂τ

(1.8)

and G is the generator of the state-variable process given by (1.4.3). This result is known

as the Feynman-Kac Theorem and is proven in (Karatzas & Shreve, 1991) under the

assumption that there are constants M 0> and 1µ ≥ such that for every dx ∈¡ we have

( )2

0 Tmax u(x, ) M 1 x µ

≤τ≤τ ≤ + . (1.9)

We will extend this result by replacing (1.9) with a less restrictive bound using the

following lemma, which appears as Problem 3.4.12 in (Karatzas & Shreve, 1991).

Lemma 1 – Let t t 0 tM ,

≤ < ∞G be a continuous, real-valued martingale such that 0M 0=

almost surely. Also, let tC be a continuous, real-valued process of bounded variation

such that t tC M+ ≤ ρ almost surely for some 0ρ > and every t 0≥ . Then, for every

n 2> ρ , the semimartingale t t tY C M≡ + satisfies

( ) ( )21

2t

0 t T

nP max Y n 3 2 exp

8−

≤ ≤

≥ ≤ πρ − ρ

. (1.10)

33

proof : We will first construct a time-changed Brownian Motion as in (Karatzas &

Shreve, 1991). For every t 0≥ define the stopping time

s

t

inf s 0 : M t if 0 t M

if t M

∞

∞

≥ > ≤ <τ ≡ ∞ ≥

. (1.11)

Without loss of generality1, assume that our probability space is rich enough to contain a

standard one-dimensional Brownian motion tt 0 t

B , τ ≤ < ∞% G that is independent of

1 See Remark 3.4.1 in (Karatzas & Shreve, 1991) for a technical justification.

t 0 tY

≤ < ∞ and define

tt t t MB B B M

∞τ∧≡ − +% % (1.12)

for every t 0≥ . It follows from Problem 4.7 of (Karatzas & Shreve, 1991) that

tt 0 t

B , τ ≤ < ∞G is a standard one-dimensional Brownian motion such that the filtration

t 0 tτ ≤ <∞

G satisfies the usual conditions and for every [ ]t 0,T∈ we have

t

t MM B= a.s. (1.13)

For every positive integer n, we define the stopping time

n t

nR inf t 0 : B

≡ ≥ ≥ 2 . (1.14)

Let n 2> ρ and note that

t t t t t

nM Y C Y Y

2≥ − ≥ − ρ ≥ − (1.15)

for every [ ]t 0,T∈ . Hence,

[ ]t t n n0 t T 0 t T

nmax Y n max M R RΤ≤ ≤ ≤ ≤

≥ ⊆ ≥ ⊆ Μ ≥ ⊆ ρ ≥ 2 . (1.16)

Using integration by parts, it can be verified that for every x 0> we have

2 2

x

u 1 xexp du exp

2 x 2

∞ − ≤ −

∫ . (1.17)

34

From (1.17), the symmetry of Brownian motion, and the reflection principle (Revuz &

Yor, 1991), it follows that

( )t0 t TP max Y n

≤ ≤≥ ( )n t t0 t 0 t

n nP R P max B 2P maxB

2 2≤ ≤ρ ≤ ≤ρ

≤ ≤ ρ = ≥ ≤ ≥

(1.18)

2

n2

n 4 u4P B exp du

2 22ρ

∞

ρ

= ≥ ≤ − π ∫

( )2 21

26 n n

exp 3 2 expn 2 8 8

− ρ≤ − ≤ πρ − π ρ ρ . <

We will use this lemma to prove a version of the Feynman-Kac Theorem for the

state-variable process tX defined in Section 1.4. Recall that

( )t t stt 0 s0

X e X e dW−θ −−θ= + ∫ , (1.19)

where tW is a d-dimensional Brownian motion and θ is a positive parameter. Define

tt t 0 tY e X X Mθ≡ = + , (1.20)

where the process t t 0 t

M ,≤ < ∞

G is the continuous, real-valued martingale defined by

t s

t s0M e dWθ≡ ∫ (1.21)

for every t 0≥ . Furthermore,

( ) ( )t 2 s 2 t 2 T

t 0

1 1M e ds e 1 e 1

2 2θ θ θ= = − ≤ −

θ θ∫ . (1.22)

Hence, for every t 0≥ we have

( )2 T0 0t

1X M X e 1

2θ+ ≤ ρ ≡ + −

θ. (1.23)

Therefore, it follows from Lemma 1 that

( ) ( )21

2t

0 t T

nP max Y n 3 2 exp

8−

≤ ≤

≥ ≤ πρ − ρ

. (1.24)

35

Theorem 2 (Feynman-Kac) – Let ( )2,1 du C [0,T]∈ ×R satisfy the Cauchy problem

given by

Lu(x, ) 0τ = (1.25)

for every ( ) dx, (0,T]τ ∈ ×x R , and

u(x,0) g(x)= (1.26)

for every dx ∈R , where

L G∂

≡ + Ψ −∂τ

(1.27)

and G is the generator of the state-variable process defined by

d

2i

i 1 i

1 fGf f x

2 x=

∂≡ ∇ − θ

∂∑ (1.28)

for every f ∈ ( )2 dC R . Furthermore, assume that there exists positive constants K and h,

with 1

h8 d

<ρ

and ρ defined by (1.23), such that for every dx ∈R we have

( ) ( )2

0 Tmax u x, Kexp h x

≤τ≤τ ≤ . (1.29)

Then, u is uniquely represented by

( ) ( ) ( )xP s0

u x, E g X exp dsτ

τ τ = Ψ ∫ . (1.30)

proof : Let ( ) dx, (0,T]τ ∈ ×x R with 0X x= almost surely. We will apply ˆIto's formula

to the process s 0 sV

≤ ≤ τ defined by

( )s s sV u X , s≡ τ − E . (1.31)

where

( )s

s q0exp dq≡ Ψ∫E (1.32)

It follows that for every [ ]s 0,∈ τ we have

( ) ( ) ( )i 2s 0 q q q q q q

i

d s s

0 0i 1

V V u X , q dX u X , q dqx q

=

∂ ∂− = τ − + ∇ + + Ψ τ − ∂ ∂ ∑∫ ∫E E . (1.33)

Recalling that tX satisfies t t tdX dW X dt= − θ , we see that the first term becomes

36

( ) ( ) ( ) ( )i iq q q q q q

i i

d s s

0 0i 1

u X , q dW X u X , q dqx x

=

∂ ∂τ − − θ τ − ∂ ∂ ∑ ∫ ∫E E . (1.34)

Combining (1.25), (1.27), (1.28), (1.33), and (1.34) we deduce that

( ) ( )is 0 q q q

i

d s

0i 1

V V u X , q dWx

=

∂− = τ −∂∑∫ E (1.35)

for every [ ]s 0,∈ τ . Moreover, the expectation of the RHS of (1.35) is zero, hence

( ) ( ) ( )( )x x0 P s P s su x, V E V E u X , sτ = = = τ − E . (1.36)

Let n tS inf t 0 : Y n d≡ ≥ ≥ for every n ≥ 1, and fix ( )m 0,∈ τ . Then,

( )u x,τ ( )( )n n

xP S m n S mE u X , S m∧ ∧= τ − ∧ E (1.37)

( ) [ ]( ) ( ) [ ]( )n nn n

x xP m m P S n SS m S mE u X , m 1 E u X , S 1> ≤= τ − + τ −E E .

As n → ∞ and m ↑ τ , we see that the first term on the RHS of (1.37) approaches

( )( ) ( ) ( )x xP P s0

E u X ,0 E g X exp dsτ

τ τ τ = Ψ ∫E . (1.38)

From (1.2), (1.24), and (1.29), we see that the second term is dominated by

( ) [ ]( )n n n

xP S n S S mE u X , S 1 ≤τ − E ( ) [ ]( )n n

xP S n S mE u X , S 1 ≤≤ γ τ − (1.39)

( ) [ ]( )n n

2xP S S TKE exp h X 1 ≤≤ γ ( ) ( )2 x

nKexp hdn P S T≤ γ ≤

( ) ( )( )di2 x

t0 t Ti 1

Kexp hdn P max Y n≤ ≤

=

≤ γ ≥∑

( ) ( )2d 1

2 2

i 1

nKexp hdn 3 2 exp

8−

=

≤ γ πρ − ρ

∑

( )1

221 8 hd

3 Kd 2 exp n8

− − ρ≤ γ πρ − ρ

,

which approaches zero as n → ∞ since 1 8h d 0− ρ > . Therefore, by combining (1.38)

and (1.39) with (1.37) we deduce the desired result (1.30). <

37

The main goal of this chapter is to construct a unique solution ( )2,1 du C [0,T]∈ ×R to

this Cauchy problem that satisfies (1.29). The next section introduces the basic

terminology inherent in the potential theoretic parametrix method. In Sections 3 and 4,

the relevant potentials are studied and technical differentiability results are provided. The

fifth section provides a series representation for the fundamental solution of the Cauchy

problem and establishes some continuity results. Finally, the chapter concludes by

constructing the solution to the Cauchy problem in Section 6.

2.2 The Potential Theoretic Parametrix Method

2.2.1 The Gaussian Semigroup The solution presented in this section updates the parametrix method (Friedman,

1964) by using the modern theories of potentials and semigroups. The canonical family

of semigroups associated with the differential operator L is the Gaussian family

Z : 0 T, 0δτ < τ ≤ δ > defined by

( )d 22 x

Z x exp2 2

δτ

δδ ≡ − πτ τ (2.1)

for every ( ) dx, (0,T]τ ∈ ×R and 0δ > .2 We will omit the subscript τ when we wish to

refer to the Gaussian semigroup as a function on d (0,T]×R .

For each (0,T]τ∈ , we define the operator [ ]Zδτ ⋅ by

[ ]( ) ( ) ( )dZ f x Z f (x) Z (x y)f y dyδ δ δ

τ τ τ≡ ∗ ≡ −∫R (2.2)

for every ( )df C∈ R for which this convolution exists. Because of the bound (1.29) , it is

important that the convolution is finite for the following subset of ( )dC R :

2 The superscript δ will be dropped for the standard Gaussian semigroup (i.e. δ = 1).

38

Definition 1 – For every positive δ, let

( ) ( )2df C : f Aexp h for some positive h and A 032 dδ

δ≡ ∈ ≤ ⋅ < >

ρ RA (2.3)

denote the set of δ-admissible functions and denote 1A by A , where

( )2 T0

T 1max , X e 1 .

4d 2θ ρ ≡ + − θ

Also, the bound parameter of a δ-admissible function

f is defined by

( )2

fh inf 0 h : f Aexp h for some A 032 d

δ≡ < < ≤ ⋅ >

ρ . (2.4)

It follows that the initial function g appearing in our Cauchy problem (1.26) is δ-

admissible for some δ depending on α . The next result gives an example of a class of

admissible functions for which the convolution in (2.2) is finite.

Lemma 2 – Let (0,T]τ∈ , 0 2< ε ≤ δ , and f ε∈A . Then [ ] 2Z fδτ ε∈A . More precisely,

for every dx ∈R we have that

[ ]( )Z f xδτ ( ) ( )2

fd

exp h Z x dδτ≤ −∫R

ξ ξ ξ (2.5)

( )d2 2 2f

ff f

hexp x Cexp h x

2 h 2 h

δ δ= ≤ δ − τ δ − τ % ,

where

d2

f

C2Th

δ≡ δ − and f

f ff

hh 2h

2Th

δ≡ <

δ −% .

proof: We first note that since Zδτ and f are continuous functions on dR , it follows

from (2.5) that [ ] ( )dZ f Cδτ ∈ R . Therefore, to prove that [ ] 2Z fδ

τ ε∈A , it suffices to

verify that (2.5) holds. After completing the square, the integrand becomes

d 22 2f f

f f

h ( 2 h )exp x exp x

2 2 h 2 2 h

δ δ δ − τ δ − − πτ δ − τ τ δ − τ ξ . (2.6)

Since f 2h δ∈A we have f2 h 0δ − τ > , thus we see that (2.5) holds for every dx ∈R . <

39

The final property of the family of Gaussian semigroups that we will review is

continuity in time. Clearly, for every dx ∈R and f ε∈A with 0 2< ε ≤ δ , the maps

( )Z xδττ a and [ ]( )Z f xδ

ττ a are continuous. Furthermore, we may continuously

extend the action of the Gaussian semigroup by defining [ ]( ) [ ]( )00

Z f x limZ f x f(x)δ δτ

τ→≡ = .

In fact, this follows from the weak convergence of the measure defined by the Gaussian

semigroup to the point mass measure as 0τ → .

2.2.2 The Fundamental Solution

We begin this section by noting that for a given function ( )dF C (0,T]∈ ×R , we will

refer to the family (0,T]

Fτ τ∈ as a continuous family of functions from ( )dC R , where

( ) ( )F x F x,τ ≡ τ for every ( ) dx, (0,T]τ ∈ ×R .

Definition 3 – A fundamental solution for L is a continuous family ( ]0,Tτ τ∈Γ of

functions from ( )dC R such that for every (0,T]τ∈ and f ∈A we have:

(i) [ ]( ) ( )L f L f 0τ τΓ ≡ Γ ∗ =

(ii) [ ] ( )00

f lim f fττ→

Γ ≡ Γ ∗ = .

It will be shown that the solution to the Cauchy problem given by (1.25) and (1.26)

may be represented by [ ]( )u(x, ) g xττ = Γ , (2.7)

where the fundamental solution τΓ shall be constructed using the following parametrix

method. Consider the following Cauchy problem for a given f ∈A :

pL u(x, ) 0τ = (2.8)

u(x,0) f(x)= , (2.9)

where the principal part pL of the differential operator L is defined by

2p

1L

2

∂≡ ∇ −

∂τ. (2.10)

40

It follows from the Feynman-Kac Theorem that the solution to this simple heat equation

may be represented by

( ) [ ]u x, Z fττ = . (2.11)

A fundamental solution for pL will be known as a parametrix associated with L. In

our case, it follows that the Gaussian semigroup Zτ is a parametrix for L. This result will

be used to motivate an expression for a fundamental solution for L that satisfies (2.7).

