Differential and Integral Equations Volume 16, Number 7, July 2003, Pages 787–811

NONLINEAR INTEGRO-DIFFERENTIAL EVOLUTIONPROBLEMS ARISING IN OPTION PRICING: A

VISCOSITY SOLUTIONS APPROACH

Anna Lisa AmadoriIstituto per le Applicazioni del Calcolo “M. Picone”

Consiglio Nazionale delle RicercheViale del Policlinico 137, I–00161 Roma, Italy

(Submitted by: G. Da Prato)

Abstract. A class of nonlinear integro-differential Cauchy problemsis studied by means of the viscosity solutions approach. In view offinancial applications, we are interested in continuous initial data withexponential growth at infinity. Existence and uniqueness of solution isobtained through Perron’s method, via a comparison principle; besides,a first order regularity result is given. This extension of the standardtheory of viscosity solutions allows to price derivatives in jump–diffusionmarkets with correlated assets, even in the presence of a large investor,by means of the PDE’s approach. In particular, derivatives may beperfectly hedged in a completed market.

1. Introduction

This paper is devoted to the study of the Cauchy problem related tointegro–differential equations of the following type

∂tu + F (x, t, u, Iu, Du, D2u) = 0, (x, t) ∈ Rn× (0, T ], (1.1)

u(x, 0) = uo(x), x ∈ Rn. (1.2)

The integral term Iu is given by

Iu(x, t) =∫

RnM

(u(x + z, t), u(x, t)

)dµx,t(z), (1.3)

where µx,t are bounded positive Radon measures fulfilling

lim(y,s)→(x,t)

∫Rn

ϕ(z)dµy,s(z) =∫

Rnϕ(z)dµx,t(z), (1.4)

Accepted for publication: October 2002.AMS Subject Classifications: 35D05, 35D10, 35K55, 45K05, 49L25, 91B24, 60J75.

787

788 Anna Lisa Amadori

for any continuous function with bounded support ϕ, and M is a Lipschitzcontinuous function, nondecreasing with respect to its first argument:

|M(u, v) − M(u′, v′)| ≤ m (|u − u′| + |v − v′|) ,M(u, u) = 0, M(u, v) ≤ M(u′, v) if u ≤ u′.

(1.5)

All results may be trivially extended to any vector valued nonlocal termIu =

(I1u, · · · , Ipu

).

We make the following general assumptionsF.0) uo ∈ C(Rn),F.1) F ∈ C ([0, T ] × R

n× R × Rn× Sn) is degenerate elliptic, namely

F (x, t, u, p, X) ≥ F (x, t, u, p, Y ) if X ≤ Y,

F.2) F is quasi–monotone as a function of u, namely there exists g ∈C([0, T ]) such that g(0) = 0,

∫ 10

drg(r) = +∞ and

F (x, t, u, p, X) ≥ F (x, t, v, p, X) − g(u − v) if u ≥ v,

F.3) F is monotone non-increasing with respect to the nonlocal term Iu:

F (x, t, u, I, p, X) ≥ F (x, t, u, J, p, X) if I ≤ J.

Here Sn stands for the set of symmetrical n × n matrices.Integro–differential parabolic operators of type

∂tu − div(A(x, t, u, Du)Du

)+ b(x, t, u, Du) = J u, (1.6)

where A is a positive definite matrix and

J u =∫

Rn

[u(x + z, t) − u(x, t) − z · Du(x, t)

]µx,t(dz), (1.7)

have been extensively studied, see for instance Garroni and Menaldi [10] andthe references therein. Even though the nonlocal terms arising in derivativespricing may always be written in the form (1.7) (see later on Remark 2.6)their technique may not be applied to many equations of financial interest,for two reasons. The first one, which does not seem to be overcome, is thatthey deal at most with weak degeneracy, namely A ≥ ω(x)In, with ω > 0almost everywhere. The second one is that it is not clear how to handle anonlinear dependence on the jump term J .

