The density of the supremum of a stable processmath.yorku.ca/~dhackman/astable.pdf · The density of the supremum of a stable process Daniel Hackmann Department of nancial mathematics

Introduction Levy processes Stable processes The problem A solution

The density of the supremum of a stableprocess

Daniel Hackmann

Department of financial mathematics and applied number theoryJohannes Kepler University

Linz, Austria

August 25–27, 2015

Joint work with Alexey Kuznetsov

Density stable supremum Daniel Hackmann 0/25


1 Introduction

2 Levy processes: A quick introduction

3 Stable processes

4 The problem

5 A solution



Overview

Introduce Levy processes and stable processes

Introduce the problem, the history and a general approach

Present a solution which has an interesting connection to thecontinued fraction representation of irrational numbers



Overview






Overview






Outline

1 Introduction


3 Stable processes

4 The problem

5 A solution



Definition

A Levy process is an R-valued stochastic process X = Xt : t ≥ 0defined on a probability space (Ω,F ,P) that possesses the followingproperties:

(i) The paths of X are right continuous with left limits P-a.s.

(ii) X0 = 0 P-a.s.

(iii) For 0 ≤ s ≤ t, Xt −Xs is independent of Xu : u ≤ s.(iv) For 0 ≤ s ≤ t, Xt −Xs is equal in distribution to Xt−s.



Applications

Levy processes are popular because they are general enough torepresent real world phenomena, but tractable enough so that we mayobtain meaningful results.

We find applications of Levy processes in many fields, the naturalsciences, operations management, actuarial science, and mathematicalfinance.



One-sided processes

Definition

Processes which are almost surely increasing are called subordinators.Processes which are not subordinators but have no negative jumps arecalled spectrally positive.



Infinite divisibility

A random variable ξ is infinitely divisible if for any n ∈ N it satisfies

ξd= ξn1 + . . .+ ξnn ,

for i.i.d. ξni : i = 1, . . . n. For a Levy process X we have

Xt =(Xt −X (n−1)t

n

)+ . . .+

(X 2t

n−X t

n

)+X t

n,

and so, by the independent and stationary increments property, wehave that Xt, in particular X1, is infinitely divisible. Conversely, forany inf. div. random variable ξ, we can find a Levy process X such

that X1d= ξ.



Outline

1 Introduction


3 Stable processes

4 The problem

5 A solution



Stable processes

A (strictly) stable random variable is defined as a random variablewhich, for some α ∈ (0, 2] satisfies

nαξd= ξ1 + . . .+ ξn, n ∈ N,

where the ξi are independent and distributed like ξ. Therefore, stablerandom variables are infinitely divisible, and so, we can speak alsoabout stable Levy processes X.

A natural way to categorize stable processes is by the parameters αand ρ = P(X1 > 0).



Admissible parameters

We will study processes with parameters in the admissible set

A = α ∈ (0, 1), ρ ∈ (0, 1) ∪ α = 1, ρ = 12 ∪ α ∈ (1, 2), ρ ∈ [1− α−1, α−1].

ρ

α



Stable processes

A key feature of such processes is the self-similarity property:

λ1/αXt : t ≥ 0 d= Xλt : t ≥ 0, λ > 0,

i.e. scaling the process in space is equivalent to a scaling in time. Astandard example: the well known scaling of Brownian motion i.e.√tB1

d= Bt.



More on the parameters

subordinator ⇔ ρ = 1, spectrally negative ⇔ ρ = 1/α

-subordinator ⇔ ρ = 0 spectrally positive ⇔ ρ = 1− 1/α

ρ

α



The Doney classes

If we generalize the the spectrally negative, ρ = 1/α, and spectrallypositive, ρ = 1− 1/α, cases we get the processes in the Doney classesCk,lk,l∈Z. A stable process X ∈ Ck,l for some k, l ∈ Z if itsparameters statisfy

ρ =l

α− k.



The Doney classes

The line corresponds to X ∈ C1,1.

α = 3/4 corresponds to exactly 2 possible ρ such that the process withparameters (α, ρ) is in a Doney class (see •). In general, α = m/n cancorrespond to exactly m− 1 parameters ρ such that X is in a Doneyclass. These have the form C(α) = j/m : j = 1, 2, . . . ,m− 1.

ρ

α



The Doney classes

The Doney classes correspond to a dense set of parameters.

ρ

α



Outline

1 Introduction


3 Stable processes

4 The problem

5 A solution



The problem

For a stable process with admissible parameters define

St := sup0≤s≤t

Xt,

which is known as the running supremum process.

Q: Can we find an explicit expression for the density p(x) of S1?



The general problem

A: In some special cases:

Darling 1956: a simple expression for the Cauchy process(α, ρ) = (1, 1/2)

Bingham 1973: an absolutely convergent series representation forthe spectrally negative case

Bernyk, Dalang, and Peskir 2008: an absolutely convergent seriesrepresentation for the spectrally positive case

Kuznetsov 2011: a full asymptotic expansion at 0 and ∞ and anabsolutely convergent series representation for X ∈ Ck,lHubalek and Kuznetsov 2011: an absolutely convergent series foralmost all processes where α /∈ Q



The general problem








The general problem








The general problem





Kuznetsov 2011: a full asymptotic expansion at 0 and ∞ and anabsolutely convergent series representation for X ∈ Ck,l

Hubalek and Kuznetsov 2011: an absolutely convergent series foralmost all processes where α /∈ Q



The general problem








The approach to the general problem

The key is to build a connection with the positive Wiener-Hopf factorof the underlying process. For a stable process X with supremumprocess S this is defined as

ϕ(z) := E[exp(−zSe(1))], Re(z) ≥ 0,

where Se(1) is the running supremum evaluated at e(1) anindependent exponential random variable with mean 1.