Definition 4 – The Gaussian potential is a family of operators 0 TU τ ≤τ≤

defined by

[ ] [ ]s0U f Z f ds

τ

τ τ−≡ ∫ (2.12)

for every ( )df C∈ R for which this integral is defined. For the continuous family

0 Tf fτ <τ≤

≡ of functions from ( )dC R , we define

[ ] [ ]s s0U f Z f ds

τ

τ τ−≡ ∫ . (2.13)

As with the Gaussian semigroup, we will omit the subscript τ when we wish to refer to

the Gaussian potential as a function on d (0,T]×R .

Recall that our state-variable process satisfies

t t tdX dW X dt= − θ . (2.14)

Hence, if θ and Ψ are identically zero, then it follows from (1.5) and (2.14) that

PL L= . This inspires us to look for a fundamental solution for L that satisfies (2.7) in

terms of the Gaussian potential of some continuous family 0 Tτ <τ≤ϕ ≡ ϕ of functions

from ( )dC R , as follows:

[ ]Z Uτ τ τΓ = + ϕ . (2.15)

The function ϕ will be determined by condition (i) of Definition 3. This condition

implies that for every (0,T]τ∈ and f ∈A we have

[ ]( ) [ ]( ) [ ][ ]( ) ( ) [ ]( )0 L f L Z f L U f L Z f L U fτ τ τ τ τ= Γ = + ϕ ≡ ∗ + ϕ ∗ . (2.16)

41

Consequently, we are led to an investigation of the differential properties of the

Gaussian semigroup and the Gaussian potential. In particular, we will show in Section 3

that the differential operator L may be passed through the convolution

[ ]Z fτ . More explicitly, for every ( ) dx, (0,T]τ ∈ ×R and f ∈A we will show that

[ ]( )( ) ( ) ( )dL Z f x LZ x f dτ τ= −∫R

ξ ξ ξ . (2.17)

In Section 4, we will derive sufficient conditions on ϕ so that the differential operator

L may be passed through the convolution [ ][ ]LU fτ ϕ . More precisely, it will be shown

that [ ]U τ ϕ is well-defined provided that there exists positive constants C and δ

with 1δ ≤ such that for every ( ) dx, (0,T]τ ∈ ×R we have

( ) ( )x CZ xδτ τϕ ≤ . (2.18)

In this case, we shall say that ϕ is Z -boundedδ . Furthermore, if ϕ is locally, uniformly

Hölder continuous in space, then we will prove that

[ ][ ]( ) [ ][ ]( ) [ ]s s0L U f L Z f ds f

τ

τ τ− τϕ = ϕ − ϕ∫ (2.19)

for every (0,T]τ∈ and f ∈A , where

[ ]f fτ τϕ ≡ ϕ ∗ (2.20)

and

[ ][ ]( ) [ ]( )s s s sL Z f L Z fτ− τ−ϕ ≡ ϕ ∗ . (2.21)

In Section 5, we will construct a series representation for a function ϕ satisfying

(2.16) and prove that it satisfies the conditions above, which have been asserted to be

sufficient for ϕ to satisfy (2.19). We will define ϕ in terms of the following potential:

42

Definition 5 – The L-potential is a family of operators 0 TVτ ≤τ≤

defined by

[ ] [ ]( )s0V f L Z f ds

τ

τ τ−≡ ∫ (2.22)

for every ( )df C∈ R for which this integral is defined. For the continuous family

0 Tf fτ <τ≤

≡ of functions from ( )dC R , we define

[ ] [ ]( )s s0V f L Z f ds

τ

τ τ−≡ ∫ . (2.23)

As with the Gaussian Potential, we will omit the subscript τ when we wish to refer to

the L-potential as a function on d (0,T]×R . It follows that (2.19) may be rewritten as

[ ][ ]( ) [ ][ ] [ ]L U f V f fτ τ τϕ = ϕ − ϕ , (2.24)

where [ ][ ] [ ]V f V fτ τϕ ≡ ϕ ∗ . (2.25)

Combining (2.24) with (2.16), we deduce that ϕ must satisfy the following Volterra

integral equation for every (0,T]τ∈ and f ∈A :

[ ] [ ]( ) [ ][ ]f L Z f V fτ τ τϕ = + ϕ . (2.26)

It is shown in Section 5 that the solution of (2.26) may be represented by the series

[ ]m

m 0

V LZ∞

τ τ=

ϕ = ∑ , (2.27)

where mVτ is an operator defined by

0V Iτ ≡ (2.28) and m 1 mV V V+

τ τ τ≡ o (2.29)

for every (0,T]τ∈ and m 0≥ .

43

2.3 Preliminary Technical Results 2.3.1 Differentiability of the Gaussian Semigroup In this sub-section we begin presenting the technical results that are required to

complete the construction of a fundamental solution as outlined in Section 2. The main

result is that for every ( ) dx, (0,T]τ ∈ ×R and f ∈A we have

[ ]( )( ) ( ) ( )dL Z f x LZ x f dτ τ= −∫R

ξ ξ ξ . (3.1)

This will require a few preliminary results.

Lemma 1 – Let ( ) dx, (0,T]τ ∈ ×R , and let n, m, ε, and δ be positive constants with

ε < δ . For every constant µ with 2m n j

02

− +≤ µ ≤ , there exists a positive constant A

such that for j = 0,1, and 2 we have

( )n j

2m n j 2m ji

x Z A(x) Z x

x x

δεττ− + − µµ

∂≤

τ ∂ τ (3.2)

Furthermore, for every constant µ with 2m n 2

02

− +≤ µ ≤ , there exists a positive

constant B such that

( )n

2m n 2 2m

x Z B(x) Z x

x

δεττ− + − µµ

∂≤

τ ∂τ τ. (3.3)

proof : Let n, m, δ, and ε be positive with ε < δ and define ( )2

xW x,

2τ ≡

τ. Then for

every µ with 2m n

02

−≤ µ ≤ there is a positive constant A such that

44

n

m

xZ (x)δ

ττ

d 22nm x

x exp2 2

− δδ = τ − πτ τ

(3.4)

( ) ( )( )

( )dm2

2m n 2

2W(x, ) exp W(x, )exp W(x, )

2x

−µ

− − µµ

τ − δ − ε τ δ = −ε τ πτ τ

( )2m n 2

AZ x

xετ− − µµ

≤τ

.

This establishes (3.2) for j 0= . For j 1= we deduce from (3.4) that

( )n

mi

x Zx

x

δτ∂

τ ∂ ( ) ( )

n n 1

im m 1

x xxZ x Z x

+δ δτ τ+

δ= ≤ δ

τ τ τ (3.5)

( )2m n 1 2

AZ x

xετ− + − µµ

≤τ

%

holds for every 2m n 1

02

− +≤ µ ≤ . Similarly,

( )n 2

m 2i

x Zx

x

δτ∂

τ ∂ ( ) ( )

n n 2

im mi

x x xZZ x x 1 Z x

x

δδ δττ τ

δ δ ∂ δ= + ≤ − τ τ ∂ τ τ τ (3.6)

( )n 2 n

m 2 m 1

x xZ x

+δτ+ +

≤ δ δ +

τ τ ( )2m n 2 2

AZ x

xετ− + − µµ

≤τ

holds for every 2m n 2

02

− +≤ µ ≤ , which implies (3.2) for j = 2. Finally, we obtain

(3.3) from the following inequality:

( )n

m

x Zx

δτ∂

τ ∂τ ( ) ( )

n 2 n 2 n

m 2 m 2 m 1

x x x d xdZ x Z x

2 2 2 2

+δ δτ τ+ +

δ δ= − ≤ +

τ τ τ τ τ (3.7)

( ) ( )n 2 n

d 2m n 2 2m 2 m 1

T x d x AZ x Z x

2 2 x

+δ ετ τ+ − + − µ+ + µ

δ≤ + ≤

τ τ τ . <

We will now apply this lemma to obtain a result on the differentiability of the

Gaussian semigroup.

45

Theorem 2 – For every 0δ > and f δ∈A we have [ ] ( )2,1 dZ f C (0,T]δ ∈ ×R . In fact,

[ ]( ) ( )j j

j ji i

dZ f x Z x f ( )d

x xδ δτ τ

∂ ∂= −

∂ ∂∫Rξ ξ ξ (3.8)

and

[ ]( ) ( )d

Z f x Z x f ( )dδ δτ τ

∂ ∂= −

∂τ ∂τ∫Rξ ξ ξ (3.9)

for every ( ) dx, (0,T]τ ∈ ×R and j 1,2∈ . Furthermore, [ ]j

ji

Z fx

δτ

∂∂

and [ ]Z fδτ

∂∂τ

are

functions in 2δA for every (0,T]τ∈ .

proof: Let 0δ > and f δ∈A . Clearly, the integrands above are continuous functions

of ( )x,τ on d (0,T]×R . Hence, to prove assertions (3.8) and (3.9) as well as show that

[ ] ( )2,1 dZ f C (0,T]δ ∈ ×R , it suffices to show that the integrals in these assertions are

locally, uniformly bounded with respect to ( ) dx, (0,T]τ ∈ ×R .

Let f δ∈A , j 1,2∈ , and dB : 1≡ ∈ <Rξ ξ . From (3.8) we obtain

( ) ( ) ( )C

j j j

j j ji i i

d B BZ x f ( ) d Z x f ( ) d Z x f ( ) d

x x xδ δ δτ τ τ

∂ ∂ ∂− ≤ − + −∂ ∂ ∂∫ ∫ ∫R

ξ ξ ξ ξ ξ ξ ξ ξ ξ . (3.10)

Clearly, the first integral on the RHS of (3.10) is bounded uniformly in

( ) dx, (0,T]τ ∈ ×R . For the second integral, consider a compact subset dM ⊆ R and let

( )x, M (0,T]τ ∈ × . We deduce from Lemma 1 with 0µ = that for every positive ε < δ

there exists a positive constant A such that

( )C

j

jiBZ x f ( )d

xδτ

∂ −∂∫ ξ ξ ξ ( ) ( )

C

j 2

fB

A x Z x exp h x d− ε

τ≤ − −∫ ξ ξ ξ (3.11)

( ) ( )C

2

fB

A exp h x Z x dετ≤ −∫ ξ ξ .

Since we may choose ε arbitrarily close to δ, we may assume that f ε∈A . It follows

from Lemma 2.2 that there exists positive constants C and f fh 2h<% such that

46

( ) ( ) ( ) ( )C

2 2 2f f f

Bexp h x Z x d Cexp h x Cexp h Nε

τ − ≤ ≤∫ % %ξ ξ , (3.12)

where ( )N diam M= .

By combining (3.10), (3.11), and (3.12), we deduce that the integral in (3.8) is locally,

uniformly bounded with respect to ( ) dx, (0,T]τ ∈ ×R . Moreover, this result may also be

established for the integral in (3.9) by repeating the proof using (3.3) to obtain the

analogue of (3.11). Consequently, we have shown that assertions (3.8) and (3.9) hold for

every ( ) dx, (0,T]τ ∈ ×R and have proven that [ ] ( )2,1 dZ f C (0,T]δ ∈ ×R . Finally, it

follows from (3.12) that [ ]j

ji

Z fx

δτ

∂∂

and [ ]Z fδτ

∂∂τ

are in 2δA for every (0,T]τ∈ . <

We proceed to prove the main result of this sub-section.

Theorem 3 – For every f ∈A we have [ ] ( )dLZ f C (0,T]∈ ×R . In fact,

[ ]( )( ) ( )dL Z f x LZ x f ( )dτ τ= −∫R

ξ ξ ξ (3.13)

for every ( ) dx, (0,T]τ ∈ ×R . Furthermore, [ ] 2LZ fτ ∈A for each (0,T]τ∈ .

Remark: The differential operator L acts in the variables ( ) dx, (0,T]τ ∈ ×R .

proof: Clearly, the integrand in (3.13) is continuous in ( ) dx, (0,T]τ ∈ ×R . Hence, to

prove (3.13) as well as show that [ ] ( )dZ f C (0,T]δ ∈ ×R , it suffices to show that the

integral in (3.13) is locally, uniformly bounded in ( ) dx, (0,T]τ ∈ ×R .

We begin by proving that for every compact dM ⊆ R there is a C 0>% such that

( ) ( )1

2LZ x C Z x−

δτ τ− ≤ τ −%ξ ξ (3.14)

for every ( )x, M (0,T]τ ∈ × and d∈Rξ . Let dM ⊆ R be compact with diameter N,

( )x, M (0,T]τ ∈ × , and d∈Rξ . It follows from Lemma 1 with 1

2µ = that there is a

C 0> such that for every positive 1δ < we have

47

LZ (x )τ − ξ ( )d

ii 1 i

x Z (x ) x Z (x )x

τ τ=

∂≤ θ − + Ψ −∂∑ ξ ξ (3.15)

d

i 1 i

N Z (x ) Z (x )x

τ τ=

∂≤ θ − + γ −∂∑ ξ ξ

( ) ( ) ( )1 1

2 2C Z x Z x , C Z x− −δ δ δ

τ τ τ≤ τ − + γ − τ ≤ τ −%ξ ξ ξ .

Now, let ( )x, M [q,T]τ ∈ × for some q 0> and 1δ < be a positive constant such that

fh32 d

δ<

ρ. It follows from (3.15) and Lemma 2.2 that there exists positive constants 1C ,

2C , and f fh 2h<% such that

( ) ( )d

LZ x f dτ −∫Rξ ξ ξ ( ) ( )

122

1 fd

C Z x exp h d−

δτ≤ τ −∫R

ξ ξ ξ (3.16)

( ) ( )1 1

2 22 22 f 2 fC exp h x C q exp h N

− −≤ τ ≤% % .

We deduce that the integral in (3.13) is locally, uniformly bounded with respect to

( ) dx, (0,T]τ ∈ ×R from which we conclude that [ ]( ) ( )dL Z f C (0,T]∈ ×R and (3.13)

holds for every ( ) dx, (0,T]τ ∈ ×R . Also, the bound (3.16) implies that [ ]( ) 2L Z fτ ∈A

for each (0,T]τ∈ . <

This result is the first step towards the construction of the fundamental solution of our

Cauchy problem. Recall that we wish to express the fundamental solution in the form

[ ]Z Uτ τ τΓ = + ϕ (3.17)

for some continuous family 0 Tτ <τ≤ϕ ≡ ϕ of functions from ( )dC R that satisfies

[ ]( ) [ ]( ) [ ][ ]( )0 L f L Z f L U fτ τ τ= Γ = + ϕ (3.18)

for every (0,T]τ∈ and f ∈A . Theorem 3 allows us to pass the differential operator L

through the convolution in the first term on the RHS of (3.18).

48

2.3.2 Basic Potential Theory Before turning our attention to the second term on the RHS of (3.18), we will

determine two classes of functions for which the Gaussian and L-potentials are defined.

Naturally, the class of admissible functions is a good place to start.

Definition 4 – For every 0δ > , the bound parameter of a family 0 Tf fτ <τ≤

≡ of

functions from δA is defined by f f0 T

h sup h32 dτ

<τ≤

δ≡ ≤

ρ. Furthermore, the family f is said

to be uniformly bounded if fh32 d

δ<

ρ.

Theorem 5 – Let 2ε ≤ and 0 Tf fτ <τ≤

≡ be a continuous, uniformly bounded family of

functions from εA . Then, [ ]U f and [ ]V f are functions in ( )dC (0,T]×R . Also, for

each (0,T]τ∈ we have that [ ]U fτ and [ ]V fτ are elements of 2εA .

proof: Recall from Definitions 2.4 and 2.5 that for every ( ) dx, (0,T]τ ∈ ×R we have

[ ]( ) [ ]( )s s0U f x Z f x ds

τ

τ τ−≡ ∫ (3.19)

and

[ ]( ) [ ]( )( )s s0V f x L Z f x ds

τ

τ τ−≡ ∫ . (3.20)

From Lemma 2.2 and Theorem 3 we see that the integrands in (3.19) and (3.20) are

continuous for every ( ) dx, ,s (0,T] (0, )τ ∈ × × τx R . Therefore, the desired assertions

follow from Lemma 2.2 and (3.16). In fact, we have that

[ ]( ) [ ]( ) ( ) ( )2 2

s s f f0 0U f x Z f x ds Cexp h x ds CTexp h x

τ τ

τ τ−≤ ≤ ≤∫ ∫ % % (3.21)

and

[ ]( )V f xτ [ ]( )( ) ( ) ( )1

22

s s f0 0L Z f x ds Cexp h x s ds

τ τ −τ−≤ ≤ τ −∫ ∫%% (3.22)

( )2

f2C Texp h x≤ %%

for every ( ) dx, (0,T]τ ∈ ×R , where C 0> , C 0>% , and f fh 2h<% . <

49

We will use this result in conjunction with the following class of functions:

Definition 6 – Let 0δ > . A family 0 Tτ <τ≤ψ ≡ ψ of functions from ( )dC R is called

Z -boundedδ , if there exists a positive constant C such that

( ) ( )x CZ xδτ τϕ ≤ (3.23)

for every ( ) dx, (0,T]τ ∈ ×R .

Consider a continuous family 0 Tτ <τ≤ψ ≡ ψ of Z - boundedδ functions and an

admissible function f δ∈A . It follows from Lemma 2.2 that

[ ]fτψ ( ) ( )d

x f dτ≤ ψ −∫Rξ ξ ξ (3.24)

( ) ( ) ( )2 2

f fd

C Z x exp h d Cexp h xδτ≤ − ≤∫R

%%ξ ξ ξ ,

where C 0> and f fh 2h<% . Hence, [ ] 0 T

fτ <τ≤ψ is a continuous, uniformly bounded

family of functions from 2δA . In particular, it follows from Theorem 5 that

[ ] ( )dU f C (0,T] ψ ∈ × R and [ ] 2U fτ δ ψ ∈ A for each (0,T]τ∈ and 1δ ≤ .

The next goal is to show that ψ is in the domain of the Gaussian potential and

combine our results to obtain

[ ] [ ][ ]U f U fτ τ ψ = ψ (3.25)

for every (0,T]τ∈ . The following lemma is the first step towards this goal and

essentially follows from the semigroup property of the Gaussian kernel.

Lemma 7 – For every a and b with a,b 1−∞ < < and positive δ we have

( ) ( ) ( ) ( )a b 1 a bs s

d0( s) s Z x Z d ds B 1 a,1 b Z x− − δ δ − − δ

τ− ττ − − = − − ττ

∫ ∫Rd ξ ξ ξ . (3.26)

where B denotes the beta function.

50

proof : We begin by substituting w in place of y where

i i i

sw x

2( s s 2( s

δτ δ≡ +τ − ) τ − )τ

ξ . (3.27)

After some algebraic manipulation, it is easy to verify that

( )

2 2 22x x

w2 s 2s 2

δ − δ δ+ = +

τ − τξ ξ

. (3.28)

Denoting the integral in (3.26) by I, we obtain the desired result

I ( ) ( )d

2d d d2 a b2 2 2

d 0

x 2s sexp exp w dw ( s) s ds

2 2

− − − −τ δ τ − δ = − − τ − π τ δτ ∫∫R

(3.29)

( ) ( )a b 1 a b a b

0 0Z x ( s) s ds Z x (1 ) dδ − − − − δ − −

τ τ

τ 1= τ − = τ − ρ ρ ρ∫ ∫

( ) ( )1 a bB 1 a,1 b Z x− − δτ= − − τ ,

where the substitution s

ρ =τ

was used in the second equality. <

Theorem 8 – Let 0 1< δ ≤ and 0 Tτ <τ≤ψ ≡ ψ be a continuous family of Z - boundedδ

functions. Then, [ ]U τ ψ is well-defined for every (0,T]τ∈ . In fact, [ ] 0 T

Uτ <τ≤ψ is a

continuous family of Z - boundedδ functions.

proof: From Lemma 7, it follows that there exists a positive constant C such that

[ ]Uτ ψ [ ] ( ) ( )s s s sd0 0

Z ds Z x d dsτ− τ−

τ τ≤ ψ ≤ − ψ∫ ∫ ∫R

ξ ξ ξ (3.30)

( ) ( ) ( )s sd0

C Z x Z d ds CTZ xδ δ δτ− τ

τ≤ − ≤∫ ∫R

ξ ξ ξ

for every (0,T]τ∈ . <

This result will now be used to establish (3.25).

51

Corollary 9 – Let 0 1< δ ≤ , f δ∈A , and 0 Tτ <τ≤ψ ≡ ψ be a continuous family of

Z - boundedδ functions. Then, for every (0,T]τ∈ we have

[ ] [ ][ ]U f U fτ τ ψ = ψ . (3.31)

proof: It follows from (3.24) that [ ] 0 T

fτ <τ≤ψ is a continuous, uniformly bounded

family of functions from 2 2δ ⊆A A . Consequently, we deduce from Theorem 5 that the

LHS of (3.31) is well-defined. In fact, [ ] ( )dU f C (0,T] ψ ∈ × R and [ ] 2U fτ δ ψ ∈ A

for each (0,T]τ∈ . Fix ( )x, (0,T]τ ∈ . From Theorem 8 we have that [ ]U τ ψ is well-

defined. Furthermore, we apply Fubini’s theorem to obtain

[ ][ ]( )U f xτ ψ [ ] ( ) [ ]( )s s0U f x Z ds f(x)

τ

τ τ−= ψ ∗ = ψ ∗∫ (3.32)

[ ]( ) ( )d s s0Z x f dsd

τ

τ−= ψ −∫ ∫Rξ ξ ξ

( ) ( ) ( )d d s s0Z x y y f dydsd

τ

τ−= − − ψ∫ ∫ ∫R Rξ ξ ξ

( ) ( ) ( )d d s s0Z x y y f dyd ds

τ

τ−= − − ψ∫ ∫ ∫R Rξ ξ ξ

( ) ( ) ( )d d s s0Z x w w f dwd ds

τ

τ−= − ψ −∫ ∫ ∫R Rξ ξ ξ

( ) ( ) ( )( )d ds s0Z x w w f d dwds

τ

τ−= − ψ −∫ ∫ ∫R R ξ ξ ξ

( ) [ ]d s s0Z x w f dwds

τ

τ−= − ψ∫ ∫R

[ ] ( ) [ ] ( )s s0Z f x ds U f x

τ

τ− τ = ψ = ψ ∫ . <

The final objective of this section is to prove that continuous, Z - boundedδ

functions are in the domain of the L-potential for every 1δ ≤ . In particular, the analogue

of (3.25) for the L-potential will be established. We will first prove a result for

continuous, Z - boundedδ functions that is similar to Theorem 3.

52

Lemma 10 – Let 0 1< δ ≤ and 0 Tτ <τ≤ψ ≡ ψ be a continuous family of Z - boundedδ

functions. Then, [ ]( ) ( )dL Z C (0,T]ψ ∈ ×R . In fact, for every ( ) dx, (0,T]τ ∈ ×R we have

[ ]( )( ) ( )dL Z x LZ x ( )dτ τ τψ = − ψ∫R

ξ ξ ξ . (3.33)

proof : We proceed as in the proof of Theorem 3. Clearly, the integrand in (3.33) is a

continuous function of ( )x,τ on d (0,T]×R . Hence, to prove (3.33) as well as show that

[ ]( ) ( )dL Z C (0,T]ψ ∈ ×R , it suffices to show that the integral in (3.33) is locally,

uniformly bounded with respect to ( ) dx, (0,T]τ ∈ ×R .

Let ( )x, M [q,T]τ ∈ × for some q 0> and compact dM ⊆ R . It follows from (3.29)

and (3.15) that there exists a positive constant C such that

( ) ( ) ( )d d

1 1 1

2 2 2LZ x ( ) d C Z x Z ( )d C Z x Cq C− − −δ δ δ

τ τ τ τ τ− ψ ≤ τ − = τ ≤∫ ∫R R%ξ ξ ξ ξ ξ ξ , (3.34)

where ( ) ( ) [ ] C sup Z x : x, M q,Tδτ≡ τ ∈ ×% . We deduce that the integral in (3.33) is

locally, uniformly bounded with respect to ( ) dx, (0,T]τ ∈ ×R from which we obtain the

desired conclusions. <

This lemma allows us to prove the following L-potential analogue of Theorem 8.

Theorem 11 – Let 0 1< δ ≤ and 0 Tτ <τ≤ψ ≡ ψ be a continuous family of Z - boundedδ

functions. Then, [ ]Vτ ψ is well-defined for every (0,T]τ∈ . In fact, [ ] 0 T

Vτ <τ≤ψ is a

continuous family of Z - boundedδ functions.

proof: From (3.15) and Lemmas 10 and 7, it follows that there exists a C 0> such that

[ ] [ ]( ) ( ) ( )ds s s s0 0V L Z ds LZ x d ds

τ τ

τ τ− τ−ψ ≤ ψ ≤ − ψ∫ ∫ ∫Rξ ξ ξ (3.35)

( ) ( ) ( )d

1

2s s0

C s Z x Z d dsτ − δ δ

τ−≤ τ − −∫ ∫R ξ ξ ξ

( ) ( )1CB 1, Z x CZ x

2δ δτ τ

= τ ≤

%

for every (0,T]τ∈ . <

53

The final result in this section is the L-potential analogue of Corollary 9.

Corollary 12 – Let 0 1< δ ≤ , f δ∈A , and 0 Tτ <τ≤ψ ≡ ψ be a continuous family of

Z - boundedδ functions. Then, for every (0,T]τ∈ we have

[ ] [ ][ ]V f V fτ τ ψ = ψ . (3.36)

proof: We proceed as in the proof of Corollary 9. It follows from (3.24) that

[ ] 0 T

fτ <τ≤ψ is a continuous, uniformly bounded family of functions from 2 2δ ⊆A A .

Consequently, we deduce from Theorem 5 that the LHS of (3.31) is well-defined. In fact,

[ ] ( )dV f C (0,T] ψ ∈ × R and [ ] 2V fτ δ ψ ∈ A for each (0,T]τ∈ . Fix ( )x, (0,T]τ ∈ .

From Theorem 11 we have that [ ]Vτ ψ is well-defined. Furthermore, we apply Fubini’s

theorem to obtain

[ ][ ]( )V f xτ ψ [ ] ( ) [ ]( )( )s s0V f x L Z ds f(x)

τ

τ τ−= ψ ∗ = ψ ∗∫ (3.37)

[ ]( )( ) ( )d s s0L Z x f dsd

τ

τ−= ψ −∫ ∫Rξ ξ ξ

( ) ( ) ( )d d s s0LZ x y y f dydsd

τ

τ−= − − ψ∫ ∫ ∫R Rξ ξ ξ

( ) ( ) ( )d d s s0LZ x y y f dyd ds

τ

τ−= − − ψ∫ ∫ ∫R Rξ ξ ξ

( ) ( ) ( )d d s s0LZ x w w f dwd ds

τ

τ−= − ψ −∫ ∫ ∫R Rξ ξ ξ

( ) ( ) ( )( )d ds s0LZ x w w f d dwds

τ

τ−= − ψ −∫ ∫ ∫R R ξ ξ ξ

( ) [ ]d s s0LZ x w f dwds

τ

τ−= − ψ∫ ∫R

[ ]( )( ) [ ] ( )s s0L Z f x ds V f x

τ

τ− τ = ψ = ψ ∫ . <

54

2.4 The Derivatives of the Gaussian Potential

In this section we turn our attention to the second term on the RHS of (3.19). In

particular, we will derive the following relationship between the Gaussian and L-

potentials expressed in (2.24):

[ ]( ) [ ] [ ]L U f V f fτ τ τ ψ = ψ − ψ (4.1)

for every f δ∈A , 1δ ≤ , and continuous family of Zδ -bounded functions 0 Tτ <τ≤ψ ≡ ψ

for which [ ]fψ is locally, uniformly Hölder continuous in space with exponent 1β < .