For a general process it is not easy to obtain an explicit expression forϕ(z), but for stable processes we now have this information due toDarling 1956 (ρ = 1/2), Doney 1987 (X ∈ Ck,l), and Kuznetsov 2011(general formula).



The approach

The Mellin transforms

M(S1, w) := E[Sw−11 ] =

∫R+

xw−1p(x)dx, 1− αρ < Re(w) < 1 + α,

and

Φ(w) :=

∫R+

zw−1ϕ(z)dz, 0 < Re(w) < αρ

of the two functions are related by the identity.

Φ(w) = Γ(w)Γ(

1− w

α

)M(S1, 1− w), 0 < Re(w) < αρ.



The approach

Since Kuznetsov 2011 we know M(S1, w) explicitly. The question iscan we invert to obtain p(x)?

For X ∈ Ck,l the answer is yes for all pairs (α, ρ) → abs. conv.series.

Analytical inversion when α ∈ Q and X /∈ Ck,l seems difficult,since M(S1, w) has poles of order greater than one andcomputing residues seems impossible. However, Kuznetsov 2013demonstrates that numerical inversion is accurate and reasonablysimple.

When α /∈ Q we have simple real poles and we can calculateresidues. → abs. conv. double series, but not for all α.



The approach







The approach







α ∈ (1, 2)\Q

Mellin inversion gives, after a considerable amount of work,

p(x) = xαρ−1∑

m+α(n+ 12 )<k

m≥0, n≥0

am,nxm+αn + ek(x),

where am,n are the residues of M(S1, w) corresponding to the poless−m,n and

|ek(x)| < (A(1 + x))k × e−εk ln(k) ×k∏l=1

∣∣∣∣sec

(πl

α

)∣∣∣∣ .



Outline

1 Introduction


3 Stable processes

4 The problem

5 A solution



Continued fractions

Recall that every real number x has a continued fraction representation

x = [a0; a1, a2, . . . ] = a0 +1

a1 +1

a2 + . . .

. (1)

Irrational numbers have infinite continued fraction representations.

For irrational x truncating (1) after k steps results in a rationalnumber pk/qk(x) := [a0; a1, a2, ..., ak] called the kth convergent.

The kth convergent is the best rational approximation of x among allrational numbers with denominator less than or equal to qk(x).



Continued fractions

We have the recursive relationship qk = akqk−1 + qk−2, and theestimate ∣∣∣∣x− pk

qk

∣∣∣∣ < 1

qkqk+1

Definition

The set L is composed of irrational numbers x for which there exists b > 1such that ak > bqk for infinitely many k.

L is closed under addition and multiplication by rational numbers, and

x ∈ L ⇔ x−1 ∈ L.



Back to our problem

Suppose now that α ∈ L in which case so is 2/α. Then, based on ourprevious discussion, it it reasonable to assume pk/qk(2/α) approximates2/α very closely. Accordingly, we will have difficulty bounding∣∣∣sec

(πqkα

)∣∣∣ =

∣∣∣∣sec

(π

2× qk ×

2

α

)∣∣∣∣by an exponential function of qk. In fact, we can show quite easily that for

α = [a0; a1, a2, . . . ], a0 = 0, a1 = 1, and ak+1 = 2q2k we have∣∣∣sec(πqkα

)∣∣∣ > 2q2k

π.



A solution

The key to establishing a conditionally convergent series, is to showthat

qk−1∏l=1

∣∣∣∣sec

(πl

α

)∣∣∣∣ ≤ C6qk ,

where qk = qk(2/α).



A solution

Then we replace the k with qk − 1 in our previous calculation:

p(x) = xαρ−1∑

m+α(n+ 12 )<qk−1

m≥0, n≥0

am,nxm+αn + eqk−1(x),

where we recall that

|eqk−1(x)| < (A(1 + x))qk−1 × e−ε(qk−1) ln(qk−1) ×qk−1∏l=1

∣∣∣∣sec

(πl

α

)∣∣∣∣ .An analogous approach works for α ∈ (0, 1) which leads us to toconclude...



Main result

Theorem

There exists a conditionally convergent double series for p(x) valid forall irrational α.



D. Hackmann and A. Kuznetsov.

A note on the series representation for the density of the supremum of astable process.

Electron. Commun. Probab., 18:no. 42, 1–5, 2013.

F. Hubalek and A. Kuznetsov.

A convergent series representation for the density of the supremum of astable process.

Elec. Comm. in Probab., 16:84–95, 2011.

A. Kuznetsov.

On extrema of stable processes.

The Annals of Probability, 39(3):1027–1060, 2011.

A. Kuznetsov.

On the density of the supremum of a stable process.

Stochastic Processes and their Applications, 123(3):986–1003, 2013.

www.danhackmann.com


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The density of the supremum of a stable processmath.yorku.ca/~dhackman/astable.pdf · The density of the supremum of a stable process Daniel Hackmann Department of nancial mathematics