Theorem 1 – Let 1δ ≤ and (0,T]

f fτ τ∈= be a uniformly bounded, continuous family of

functions from δA that is locally, uniformly Hölder continuous in space with exponent

1β < . Then, for j 1,2∈ we have that [ ] ( )j

dji

U f C (0,T]x

∂∈ ×

∂R . In fact,

[ ] [ ]j j

s sj ji i0U f Z f ds

x xτ τ−

τ∂ ∂=∂ ∂∫ (4.2)

and [ ]j

2ji

U fx τ δ∂

∈∂

A for every (0,T]τ∈ .

proof: From Theorem 3.2 we have that ( ) [ ]( )j

s sji

x, ,s Z f xx τ−∂

τ∂

a is continuous on

( ) dE x, ,s : x ,0 s T≡ τ ∈ < < τ ≤R ; However, there is a singularity at s = τ . Hence, to

prove (4.2) as well as show that [ ] ( )j

dji

U f C (0,T]x

∂∈ ×

∂R , it suffices to show that the

integral in (4.2) is locally, uniformly bounded in ( ) dx, (0,T]τ ∈ ×R . We break up the

integration in (4.2) as follows:

[ ] [ ] [ ]j j j

s s s s s sj j ji i i

q

0 0 qZ f ds Z f ds Z f ds

x x xτ− τ− τ−

τ τ∂ ∂ ∂= +∂ ∂ ∂∫ ∫ ∫ , (4.3)

where q (0,T]∈ is fixed.

55

Let ( ) ( ) qx, ,s E x, ,s E : x M, s qτ ∈ ≡ τ ∈ ∈ ≤ < τ where dM ⊆ R is compact. For

the first integral on the RHS of (4.3), we apply Theorem 3.2, Lemma 3.1, and Lemma 2.2

to obtain the estimate

[ ]( )j

s sji

Z f xx τ−∂∂

( ) ( )d

j

s sji

Z x f dx τ−∂≤ −∂∫R

ξ ξ ξ (4.4)

( ) ( ) ( )d

j2

21 f sC s exp h Z x d

− ετ−≤ τ − −∫R

ξ ξ ξ

( )( )j

22

2 fC exp h x s−≤ τ −% ,

where f fh 2h<% . This implies that

[ ]( ) ( ) ( ) ( )j j

2 22

s s 2 f 3 fji

q q

0 0Z f x ds C exp h x s ds C exp h x

x−

τ−∂ ≤ τ − ≤∂∫ ∫% % . (4.5)

Now, let ( ) ( ) qx, ,s E x, ,s E : x M,0 q sτ ∈ ≡ τ ∈ ∈ ≤ < < τ% . For the second integral

on the RHS of (4.3), we define dB : 1≡ ∈ <Rξ ξ and rewrite (4.4) as

[ ]( )j

s sji

Z f xx τ−∂∂

( ) ( )C

j j

s s s sj ji iB BZ x f ( ) d f (x) Z x d

x xτ− τ−∂ ∂≤ − + −∂ ∂∫ ∫ξ ξ ξ ξ ξ (4.6)

( ) ( ) ( )j

s s sjiBZ x f f x d

x τ−∂+ − −∂∫ ξ ξ ξ .

For the first integral on the RHS of (4.6), it follows from (3.11) and (3.12) that

( ) ( )C

j2

s s 4 fjiBZ x f ( ) d C exp h x

xτ−

∂ − ≤∂∫ %ξ ξ ξ . (4.7)

We use the divergence theorem to obtain the following bound for the second integral:

( ) ( ) ( )j j 1

s s ij j 1i iB BZ x d Z x cos dS

x x

−

τ− τ−−∂

∂ ∂− = − γ

∂ ∂∫ ∫ ξξ ξ ξ (4.8)

( )j 1

sj 1iB

Z x dSx

−

τ−−∂

∂≤ −

∂∫ ξξ ,

56

where iγ is the angle between iξ and the outwardly directed normal to the boundary ∂B,

and dSξ is the ( )d 1− -dimensional volume element on ∂B. Since x − ξ = 1 on ∂B, we

use Lemma 3.1 with j 1

2

−µ = to deduce that

( ) ( )j j 1

s sj j 1i iB BZ x d Z x dS

x x

−

τ− τ−−∂

∂ ∂− ≤ −∂ ∂∫ ∫ ξξ ξ ξ (4.9)

( )2

d j 1

25

B

xC s exp dS

s

+ −−

∂

−≤ τ − −δ

τ − ∫ ξξ

d j 1

2

6 7B

C exp dS Cs s

+ −

∂

δ δ ≤ − ≤ τ − τ − ∫ ξ ,

where we used the fact that the expression in square brackets is uniformly bounded. It

follows that the second term on the RHS of (4.6) is bounded by

( ) ( ) ( )j

2s s 8 fj

iBf x Z x d C exp h x

x τ−∂ − ≤∂∫ ξ ξ . (4.10)

Since f is locally, uniformly Hölder continuous in space with exponent 1β < , we

have that there exists a positive constant 9C such that

( ) ( )s s 9f x f C xβ− ≤ −ξ ξ (4.11)

for every x and ξ in M and q s T≤ ≤ . Without loss of generality, we may assume that

B M⊆ . It follows from Lemma 3.1 with 1 12

β− < µ < that the final integral on the RHS

( ) ( ) ( )( )

2

j

s s s 9 d j 2jiB B

xexp

sZ x f f x d C d

x s xτ− + − µ−βµ

−−δ τ −∂ − − ≤

∂ τ −∫ ∫% ξ

ξ ξ ξ ξ− ξ

(4.12)

( )

( )910d j 2

B

C dC s

s x

−µµ + − µ−β≤ ≤ τ

τ −∫ ξ −− ξ

.

57

Combining (4.7), (4.10), and (4.12) with (4.6), we obtain

[ ] ( ) ( ) ( )j

2 2

s s 4 f 8 f 10ji

Z f C exp h x C exp h x C sx

−µτ−

∂ ≤ + + τ∂

% − (4.13)

( ) ( )2

11 fC s exp 2h x−µ≤ τ − .

Since 1µ < , it follows that

[ ] ( ) ( ) ( )j

2 2s s 11 f 12 fj

iq qZ f ds C exp 2h x s ds C exp 2h x

x−µ

τ−

τ τ∂ ≤ τ − ≤∂∫ ∫ . (4.14)

Therefore, we deduce from (4.3), (4.5), and (4.14) that for every ( )x, M (0,T]τ ∈ ×

we have

[ ]( ) ( ) ( ) ( )j

2 2 2

s s 3 f 12 f 13 fji0Z f x ds C exp h x C exp 2h x C exp 2h x

x τ−

τ ∂ ≤ + ≤∂∫ % (4.15)

We conclude that [ ] ( )j

dji

U f C (0,T]x

∂∈ ×

∂R , [ ]

j

2ji

U fx

τ δ∂

∈∂

A , and (4.2) holds for every

(0,T]τ∈ . < We have a similar result for the time derivative of the Gaussian potential.

Theorem 2 – Let 1δ ≤ and (0,T]

f fτ τ∈= be a uniformly bounded, continuous family of

functions from δA that is locally, uniformly Hölder continuous in space with exponent

1β < . Then, we have that [ ] ( )dU f C (0,T]∂

∈ ×∂τ

R . In fact, we have

[ ] [ ]s s0

U f Z f ds fτ τ− τ

τ∂ ∂= +∂τ ∂τ∫ (4.16)

from which we conclude that [ ] 2U fτ δ∂

∈∂τ

A for every (0,T]τ∈ .

proof: Let dM ⊆ R be compact, ( )x, M (0,T]τ ∈ × , and 0∆τ > . We apply the mean

value theorem to the following difference quotient to obtain

58

[ ]( )U Uf xτ+∆τ τ−

∆τ [ ]( ) [ ]( )s s s s

0 0

1Z f x ds Z f x dsτ+∆τ− τ−

τ+∆τ τ = −

∆τ ∫ ∫ (4.17)

[ ]( ) [ ]( )s ss s s

0

1 Z ZZ f x ds f x dsτ+∆τ− τ−

τ+∆τ−

τ+∆τ τ

τ

− = + ∆τ ∆τ ∫ ∫

[ ]( ) [ ]( )s s s s0

1Z f x ds Z f x dsτ+∆τ− τ−

τ+∆τ τ

τ

∂= +∆τ ∂τ∫ ∫ %

for some ( ),τ∈ τ τ + ∆τ% . In case the notation in the last line is unclear, we note that

[ ]( ) [ ]( )s s s sZ f x Z f xτ− τ−τ=τ

∂ ∂=

∂τ ∂τ%% (4.18)

We will show that the RHS of (4.17) approaches the RHS of (4.16) as ∆τ decreases

to zero. A symmetric argument can then be made to show that this result also holds as

∆τ increases to zero. This will establish (4.16) since the limit of the LHS of (4.17) as

0∆τ → is the LHS of (4.16).

Since [ ]( )s ss Z f xτ+∆τ−a is continuous on [ ],τ τ + ∆τ , it follows from the mean value

theorem for integrals that the first integral on the RHS of (4.17) may be rewritten as

[ ]( ) [ ]( )s s s s

1Z f x ds Z f xτ+∆τ− τ+∆τ−

τ+∆τ

τ=

∆τ∫ % % (4.19)

for some ( )s ,∈ τ τ + ∆τ% . Letting 0∆τ → yields

[ ]( ) [ ]( ) ( )s s s s0 0

1Z f x ds Z f x f xlim limτ+∆τ− τ+∆τ− τ

τ+∆τ

∆τ→ ∆τ→τ= =

∆τ∫ % % . (4.20)

For the second integral on the RHS of (4.17), it remains to be shown that

[ ]( ) [ ]( )s s s s0 0 0Z f x ds Z f x dslim τ− τ−

τ τ

∆τ→

∂ ∂=∂τ ∂τ∫ ∫% . (4.21)

59

Let 0ε > , 1 12

β− < µ < ,

( )1

1

1

11

2 3C

−µ− µ ε ε ≡

, and ( )1 ,ετ ∈ τ − ε τ , where the positive

constant C will be determined below. Consider

I [ ]( ) [ ]( )s s s s0 0

Z f x ds Z f x dsτ− τ−

τ τ∂ ∂≡ −∂τ ∂τ∫ ∫% (4.22)

[ ]( ) [ ]( ) [ ]( )e e

s s s s s s s0

Z Z f x ds Z f x ds Z f x dsε

τ− τ− τ− τ−

τ τ τ

τ τ

∂ ∂ ∂ ∂ ≤ − + + ∂τ ∂τ ∂τ ∂τ ∫ ∫ ∫% % .

For the first integral on the RHS of (4.22), fix s (0, ]ε∈ τ and note that

s 0ετ − > τ − τ >% . We deduce from Theorem 3.2 that [ ]( )q s sq Z f x−∂∂τ

a is continuous on

[ ],Tτ . It follows that there exists a 2 1ε ≤ ε such that for 2∆τ < ε we have

[ ]( )s s sZ Z f x3Tτ− τ−

∂ ∂ ε − < ∂τ ∂τ % (4.23)

from which we deduce

[ ]( )s s s0

Z Z f x ds3

ε

τ− τ−

τ ∂ ∂ ε − < ∂τ ∂τ ∫ % . (4.24)

To evaluate the second and third integrals on the RHS of (4.22), recall that Zτ is a

parametrix for L. This implies that

2Z 1Z

2τ

τ∂

= ∇∂τ

. (4.25)

Let 1 12

β− < µ < and N denote the diameter of M. Combining (4.25) with (4.13) yields

[ ]( ) ( )( ) ( )( ) ( )2 2ss f f

Zf x Cexp 2h x s Cexp 2h N s C s

−µ −µ −µτ−∂≤ τ − ≤ τ − ≤ τ −

∂τ% % (4.26)

for every [ ]s ,Tε∈ τ , where C% is some positive constant and ( )2fC Cexp 2h N≡ % .

60

We use this estimate to see that for ∆τ < δ and q ,∈ τ τ% we have

[ ]( )q s sZ f x dsε

−

τ

τ

∂∂τ∫ ( ) ( ) ( )1 1C

C q s ds q q1

ε

−µ −µ −µε

τ

τ ≤ − = − τ − − τ − µ∫ (4.27)

( ) ( ) ( ) 11C Cq q

1 1

−µ−µε ε≤ − τ ≤ − τ + τ − τ − µ − µ

( ) ( )1 1C C2

1 1 3

−µ −µε

ε≤ ∆τ + τ − τ < δ = − µ − µ

.

Combining the estimates (4.24) and (4.27) with (4.22) yields I < ε when ∆τ < δ% from

which we deduce (4.21). It follows from (4.20) and (4.21) that

[ ]( ) [ ]( )s s s s0 0

1Z f x ds Z f x dslim τ+∆τ− τ−∆τ↓

τ+∆τ τ

τ

∂+ ∆τ ∂τ ∫ ∫ % (4.28)

[ ]( ) ( )s s0

Z f x ds f xτ− τ

τ ∂= +∂τ∫ .

A symmetric argument can be made to establish the analogous result for the left-handed

limit. Combining this with (4.17) yields the desired result (4.16).

The remaining assertions follow from Theorem 3.2 and (4.16). In fact, it follows that

[ ]( )U f xτ∂∂τ

[ ]( ) ( )s s0

Z f x ds f xτ− τ

τ ∂≤ +∂τ∫ (4.29)

( ) ( ) ( )2 2 2

1 f 2 f 3 fC exp 2h x C exp h x C exp 2h x≤ τ + ≤

for every ( ) dx, (0,T]τ ∈ ×R from which we deduce that [ ] ( )j

dji

U f C (0,T]x

∂∈ ×

∂R and

[ ]j

2ji

U fx τ δ∂

∈∂

A . <

61

We combine Theorems 1 and 2 to obtain the desired relation (4.1) between the

Gaussian and L-potentials. Let 0 Tτ <τ≤ψ ≡ ψ be a continuous family of Z - boundedδ

functions and f δ∈A for some positive 1δ ≤ . It follows from (3.25) that [ ] 0 T

fτ <τ≤ψ is

a continuous, uniformly bounded family of functions from 2δA . Assuming that [ ]fψ is

locally, uniformly Hölder continuous in space with exponent 1β < , we deduce from

Theorems 1 and 2 that for every ( )x, (0,T]τ ∈ we have

[ ]( ) [ ]( ) [ ] [ ] [ ]s s0L U f L Z f ds f V f f

τ

τ τ− τ τ τ ψ = ψ − ψ = ψ − ψ ∫ . (4.30)

2.5 A Series Representation of the Fundamental Solution We will now apply relation (4.30) to the problem of representing the fundamental

solution for L by [ ]Z Uτ τ τΓ = + ϕ (5.1)

for some continuous family 0 Tτ <τ≤ϕ ≡ ϕ of functions from ( )dC R that satisfies

[ ]( ) [ ]( ) [ ][ ]( )0 L f L Z f L U fτ τ τ= Γ = + ϕ (5.2)

for every (0,T]τ∈ and f ∈A . Let f ∈A and assume that ϕ is Z - boundedδ for some

1δ ≤ . If [ ]fϕ is locally, uniformly Hölder continuous in space (with exponent 1β < ),

then we deduce from Corollary 3.9 and (4.30) that

[ ][ ]( ) [ ]( ) [ ] [ ]L U f L U f V f fτ τ τ τ ϕ = ϕ = ϕ − ϕ . (5.3)

Combining this with (5.2) implies that ϕ must satisfy the following Volterra integral

equation for every (0,T]τ∈ and f ∈A :

[ ] [ ]( ) [ ]f L Z f V fτ τ τ ϕ = + ϕ . (5.4)

62

2.5.1 Convergence and Continuity

The goal of this sub-section is to show that the solution to (5.4) is given by

[ ]m

m 0

V LZ∞

τ τ=

ϕ = ∑ , (5.5)

where mVτ is an operator defined by

0V Iτ ≡ (5.6) and m 1 mV V V+

τ τ τ≡ o (5.7)

for every (0,T]τ∈ and m 0≥ . We will see in the proof of Theorem 2 below that

[ ]mV LZτ is well-defined for every (0,T]τ∈ and m 0≥ . Furthermore, it is shown that

the series representation (5.5) is a continuous family of Zδ -bounded functions for every

1δ < . Finally, our objective is obtained in Corollary 3, where we deduce that (5.5) is a

solution to (5.4). We begin with a preliminary result:

Lemma 1 – LZ is Zδ -bounded for every positive 1δ < .

proof : Let0 1< δ < . From the definition of L we have

( ) ( ) ( ) ( ) ( ) ( )d

2

ii 1 i

LZ x x Z x x Z x x x Z xxτ τ τ τ

=

∂ θ = −θ + Ψ = + Ψ ∂ τ ∑ (5.8)

and

21

(x) x2

+ Ψ = γ − σ

% . (5.9)

From Lemma 3.1 with 0µ = , it follows that there is an A 0> such that for every

( ) dx, (0,T]τ ∈ ×R we have

( )0 LZ (x) AZ (x) Z x CZ (x)δ δτ τ τ τ≤ ≤ θ + γ ≤ , (5.10)

where C A≡ θ + γ . <

63

Throughout the remainder of this section, let ϕ be given by the series (5.5).

Theorem 2 – The series ϕ converges absolutely. Moreover, ϕ is a continuous, Zδ -

bounded function for every 1δ < .

proof : Let 0 1< δ < . The first step is to prove by induction on m that [ ] m

0 TV LZτ <τ≤

is a continuous family of Z - boundedδ functions such that for some C 0> we have

[ ] ( ) ( )m m 1

m 2C

V LZ Z xm 1

2

−δ

τ τ≤ πτ+ Γ

. (5.11)

From Lemma 1 we see that [ ] 0

0 TV LZτ <τ≤

is a continuous family of Z - boundedδ

functions that satisfies (5.11) for some C 0> . To use induction, assume that this holds

for some m 0≥ . Since [ ] m

0 TV LZτ <τ≤

is a continuous family of Z - boundedδ functions,

it follows from (3.36) and (5.11) that

[ ]m 1V LZ (x)+τ [ ] ( ) [ ]m m

s sd0

V V LZ x LZ (x )V LZ ( ) d dsτ τ−

τ

= ≤ − ∫ ∫Rξ ξ ξ (5.12)

( ) ( ) ( ) ( )m 1 1 m 1

2 2s s

d0

Cs s Z x Z d ds

m 1

2

+ −− δ δτ−

τ

≤ τ − π −+ Γ

∫ ∫Rξ ξ ξ

( ) ( )m 1 m

2C 1 m 1

B , Z xm 1 2 2

2

+δτ

+ = πτ + Γ π

( ) ( )m 1 m

2C

Z xm 2

2

+δτ= πτ

+ Γ

.

Therefore, [ ] m

0 TV LZτ <τ≤

is a continuous family of Z - boundedδ functions that

satisfies (5.11) for every m 1≥ . To obtain the desired bound on ϕ, we will break the

series apart as follows:

64

[ ]( ) [ ]( ) [ ]( )m m m

m 1 m 1 m 2mevenmodd

(x, ) V LZ x V LZ x V LZ xτ τ τ

∞ ∞ ∞

= = =

ϕ τ ≤ ≤ +∑ ∑ ∑ . (5.13)

For the odd series we have

[ ]( ) [ ]( ) ( ) ( )( )

n2n 1m 2n 1

m 1 n 0 n 0modd

CV LZ x V LZ x Z x

n 1

++ δ

τ τ τ

∞ ∞ ∞

= = =

πτ= ≤ Γ +

∑ ∑ ∑ (5.14)

( ) ( ) ( ) ( ) ( )n2n

21

n 0

CCZ x, Cexp C Z x, C Z x,

n!δ δ δ

∞

=

πτ≤ τ ≤ π τ τ ≤ τ

∑ ,

where ( )21C Cexp C T≡ π . For the even series we have

[ ]( ) [ ]( ) ( ) ( )1

n2n 2m 2n

m 1 n 1 n 1meven

CV LZ x V LZ x Z x

1n

2

−δ

τ τ τ

∞ ∞ ∞

= = =

πτ = ≤

Γ +

∑ ∑ ∑ (5.15)

( ) ( )( ) ( ) ( )

1n n2n 2n2

2

n 1 n 0

C C2Z x 2Z x C

n 1 ! n!

−δ δτ τ

∞ ∞

= =

πτ πτ ≤ ≤ πτ − ∑ ∑

( ) ( ) ( )2 222C exp C Z x C Z xδ δ

τ τ≤ πτ π τ ≤ ,

where ( )2 22C 2C Texp C T≡ π π . By inserting (5.14) and (5.15) into (5.13) we conclude

that ϕ is Zδ -bounded. Furthermore, since [ ] ( )m dV LZ C (0,T]∈ ×R for every m 1≥ and

the series representation for ϕ is uniformly bounded on compact subsets of d (0,T]×R ,

we also have that ( )dC (0,T]ϕ∈ ×R . <

Corollary 3 – ϕ is a solution to the following Volterra integral equation for every

(0,T]τ∈ and f ∈A :

[ ] [ ]( ) [ ]f L Z f V fτ τ τ ϕ = + ϕ . (5.16)

65

proof : It follows from Theorem 2 that for every (0,T]τ∈ we have

[ ]Vτ ϕ [ ]( ) [ ]( ) [ ]m ms s s s

m 0 m 00 0L Z ds L Z V LZ ds V V LZ

∞ ∞

τ− τ− τ= =

τ τ = ϕ = = ∑ ∑∫ ∫ (5.17)

[ ] [ ] [ ]m 1 m 0

m 0 m 1

V LZ V LZ V LZ LZ∞ ∞

+τ τ τ τ τ τ

= =

= = = ϕ − = ϕ −∑ ∑ .

Therefore, we deduce from Corollary 3.12 that for every (0,T]τ∈ and f ∈A we have

[ ] [ ][ ] ( )[ ]V f V f LZ fτ τ τ τ ϕ = ϕ = ϕ − . (5.18)

Hence, we conclude that (5.16) holds for the series representation (5.5) for ϕ. <

We would like to be able to claim that

[ ]( )L f 0τΓ = , (5.19)

where [ ]Z Uτ τ τΓ = + ϕ (5.20)

for every (0,T]τ∈ and f ∈A . However, it remains to be shown that [ ]fϕ is locally,

uniformly Hölder continuous in space with exponent 1β < . As stated in the beginning of

this section, this condition combined with Theorem 2 and Corollary 3 is sufficient to

conclude that (5.19) holds for every (0,T]τ∈ and f ∈A .

2.5.2 Hölder Continuity The first objective of this sub-section is to show that ϕ is uniformly Hölder

continuous in space with exponent 1β < . We deduce from (5.17) that

( ) ( ) ( ) ( ) [ ]( ) [ ]( )x y LZ x LZ y V x V yτ τ τ τ τ τϕ − ϕ ≤ − + ϕ − ϕ . (5.21)

for every x and y in dR and (0,T]τ∈ . Our goal will be obtained by proving the desired

result for each of the terms on the RHS of (5.21). We begin with a preliminary lemma:

66

Lemma 4 – Assume that 2

x y− < τ . Suppose that ζ lies along the line segment

connecting x and y. Then, for any positive constant δ there is a C 0> such that

( ) ( )Z CZ yδ δτ τζ ≤ . (5.22)

proof : Since ζ lies on the line segment connecting x and y, we have ζ = λx + (1 – λ)y

for some λ ∈ (0,1). Hence,

2 2 2

x (1 )y (x y) yζ = λ + − λ = λ − + (5.23)

2 2

y x y≥ − λ − 2y≥ −λτ .

It follows that

( )( )

( )d d 222 2 y

Z exp exp CZ y2 2 2 2

δ δτ τ

δ −λτ δ ζδ δ ζ = − ≤ − = πτ τ πτ τ

, (5.24)

where C exp2

δλ ≡

.

Theorem 5 – Let x and y be elements of the compact set dM ⊆ R , d∈Rξ (0,T]τ∈ ,

0 1< β < , and 0 1< δ < . Then, there exists a positive constant C such that

( ) ( )( )1

2x yL Z (x ) L Z (y ) C x y Z x Z y

+β−β δ δ

τ τ τ τ− − − ≤ − τ − + −ξ ξ ξ ξ , (5.25)

where the subscripts x and y denote the spatial variable of the action of the operator L.

proof: We divide this proof into two cases:

Case 1 – 2

x y− ≥ τ

From (3.15) it follows that for some C 0> we have

( ) ( )1 1

2 2 2xL Z x ) C Z x C x y Z x

β +β +β− −βδ δ

τ τ τ( − ≤ τ τ − ≤ − τ −ξ ξ ξ . (5.26)

A similar inequality holds for yL Z (y )τ − ξ , thus we obtain (5.25) in this case.

67

Case 2 – 2

x y− < τ

Throughout this case we will use the following result for every [ ]a 0,1∈ + β :

a 1 a 1 1

12 2 2 2x y x y x y C x y+β− +β +β− − −−β β β

τ − = τ − τ − ≤ τ −

. (5.27)

Consider the inequality

x y i ii i

d

i 1

L Z (x ) L Z (y ) x Z (x ) y Z (y )x y

τ τ τ τ

=

∂ ∂− − − ≤ θ − − −

∂ ∂∑ξ ξ ξ ξ (5.28)

( ) ( ) ( ) ( )x Z x y Z yτ τ+ Ψ − − Ψ −ξ ξ .

We first break up the summation and find that

i i i ii i i

d d

i 1 i 1

x Z (x ) y Z (y ) x y Z (x )x y xτ τ τ

= =

∂ ∂ ∂− − − ≤ − −

∂ ∂ ∂∑ ∑ξ ξ ξ (5.29)

ii i

d

i 1

Z (x ) Z (y ) yx yτ τ

=

∂ ∂+ − − −

∂ ∂∑ ξ ξ .

For the first term on the RHS of (5.29), we see from Lemma 3.1 with1

2µ = and

(5.27) that there exists a constant 1C 0> such that

( ) ( )1 1

2 2i i 1 1

i

d

i 1

x y Z (x ) C x y Z x C x y Z xx

+β− − βδ δτ τ τ

=

∂− − ≤ τ − − ≤ τ − −

∂∑ ξ ξ ξ . (5.30)

Applying the mean value theorem to the second term on the RHS of (5.29), it follows

that there exists a point ζ lying on the line segment between x − ξ and y − ξ such that for

some 2C > 0 we have

ii i

d

i 1

Z (x ) Z (y ) yx y

τ τ

=

∂ ∂− − −

∂ ∂∑ ξ ξ ( )2

i i i2i

d

i 1

Z x y yτ

=

∂≤ ζ −

∂ζ∑ (5.31)

( )2

2 2i

d

i 1

C x y Zτ

=

∂≤ − ζ

∂ζ∑ .

68

It follows from Lemma 3.1 with 1µ = that for some 3C 0> we have

( ) ( )2

132

i

d

i 1

Z C Z− δτ τ

=

∂ζ ≤ τ ζ

∂ζ∑ . (5.32)

Since 2

x y− < τ , we deduce from Lemma 4 that

( )2

2i

d

i 1

Zτ

=

∂ζ

∂ζ∑ ( ) ( )1 1

1 2 24 4C Z y C Z y

+β −β− −− δ δ

τ τ≤ τ − = τ τ −ξ ξ (5.33)

( )1

(1 )24C x y Z y

+β− − −β δ

τ≤ τ − − ξ .

Combining this with (5.31) yields

( )1

2i 5

i i

d

i 1

Z (x ) Z (y ) y C x y Z yx y

+β− β δτ τ τ

=

∂ ∂− − − ≤ τ − −

∂ ∂∑ ξ ξ ξ . (5.34)

Finally, by combining (5.30) and (5.34) with (5.29) we obtain

( ) ( )( )1

2i i 6

i i

d

i 1

x Z (x ) y Z (y ) C x y Z x Z y ,x y

+β− β δ δτ τ τ τ

=

∂ ∂− − − ≤ τ − − + − τ

∂ ∂∑ ξ ξ ξ ξ . (5.35)

We now consider the second term on the RHS of (5.28). Recalling the definition of

Ψ given in (5.9), we have

( ) ( ) ( ) ( ) ( ) ( )x Z x y Z y Z x Z xτ τ τ τΨ − − Ψ − ≤ γ − − −ξ ξ ξ ξ . (5.36)

Applying the mean value theorem, Lemma 3.1, Lemma 4, and (5.27) to the RHS of

(5.36), it follows that there exists a point ζ lying on the line segment between x − ξ and

y − ξ such that

( ) ( ) ( ) ( )12

i i 7i

Z x Z y x y Z C x y Z− δ

τ τ τ τ∂γ − − − = γ − ζ ≤ τ − ζ

∂ζξ ξ (5.37)

( )1

28C x y Z y

+β− β δ

τ≤ τ − − ξ .

Combining (5.35), (5.36), and (5.37) with (5.28), we see that (5.25) holds for case 2. <

69

Applying this result with 0=ξ , we deduce that for every q 0> and compact

dM ⊆ R , there exists positive constants C and C% such that

( ) ( )( )1 1

2 2LZ (x) LZ (y) C x y Z x Z y Cq x y+β +β

− −β βδ δτ τ τ τ− ≤ − τ + ≤ −% (5.38)

for every x and y in M and [ ]q,Tτ∈ . Hence, LZ is locally, uniformly Hölder continuous

in space with exponent 1β < . We proceed to establish this result for the second term on

the RHS of (5.21).

Corollary 6 – [ ]V ϕ is locally, uniformly Hölder continuous in space with exponent

1β < . More precisely, for every compact dM ⊆ R and positive 1δ < , we have

[ ]( ) [ ]( ) ( ) ( )( )V x V y C x y Z x Z yβ δ δ

τ τ τ τϕ − ϕ ≤ − + (5.39)

for every x and y in M and ( ]0,Tτ∈ .

proof : Let x and y be elements of the compact set M, 0 1< β < , and 0 1< δ < . Since ϕ

is Zδ -bounded, it follows from Theorem 5, Theorem 2, and Lemma 3.7 that for some

C 0> we have

[ ]( ) [ ]( )V x V yτ τϕ − ϕ [ ]( )( ) [ ]( )( )( )s s s s0

L Z x L Z y dsτ− τ−

τ≤ ϕ − ϕ∫ (5.40)

( ) ( )d

x s y s s

0L Z (x ) L Z (y ) d dsτ− τ−

τ

≤ − − − ϕ∫ ∫Rξ ξ ξ ξ

( ) ( )d

12

s s

0C x y ( s) Z Z x

+β−β δ δτ−

τ

≤ − τ − ∫ ∫Rξ −ξ

( )sZ y d dsδτ− + −ξ ξ

( ) ( )( )1

21

CB ,1 x y Z x Z y2

−ββ δ δ

τ τ

− β = − τ +

( ) ( )( )C x y Z x Z yβ δ δ

τ τ≤ − + . <

70

We conclude from (5.21), (5.38), and Corollary 6 that for every compact dM ⊆ R

and positive 1δ < , there is a positive constant C such that

( ) ( )( )1

2(x) (y) C x y Z x Z y+β

−β δ δτ τ τ τϕ − ϕ ≤ − τ + (5.41)

for every x and y in M and ( ]0,Tτ∈ . Hence, ϕ is locally, uniformly Hölder continuous

in space with exponent 1β < . This result allows us to prove the following theorem from

which we will deduce that [ ]L f 0τΓ = for every f ∈A and ( ]0,Tτ∈ .

Theorem 7 – [ ]fϕ is locally, uniformly Hölder continuous in space with exponent 1β <

for every f ∈A . More precisely, for every compact dM ⊆ R there is a positive constant

C such that we have

[ ]( ) [ ]( ) ( ) ( )( )1

2 22f ff x f y C x y exp h x exp h y

+β−β

τ τϕ − ϕ ≤ − τ +% % (5.42)

for every x and y in M and ( ]0,Tτ∈ , where f f0 h 2h< <% .

proof: Let x and y be elements of the compact set M, 0 1< β < , and f ∈A . It follows

from (5.41) that for some positive constants C and δ with 1

12

< δ < we have

[ ]( ) [ ]( )f x f yτ τϕ − ϕ ( ) ( )( ) ( )d

x y f dτ τ≤ ϕ − − ϕ −∫Rξ ξ ξ ξ . (5.43)

( ) ( )( ) ( )122

fd

C x y Z x Z y exp h d+β

−β δ δτ τ≤ − τ − + −∫R

ξ ξ ξ ξ

Since 2f δ∈ ⊂A A , it follows from Lemma 2.2 that there exists positive constants C%

and fh% with f fh 2h<% such that

[ ]( ) [ ]( ) ( ) ( )( )1

2 22f ff x f y C x y exp h x exp h y

+β−β

τ τϕ − ϕ ≤ − τ +% %% . < (5.44)

71

2.6 The Solution to the Cauchy Problem Before proceeding to prove the final results necessary to provide the solution to the

Cauchy problem, let us summarize the work we have completed in the past few sections.

We began by looking for a fundamental solution for L of the form

[ ]Z Uτ τ τΓ = + ϕ (6.1)

for some function ( )dC (0,T]ϕ∈ ×R . It was shown in Section 4 that

[ ][ ]( ) [ ] [ ]L U f V f fτ τ τ ϕ = ϕ − ϕ , (6.2)

provided that ϕ is Zδ -bounded for some 1δ ≤ , and that [ ]fϕ is locally, uniformly

Hölder continuous in space (with exponent 1β < ) for every f ∈A . Combining

(6.1) and (6.2) with the condition that [ ]( )L f 0τΓ = for every (0,T]τ∈ and f ∈A , it

was deduced in Section 5 that ϕ must satisfy the following Volterra integral equation:

[ ] [ ]f LZ V fτ τ τϕ = + ϕ . (6.3)

Finally, it was shown that the series [ ]m

m 0

V LZ∞

τ τ=

ϕ = ∑ satisfies (6.3) and the conditions

necessary for (6.2) to hold from which we deduced that [ ]( )L f 0τΓ = for every (0,T]τ∈

and f ∈A . Thus, to conclude that τΓ is a fundamental solution for L, it suffices to show

that [ ] [ ]( )00

f lim f fττ→Γ ≡ Γ = for every f ∈A . Furthermore, from (6.1) and Corollary 3.9

we have that

[ ] [ ]( )[ ] [ ]0 0 0

lim f lim Z U f f limU fτ τ τ ττ→ τ→ τ→ Γ = + ϕ = + ϕ (6.4)

for every f ∈A . Therefore, it suffices to prove the following theorem:

Theorem 1 – For every f ∈A , we have [ ]0

limU f 0ττ→ ϕ = .

proof: Let f ∈A . It follows from Theorem 5.2 and (3.24) that [ ] 0 T

fτ <τ≤ϕ is a

72

continuous, uniformly bounded family of functions from 2A . Hence, we deduce from

Lemma 2.2 that for some positive constants C and [ ] [ ]f fh 2hϕ ϕ<% we have

[ ] ( ) [ ] [ ]( )2

s s f0

U f x Z f ds C exp h x 0 as 0τ τ− ϕ

τϕ ≤ ϕ ≤ τ → τ → ∫ % . < (6.5)

Therefore, τΓ is a fundamental solution for L. It follows that the solution to our

Cauchy problem may be represented by

( ) [ ]( ) [ ]( ) [ ] ( )u x, g x Z g x U g xτ τ τ τ = Γ = + ϕ , (6.6)

where ( )2ˆg(x) exp x≡ α is the initial condition, provided that 1ˆ

32 dα <

ρ.

Finally, we must verify that u satisfies the hypotheses of the Feynman-Kac Theorem

from which we will deduce that

( ) ( ) ( )xP s0

u x, E g X exp dsτ

τ τ = Ψ ∫ (6.7)

for every ( ) [ ]dx, 0,Tτ ∈ ×R . From (6.6), Theorem 3.2, Theorem 4.1, and Theorem 4.2

we have that u is of class ( )2,1 dC [0,T]×R . Furthermore, by assuming that 1ˆ

32 dα <

ρ,

we see that g ∈A . Hence, we deduce from Theorem 5.2 and (3.24) that [ ] 0 T

gτ <τ≤ϕ is

a continuous, uniformly bounded family of functions from 2A . Consequently, it follows

from Lemma 2.2 and Theorem 3.5 that 4u ∈A , thus there exists positive constants A and

1h

8 d<

ρ such that

( ) ( )2

0 Tmax u x, Aexp h x

≤τ≤τ ≤ (6.8)

for every dx ∈R . We conclude from the Feynman-Kac Theorem that (6.7) holds for

every ( ) [ ]dx, 0,Tτ ∈ ×R . Therefore, we deduce from (1.5) that the arbitrage price of the

risk-spread option is given by

( ) ( )( ) ( ) [ ]( ) ( )t

t t s t0C exp r exp ds g X B t,Tτπ = − ατ + κ Ψ Γ −∫ . (6.9)

73

CHAPTER 3 NUMERICAL RESULTS AND APPLICATIONS

3.1 The Fourier and Laplace Transforms The series solution of convoluted potentials presented in the previous chapter is an

elegant exhibition of the ability of the modern mathematical prose to convey a relatively

simple method in a clear, concise manner. In fact, for those pure mathematicians who are

familiar with the antiquated notation used to present the parametrix method in Friedman

(1964), the refined potential theoretic approach is a significant contribution to the

literature. However, it remains to be shown that this solution is of practical value.

Although the iterated convolutions prevent us from directly integrating the series,

they are ideal for calculating the Fourier and Laplace transforms of our solution. Recall

that these transforms are defined by

[ ]( ) ( ) ( ) ( )d

d

2f 2 exp i x f x dx−= π − ⋅∫RF ξ ξ (1.1)

and

[ ]( ) ( ) ( )0

h s exp st h t dt∞

≡ −∫L (1.2)

for every df : →R R and h : + →R R for which these integrals are defined. A useful

result is that these transforms change convolutions into products. In fact, we have

[ ] ( ) [ ] [ ]d

21 2 1 2f f 2 f f∗ = πF F F (1.3)

and [ ] [ ] [ ]1 2 1 2h h h h∗ =L L L h . (1.4)

74

Recall from (2.6.6) that the solution to our Cauchy problem may be represented by

( ) [ ]( ) [ ]( ) [ ] ( )u x, g x Z g x U g xτ τ τ τ = Γ = + ϕ , (1.5)

where ( )2ˆg(x) exp x≡ α and [ ]m

m 0

V LZ∞

τ τ=

ϕ = ∑ for every ( ) [ ]dx, 0,Tτ ∈ ×R . It follows

from (2.2.5) that the first term on the RHS of (1.5) is given by

[ ]( ) ( )d

22

ˆˆZ g x 1 2 exp x

ˆ1 2

−τ

α = − ατ − ατ . (1.6)

The remainder of this section is devoted to the Gaussian potential in (1.5). Moreover, we

will consider the case where d 1= henceforth.

Since the Fourier transform of g does not exist, we will consider the truncated

approximation

( ) ( )n

g x if x ng x

0 if x n

≤≡ >

. (1.7)

Hence, we approximate the Gaussian potential in (1.5) by the sequence [ ] n n 1U g

∞

τ = ϕ .

We proceed to take the Fourier and Laplace transforms of the approximated Gaussian

potential in the spatial and temporal variables, respectively. We recall that

[ ] [ ]n s s n0U g Z g ds

τ

τ τ− ϕ = ϕ ∫ . (1.8)

It follows from (1.3) that for every ( ) [ ], 0,Tτ ∈ ×Rξ we have

[ ] ( ) ( )( ) [ ]( )n q q n0U g 2 Z dq g

τ

τ τ− ϕ = π ϕ ∫F F F Fξ ξ ξ . (1.9)

By defining Z 0τ ≡ for every 0τ ≤ , we see that

[ ]( ) [ ]( ) [ ] [ ]( )( )s s0Z ds Z ,

τ

τ− ϕ = • ϕ τ∫ F F F Fξ ξ ξ (1.10)

where • denotes convolution in time. Thus, we deduce from (1.4) that

75

[ ] ( )nU g ,s ϕ L F ξ [ ] ( ) [ ] ( ) [ ]( )n2 Z ,s ,s g = π ϕ L F L F Fξ ξ ξ (1.11)

[ ]( ) [ ] ( )n

2

2 2 g,s

2s

π = ϕ +

F L Fξ ξξ .

From the series representation for ϕ, we see that

[ ] [ ]m

m 0

V LZ∞

=

ϕ = ∑L F L F . (1.12)

It follows from the definition of the L-potential that

[ ]( ) [ ] ( ) [ ]( )m 1 m ms s0

V LZ x V V LZ x L Z V LZ dsτ+

τ τ τ− = = ∫ (1.13)

( ) [ ]( )mx s s0

L Z x y V LZ y dydsτ ∞

τ−−∞= −∫ ∫

( ) [ ]( )mx s s0

L Z x y V LZ y dsdy∞ τ

τ−−∞= −∫ ∫

( ) [ ]( )( )( )mxL Z x y V LZ y dy

∞

−∞= − • τ∫

for every ( ) [ ]x, 0,Tτ ∈ ×R and m 0≥ . Applying (1.4) to (1.13), we deduce that

[ ] ( ) [ ]( ) [ ] ( )m 1 mxV LZ x,s L Z x y,s V LZ y,s dy

∞+

−∞ = − ∫L L L . (1.14)

Let ( ) [ ]x, 0,Tτ ∈ ×R and y∈ R . From the definition of L, we have that

( )xL Z x yτ − ( ) ( ) ( )x x y x Z x yτθ = − + Ψ − τ

(1.15)

( ) ( ) ( )2 2x x y R x Z x y2

+

τθ σ = − + − − τ

%,

where 2

Rγ≡

σ% . Ideally, we would like to express ( )xL Z x yτ − as a function of x y−

and y so that we may rewrite (1.14) as a convolution in x. However, the plus operator in

the second term on the RHS of (1.15) makes this impossible. For the first term, we will

use the following identity:

( ) ( ) ( )2x x y x y y x y− ≡ − + − . (1.16)

76

Inserting this identity into (1.15) yields

( ) ( ) ( )( ) ( ) ( )2

xL Z x y x y y x y x Z x yτ τθ − = − + − + Ψ − τ

. (1.17)

We will write the Laplace transform of ( )xL Z x y− in terms of the following functions:

( ) ( ) ( )2 ˆF x,s x Z x,s x exp x 2s ≡ θ = θ − L (1.18)

and

( ) ( ) ( ) ( )ˆG x,s x Z x,s sgn x exp x 2s ≡ θ = θ − L , (1.19)

where

( ) ( )1Z x, Z xττ ≡

τ. (1.20)

It follows from (1.17) that

( ) ( ) ( ) ( )xL Z x y F x y,s yG x y,s xτ − = − + − + Ψ L . (1.21)

Let m 0≥ . Combining (1.21) and (1.14) yields

[ ] ( ) ( ) [ ] ( )m 1 mV LZ x,s F x y,s V LZ y,s dy+∞

−∞ = − ∫L L (1.22)

( ) [ ] ( )( )mG x y,s y V LZ y,s dy∞

−∞ + − ∫ yL

( ) [ ]( ) [ ] ( )mx Z x y,s V LZ y,s dy∞

−∞ +Ψ − ∫ L Ly .

The first two integrals may be expressed as convolutions. In fact, we have

( ) [ ] ( ) [ ]( )( )m mF x y,s V LZ y,s dy F V LZ x,s∞

−∞ − = ∗ ∫ L L (1.23)

and

( ) [ ] ( )( ) [ ]( )( )( )m mG x y,s y V LZ y,s dy G x V LZ x,s∞

−∞ − = ∗ ∫ yL L . (1.24)

For the third integral on the RHS of (1.22), we have

( ) [ ]( ) [ ] ( ) ( ) [ ] [ ]( )( )m mx Z x y,s V LZ y,s dy x Z V LZ x,s∞

−∞ Ψ − = Ψ ∗ ∫ L Ly L Ly . (1.25)

77

We now compute the Fourier transform. Applying (1.3) to (1.23) and (1.24) yields

[ ] ( ) [ ] [ ] ( )m mF V LZ ,s 2 F V LZ ,s ∗ = π F L F F Lξ ξ (1.26)

and

[ ]( ) ( ) [ ] [ ] ( )m mG x V LZ ,s 2 G x V LZ ,s ∗ = π F L F F Lξ ξ (1.27)

[ ] [ ]( )( )m2 G i V LZ ,s∂ = π ∂

F F L ξξ ,

respectively. The last equality follows from the fact that

( ) [ ]( )n

n n

nx f i f

∂ = ∂F Fξ ξξ (1.28)

for every function f for which these Fourier transforms are defined.

To compute the Fourier transform of (1.25), we will also need the following property:

[ ] ( ) [ ] [ ]( )d

21 2 1 2f f 2 f f

−= π ∗F F F (1.29)

Applying (1.28), (1.29), and (1.3) to (1.25) yields

[ ] [ ]( ) ( ) ( ) [ ] [ ]( ) ( )m 2 2 mZ V LZ ,s R x Z V LZ ,s2

+σ Ψ ∗ = − ∗ %F L Ly F L Lyξ ξ (1.30)

[ ] [ ] [ ]( ) ( )2

2 mR,R2

R 1 Z V LZ ,s2 −

σ ∂ = + ∗ ∂

% F L Ly ξξ

[ ] [ ] [ ]( )( )2

2 mR,R2

R 1 * Z V LZ ,s2 −

σ ∂ = + ∂

% F F L F Ly ξξ .

Combining (1.26), (1.27), and (1.30) with (1.22) yields the following iterative relation for

every m 0≥ :

[ ] ( )m 1V LZ ,s+ F L ξ [ ] [ ] [ ] ( )m2 F i G V LZ ,s ∂ = π + ∂ F F F L ξξ (1.31)

[ ] [ ] [ ]( )( )2

2 mR,R2

R 1 * Z V LZ ,s2 −

σ ∂ + + ∂

% F F L F Ly ξξ .

78

Computing the various Fourier transforms in (1.31), we have

[ ]( )( )

2

22

2 2sF ,s

2s

−= θ

π +F ξξ

ξ, (1.32)

[ ]( ) 2

2i G ,s

2s= θ

π +F ξξ ξ , (1.33)

[ ] ( ) ( )R,R

sin R21 −

= πF ξξ ξ , (1.34)

and

[ ] 2

2 1Z

2s = π +

F L ξ . (1.35)

Inserting these results into (1.31) yields

[ ] ( )m 1V LZ ,s+ F L ξ (1.36)

( )

( ) [ ] ( )2 2 m22

22s 2s V LZ ,s

2s

θ ∂ = − + + ∂ +F Lξ ξ ξ ξξξ

( )( )

( )( ) [ ] ( )2

2 m2 2

sin y RR V LZ y,s dy

2 y 2s y

∞

−∞

− σ ∂ + + ∂ − + ∫% F Lyξ

ξ ξ .

Combining (1.11), (1.12), and (1.6), we see that (1.5) becomes

( ) ( )d

22ˆ

ˆu x, 1 2 exp xˆ1 2

− α τ = − ατ − ατ (1.37)

[ ]( ) [ ] ( )n1 1 m

2nm 0

2 2 glim V LZ ,s

2s

∞− −

→∞ =

π + + ∑

FF L F Lξ ξξ ,

where the terms of the series are given by (1.36). Although this is not as aesthetically

pleasing as (1.5), we have removed the temporal convolution in the Gaussian potential

term. Unfortunately, the spatial convolution remains in the iterative relation (1.36).

Hence, we will set aside the numerical analysis of the risk-spread option for a future

project.

79

3.2 Delta Hedging with the Risk-Spread Option The introduction of a financial derivative invariably leaves the writer of the option

with the task of hedging against the risk he incurs by assuming a short position in the new

derivative. In the case of a call option on a stock, the hedging of the option writer against

changes in the underlying stock price is commonly known as delta hedging. The delta of

a portfolio is the first derivative of the portfolio with respect to the stock price.

Moreover, the portfolio is said to be delta neutral when it is insensitive to changes in the

stock price, that is, when it has a delta of zero. If we denote the delta of a call option by

∆, then a portfolio consisting of a short position in the option and ∆ shares of the

underlying stock is delta neutral. In fact, denoting the call option by C and the portfolio

value by Π , we find C SΠ = − + ∆ which implies 0.S

∂Π= −∆ + ∆ =

∂

For more exotic derivatives, delta hedging refers to the act of protecting the writer of

the option against changes in various risk factors. Unlike the call option on a tradable

security such as a stock, the risk-spread option must be hedged against the untradable risk

factors that drive the interest rates. However, there are numerous traded bonds that are

affected by the interest rate risk factors. Assuming that the number of risk factors is d,

then a portfolio of d distinct bonds together with a short position in the risk-spread option

that is delta neutral can be constructed.

Let C denote the risk-spread option and ii

C

x

∂∆ ≡

∂ for each of the d risk factors.

Consider d distinct bonds jB with jij

i

B

x

∂δ ≡

∂ and a portfolio with value Π . The hedging

problem is to find the amount jh of bond jB holdings such that the portfolio is delta

neutral. The portfolio value is given by

d

j ji 1

C h B=

Π = − + ∑ . (2.1)

80

This implies that the bond holdings may be determined by setting

d

i j ijj 1i

h 0x =

∂Π= −∆ + δ =

∂ ∑ (2.2)

for each i. We represent this by Dh = ∆ , where

( )d

ij i , j 1D

=≡ δ , (2.3)

( )T

1 dh h ,...,h≡ , (2.4)

and

( )T

1 d,...,∆ ≡ ∆ ∆ . (2.5)

Hence, the vector of bond holdings is given by 1h D−= ∆ , provided that D is invertible.

Returning to the example of the previous sections, consider the case of two risk

factors. We will construct a hedging strategy using the risky and riskless bonds. Fix

[ ]t 0,T∈ and recall the prices of the risky and riskless bond from Section 1.4:

( )B t,T ( ) ( )( )d

2t

ˆˆ1 2 V exp V r−τ τ= − α − +ατ (2.6)

and

( )B t,T% [ ] ( )( ) ( )( )d2

ttˆ1 B t,T 1 2 V exp V

−

τ τν>= − β − λ +βτ% . (2.7)

where V

Vˆ1 2 Vτ

ττ

≡− α

,V

Vˆ1 2 Vτ

τ

τ

≡− β

% , and ( )21V 1 e

2− θτ

τ ≡ −θ

. Also, we recall that

2

t t

1r X

2= σ (2.8)

and

2

t t

1X

2λ = σ% . (2.9)

Hence,

( ) ( )ii

B ˆt,T x V B t,Tx τ

∂= −σ

∂ (2.10)

and

( ) ( )ii

Bt,T x V B t,T

x τ∂

= −σ∂

% % % . (2.11)

where tX x= almost surely.

81

From (2.10) and (2.11), we see that the matrix D in this simple example is singular.

Hence, under the assumption that that there is more than one risk factor, we see that it is

not possible to simultaneously hedge the risk-spread option against all of the risk factors

using a portfolio consisting of only riskless and risky bonds. Consequently, we must add

a different type of interest rate derivative to our portfolio to make it delta neutral.

On the other hand, if we assume that there is a single risk factor, then it follows from

(2.2), (2.10), and (2.11) that we may hedge the risk-spread option by holding Bth riskless

bonds, or Bth% risky bonds, where

( )

Bth ˆxV B t,Tτ

∆= −

σ (2.12)

and

( )

Bth

xV B t,Tτ

∆= −σ

%%% . (2.13)

Alternatively, we may be able to hedge the risk-spread option against the riskless spot

rate directly, if it can be shown that the risk-spread option only depends on the risk

factors through the riskless spot rate. In this case, we compute

( )BV B t,T

r τ∂

= −∂

(2.14)

and

( ) ( )Bt,T V B t,T

rτ

∂ σ= −

∂ σ

% % % % (2.15)

from which we deduce that holdings of either

( )

Bth

V B t,Tτ

∆= − (2.16)

in the riskless bond, or

( )

Bth

V B t,Tτ

σ∆= −σ

%% %% (2.17)

in the risky bond will provide the desired hedge.

82

3.3 Numerical Properties of the Yield Curve Recall that the state-variable process tX from Section 1.4 satisfies

t t tdX dW X dt= − θ (3.1) for some positive parameter θ, where tW is a d-dimensional Brownian motion on

( ), ,PΩ F . Continuing with the example of Section 1.4, we also recall that the riskless

spot rate and risk spread are given by

2

t t

1r X

2= σ (3.2)

and

2

t t

1X

2λ = σ% , (3.3)

respectively, where ˆ ˆ4 ( )σ ≡ α θ − α , ˆ ˆ4 ( )σ ≡ β θ − β% , ˆd

αα ≡ , and ˆ

d

ββ ≡ are positive

constants. We will show that tr follows the Cox-Ingersoll-Ross (CIR) process given by

( )t t t tdr a b r dt 2 r dM= − + σ (3.4)

for some positive constants a and b, where tM is a one-dimensional Brownian

Motion on ( ), ,PΩ F with respect to the natural filtration tG of tW (Elliot & Kopp, 1999).

In fact, from ˆIto's lemma we deduce from (3.1) and (3.2) that

d d

i i it t t t

i 1 i 1

dr X dX d X2= =

σ= σ +∑ ∑ (3.5)

d

2i it t t

i 1

dX dW X dt dt

2=

σ= σ −θσ +∑

i idt t

t ti 1 t

d X dW2 r dt X

2 X=

σ = − θ + σ ∑

( )t t ta b r dt 2 r dM= − + σ ,

where a 2≡ θ , d

b4

σ≡

θ, and

i is s

ts

d

i 1

t

0

X dWM

X=≡ ∑∫ .

83

It remains to be shown that tM is a one-dimensional Brownian Motion on ( ), ,PΩ F

with respect to tG . We first assert that tM is a continuous martingale. In fact, we have

2i

isP s

s

T

0

XE dW T

X

≤ < ∞ ∫ (3.6)

for every i. Furthermore, it follows from ˆIto's lemma that

2i

2 st s s s s s ss

s

d

i 1

t t t tt

00 0 0 0

XM 2 M dM d M 2 M dM ds 2 M dM t

X=

= + = + = +

∑∫∫ ∫ ∫ ∫ . (3.7)

We deduce that 2tM t− is a martingale from which we conclude that tM is a standard

Brownian motion with respect to tG . Hence, we have shown that tr follows the CIR

process given by (3.4). Similarly, we have that

( )t t t td a b dt 2 dMλ = − λ + σλ% % , (3.8)

where d

b4

σ≡

θ%% .

The CIR process has two properties that correspond with empirical spot rates and risk

spreads. First, it is shown in (Lamberton & Lapeyre, 1996) that the CIR processes (3.4)

and (3.8) are almost surely positive, provided that d ≥ 2 , 0r 0> , and 0 0λ > almost

surely. Unfortunately, in the one dimensional case we have that the probability that these

CIR processes vanish for infinitely many times is one.

The second ideal property of the CIR process is that of mean reversion. Consider the

following ordinary differential equation:

( )dx a b x dt= − . (3.9)

The solution of (3.9) is given by

( ) ( )x t exp at b= − + . (3.10)

84

This function exponentially decays to the mean reversion level b at the mean reversion

rate a as t tends to infinity. Comparing this with (3.4) and (3.8), we deduce that the CIR

process has a mean-reverting drift term.

We conclude this chapter with a graphical comparison of the yield curves of the

riskless and risky bonds. Recall from Section 1.4 that the riskless and risky yield curves

are given by

( ) ( ) ( )t

1 1 dˆ ˆY t,T lnB t,T V r ln 1 2 V2τ τ

≡ − = + α τ + − α τ τ (3.11)

and

( ) ( ) [ ] ( ) ( )tt

1 1 d ˆY t,T lnB t,T 1 Y t,T V ln 1 2 V2

τ τν >

≡ − = + λ +βτ+ − β τ τ % % % , (3.12)

respectively. For fixed t, we see from the definitions following (2.7) that

( ) ( )( ) ( ) ( )( )t

1 exp 2 ˆ1 dY t,T r ln 1 1 exp 2

ˆ ˆ2 2 exp 2 2

− − θτ α = + α τ + − − − θτ τ θ − α + α − θτ θ (3.13)

and

( ) [ ] ( )tY t,T 1 Y t,Tν>=% (3.14)

[ ]( )

( ) ( )( )( )tt

ˆ1 exp 21 d1 r ln 1 1 exp 2

ˆ ˆ 22 2 exp 2ν>

− − θτ β + +βτ+ − − − θτ τ θθ − β + β − θτ

.

It follows that

( )limY t,Tτ→∞

= α (3.15)

and ( )limY t,T

τ→∞= α + β% . (3.16)

In the graphs below, we present the initial riskless and risky yield curves for various

parameter values. In particular, we see that the rate of convergence in (3.15) and (3.16)

depends on the values of θ, ˆθ − α , and ˆθ − β .

85

Fig. 1 The Risky and Riskless YTM with d = 5, θ = .05, α = .09, and β = .04

4.50%

6.50%

8.50%

10.50%

12.50%

14.50%

16.50%

18.50%

20.50%

22.50%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Number of Years Until Maturity

RISKY YTM 20.8% RISKY YTM 16.8% RISKY YTM 12.81% RISKY YTM 8.81%

RISKLESS YTM 11% RISKLESS YTM 9% RISKLESS YTM 7% RISKLESS YTM 5%

86

Fig. 2 The Risky and Riskless YTM with d = 5, θ = .05, α = .09, and β = .04

4.50%

6.50%

8.50%

10.50%

12.50%

14.50%

16.50%

18.50%

20.50%

22.50%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Number of Years Until Maturity

RISKY YTM 20.8% RISKY YTM 16.8% RISKY YTM 12.81% RISKY YTM 8.81%

RISKLESS YTM 11% RISKLESS YTM 9% RISKLESS YTM 7% RISKLESS YTM 5%

87

Fig. 3 The Risky and Riskless YTM with d = 5, θ = .05, α = .09, and β = .04

4.50%

6.50%

8.50%

10.50%

12.50%

14.50%

16.50%

18.50%

20.50%

22.50%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Number of Years Until Maturity

RISKY YTM 19.91% RISKY YTM 15.91% RISKY YTM 11.92% RISKY YTM 7.92%

RISKLESS YTM 11% RISKLESS YTM 9% RISKLESS YTM 7% RISKLESS YTM 5%

88

Fig. 4 The Risky and Riskless YTM with d = 5, θ = .05, α = .09, and β = .04

4.50%

6.50%

8.50%

10.50%

12.50%

14.50%

16.50%

18.50%

20.50%

22.50%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Number of Years Until Maturity

RISKY YTM 19.91% RISKY YTM 15.91% RISKY YTM 11.92% RISKY YTM 7.92%

RISKLESS YTM 11% RISKLESS YTM 9% RISKLESS YTM 7% RISKLESS YTM 5%

89

Fig. 5 The Risky and Riskless YTM with d = 5, θ = .05, α = .09, and β = .04

4.50%

6.50%

8.50%

10.50%

12.50%

14.50%

16.50%

18.50%

20.50%

22.50%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Number of Years Until Maturity

RISKY YTM 19.91% RISKY YTM 15.91% RISKY YTM 11.92% RISKY YTM 7.92%

RISKLESS YTM 11% RISKLESS YTM 9% RISKLESS YTM 7% RISKLESS YTM 5%

90

Fig. 6 The Risky and Riskless YTM with d = 5, θ = .05, α = .09, and β = .04

4.50%

6.50%

8.50%

10.50%

12.50%

14.50%

16.50%

18.50%

20.50%

22.50%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Number of Years Until Maturity

RISKY YTM 19.91% RISKY YTM 15.91% RISKY YTM 11.92% RISKY YTM 7.92%

RISKLESS YTM 11% RISKLESS YTM 9% RISKLESS YTM 7% RISKLESS YTM 5%

91

CHAPTER 4 SUMMARY AND CONCLUSIONS

4.1 Summary of Results

The potential theoretic framework developed in Chapter 1 extends the work of Rogers

(1997) to include the case of defaultable bonds. Among the numerous examples of both

the riskless and risky spot rates of interest that may be generated from this procedure, the

familiar, tractable example of the Ornstein-Uhlenbeck process was used as an illustration.

This example is realistic in the sense that the resulting Cox-Ingersoll-Ross model of the

spot rates is strictly positive, provided the dimension of the driving Markov process is at

least two. Furthermore, this model exhibits the mean-reverting behavior that has been

observed empirically.

This example was carried forward to the Cauchy problem for the risk-spread option

treated in Chapter 2. The potential theoretic parametrix method used to develop a series

solution to the Cauchy problem represents a significant contribution to mathematical

literature. To gain an appreciation for the modern language, the interested reader should

compare this method with the parametrix method outlined in the first chapter of

(Friedman, 1964).

In Chapter 3, the Fourier and Laplace transforms were used to derive an expression

for the risk-spread option. This removed the temporal convolution in the Gaussian

92

potential term; however, the remaining spatial convolution forced us to postpone the

numerical analysis of our solution. In the second section of this chapter, it was shown

how to delta hedge the risk-spread option using a portfolio of riskless and risky bonds.

Finally, we examined the properties of the spot rates through the graphs of the yield

curves in Section 3.3.

4.2 Future Projects and Model Extensions

The work presented in this dissertation has laid down the foundation for future

development in pricing derivatives on the risky spot rate such as the risk-spread option.

It also presents a new framework within which various models of the risky spot rate may

be produced. One way to extend the model developed in this dissertation is to relax the

independence assumption used to obtain the representation (1.3.41) of the risky bond

price. Another extension that might prove interesting is to allow the mean of the

exponential random variable used in modeling the time of default to be a free parameter.

The procedure outlined in Section 1.2 for constructing a positive supermartingale to

model the state-price density may be modified by rewriting

( )( )

ttt

0

f Xe

f X−αζ = (2.1)

as

( )( )

tt t

0

f XM

f Xζ = , (2.2)

where tM is a multiplicative functional such as ( )( )t

s0exp X ds− α∫ . Naturally, this could

also be done to the risky state-price density.

93

The simple Ornstein-Uhlenbeck process that we have considered is one of many

Markov processes that may be used to obtain tractable results. A slight modification of

this example is to replace the parameter θ with a d d× matrix to allow for interactions

between the components of the Markov process. In addition, the function f in (2.1) could

be redefined using a symmetric positive-definite matrix A and a vector dc∈R :

( )Tf(x) exp (x c) A(x c)= − − . (2.3)

The interested reader is referred to (Rogers, 1997) for more examples.

Regardless of the state-variable used in this framework, the calculation of the

effective risk-spread insurance level is a nice problem that naturally follows from the

risk-spread option. Recall from Section 1.5 that for a given risk-spread insurance level γ,

the effective risk-spread insurance level γ% is the minimum rate of return that the holder

of a risky bond and a risk-spread option expects to receive. It was suggested in Section

1.5 that γ% be defined by

( )( ) ( )( ) ( ) ( )( )P T P T T 0E E r ln B 0,T CT

+ + ∂γ − λ = γ − λ − − + π

∂%% . (2.4)

An investigation of this relation will make an interesting future project.

The final extension that will be discussed follows the work of Lando (1998) in the

area of risk classes. By constructing a different risky state-price density for each risk

class, the Markov chain model used by Lando could be adapted to our potential theoretic

framework. In fact, the rate of transition between risk classes should follow a process

that is similar to the risk-spread. After establishing a model of these transition processes,

it would be interesting to price options on the transition between risk classes using the

potential theoretic parametrix method of Chapter 2.

94

REFERENCES

Björk, T. (1997) Interest rate theory. In: Financial Mathematics, Bressanone, 1996, W. Runggaldier, ed. Lecture Notes in Mathematics 1656. Springer-Verlag, New York, 53-122.

Blumenthal, R. M. & Getoor, R. K. (1968) Markov Processes and Potential Theory.

Academic Press, New York. Chung, K. L. (1974) A Course in Probability Theory. Academic Press, New York. Chung, K. L. & Williams, R. J. (1990) Introduction to Stochastic Integration.

Birkhäuser, Boston. Delbaen, F. & Scachermayer, W. (1994) A general version of the fundamental theorem of

asset pricing. Mathematische Annalen 300, 463-520. Delbaen, F. & Scachermayer, W. (1998) The fundamental theorem of asset pricing for

unbounded stochastic processes. Mathematische Annalen 312, 215-250. Elliot, R. J. & Kopp, P. E. (1999) Mathematics of Financial Markets. Springer-Verlag,

New York. Friedman, A. (1964) Partial Differential Equations of Parabolic Type. Prentice-Hall,

Englewood Cliffs, NJ. Hull, J. C. (1997) Options, Futures, and Other Derivatives. Prentice-Hall, Upper Saddle

River, NJ. Karatzas, I. & Shreve, S. E. (1991) Brownian Motion and Stochastic Calculus. Springer-

Verlag, New York. Lamberton, D. & Lapeyre, B. (1996) Introduction to Stochastic Calculus Applied to

Finance. Chapman & Hall, New York. Lando, D. (1998) On Cox processes and credit risky securities. Review of Derivatives

Research 2, 99-120.

95

Musiela, M. & Rutkowski, M. (1998) Martingale Methods in Financial Modelling. Springer-Verlag, New York.

Renardy, M. & Rogers, R. C. (1992) An Introduction to Partial Differential Equations.

Springer-Verlag, New York. Revuz, D. & Yor, M. (1991) Continuous Martingales and Brownian Motion. Springer-

Verlag, New York. Rogers, L. C. G. (1997) The potential approach to the term structure of interest rates and

foreign exchange rates. Mathematical Finance 7 (2), 157-176. Stoer, J. & Bulirsch, R. (1993) Introduction to Numerical Analysis. Springer-Verlag,

New York.

96

BIOGRAPHICAL SKETCH

At the age of seventeen, I finished my junior year of high school as well as sixty

hours of college credits at the local community college. These credits included the two-

semester freshman physics and chemistry sequences, calculus I, II, and III, and a first

course in differential equations. Since I did not require a senior year of high school, I

entered the University of Florida in the fall of 1990 without a high school diploma. I

chose to major in physics because of my passion for scientific truth and wanted a deeper

understanding of this universe into which I have been born. Since I entered the university

as a junior, I received a bachelor’s degree in May of 1992 as I turned nineteen years of

age.

After obtaining a bachelor’s degree three years ahead of schedule, I realized that I

might enjoy a couple of semesters away from academia while I considered the

appropriate path to follow in graduate school. In the fall of 1992, I took a job at 102.5

FM in St. Petersburg, where I worked until the following summer. I still had not decided

what to do about graduate school, but I was ready to leave the radio station and return to

academia in June of 1993. During that summer I worked on a project in a solid state

physics lab under the guidance of Dr. Tanner at the University of Florida. Although I

enjoyed experimental physics, I realized that a deep understanding would only come

from a more theoretical perspective.

97

In the fall of 1993, I was back at UF as a post-baccalaureate student taking the few

remaining classes needed to get a second bachelor’s degree in mathematics. During the

Spring semester of 1994, I had a revelation in Dr. Groisser’s Advanced Calculus II class.

It became apparent to me that mathematics is nothing less than the language of physics.

Furthermore, I knew that I had to obtain at least a master’s in mathematics before I could

understand physics. So, after obtaining a bachelor’s degree in mathematics in 1994, I

proceeded to enter graduate school in mathematics the following fall.

The first two years of graduate school seemed to go by rather quickly, and I received

a master’s degree in the spring of 1996. During the second year, I was influenced the

most by Dr. Dinculeanu in his two-semester course on measure and integration theory.

Dr. Dinculeanu taught me how to write a mathematical proof. His style and teaching

mannerisms have left lasting impressions in me and guided me toward my ultimate

decision to work in the area of probability theory and stochastic processes.

In the spring of my third year of graduate school, I became somewhat disillusioned

with the poor job outlook for a mathematical physicist, and took a class called The

Mathematics of Financial Derivatives with my current advisor Dr. Glover. As I was

maturing mathematically, I realized that mathematics is more than the language of

physics. Mathematics is the language of order and structure itself. All structural

concepts begin as vague, creative thoughts which can only be tested when measured, and

can only be measured when expressed in the proper mathematical framework. Therefore,

I decided that I could write a dissertation in mathematical finance as well as gain an

understanding of the probabilistic nature of the universe. Of course, the prospect of a

starting six-figure salary on Wall Street had nothing to do with this decision at all.