Therefore, it is natural to look to the theory of viscosity solutions, whichallows to deal with strongly degenerate equations. For instance, using thisapproach Alvarez and Tourin [1] studied the Cauchy problem concerning anintegro-differential equation of type

∂tu + F (x, t, u, Du, D2u) = Iu, (1.8)

nonlinear integro-differential equations 789

where the nonlocal term is given by (1.3)–(1.5), and F is a pure differen-tial second order operator, possibly degenerate elliptic. In particular, theyproved the well posedness in the class of continuous functions with lineargrowth for large values of |x|. In this paper we follow this approach and weextend the results of [1] in order to handle the wider class (1.1).

The motivation for such improvement of the known theory relies in thestudy of derivatives pricing in jump–diffusion markets with large investor,i.e., in that, cases where the agent policy effects the assets prices, since inthis framework the pricing equation has a nonlinear behavior with respectto the integral term Iu. In Section 2, we shall give more details about therelationship between the option pricing formula and integro–partial differen-tial problems of type (1.1)–(1.2). For the time being we just point out that,in view of these financial applications, we are interested into well posednessresults and local Lipschitz regularity of the solutions to the Cauchy problem,when the initial datum has exponential growth at infinity

(H.0) e−no|x|uo(x) ∈ C(Rn) ∩ L∞(Rn),

or, respectively, when the first order derivatives of the initial datum growup exponentially at infinity

(H′.0) e−n|x|uo(x) ∈ W 1,∞(Rn).

This is the case, for instance, when pricing any call option, since uo is givenby uo(x) = max

ex − eK , 0

, where K is the strike price of the option.

To this aim, we shall focus our interest into the semilinear case

∂tu + LIu + H(x, t, u, Iu, Du) = 0, (1.9)

whereLIu = −1

2tr

[σσT D2u

]+ b · Du + c u − Iu

is a linear integro–differential operator, possibly strongly degenerate.This paper is organized as follows. We begin by illustrating jump–diffusion

models of market and by relating the price of derivatives to a Cauchy problemof type (1.9)–(1.2), in Section 2.

Section 3 concerns the well posedness of integro–differential Cauchy prob-lems: the notion of viscosity solutions and the Perron’s method establishedin [1] are extended to the problem (1.1)–(1.2); next, by means of the com-parison principle stated by Theorem 3.4, the following result is attained.

Theorem 1.1 (Well posedness). We assume that the initial datum satis-fies H.0 and that the semilinear integro–differential equation (1.9) has thefollowing regularity properties

790 Anna Lisa Amadori

H.1) σij , bj , c are bounded and continuous functions of x, t, and σij , bj arelocally Lipschitz continuous w.r.t. x (uniformly w.r.t. t).

H.2) The function H is continuous with respect to all the variables andmonotone nonincreasing with respect to the forth one. For all R > 0,there exists a modulus of continuity ωR such that∣∣H(x, t, u, I, p) − H(y, t, u, I, p)

∣∣ ≤ ωR ((1 + |u| + |I| + |p|)|x − y|) (1.10)

if |x|, |y| ≤ R. Moreover, there exists l′ ≥ 0 such that

H(x, t, u, I, p) − H(x, t, v, J, q) ≤ l′ [|u − v| + |I − J | + |p − q|] , (1.11)

for all (x, t) ∈ Rn× [0, T ], u, v, I, J ∈ R, and p, q ∈ R

n. Lastly, thereexists n1 ≥ 0 such that

e−n1|x|H(x, t, 0, 0, 0) ∈ C(Rn× [0, T ]) ∩ L∞(0, T ; L∞(Rn)

). (1.12)

H.3) There exists n ≥ max(no, n1) such that 1+en|·| belongs to L1(Rn; µx,t)for all x, t and that∫ (

1 + en|z|)

dµx,t(z) ∈ C(Rn× [0, T ]) ∩ L∞(0, T ; L∞(Rn)

). (1.13)

Then the problem (1.9)–(1.2) admits a continuous viscosity solution u ful-filling

e−max(no,n1)|x|u(x, t) ∈ L∞(0, T ; L∞(Rn)

),

which is unique in the classv ∈ C

(R

n× (0, T ))

: e−n|x|v(x, t) ∈ L∞(0, T ; L∞(Rn)

).

Finally, Section 4 is devoted to the study of the regularity properties ofthe viscosity solutions. In view of the financial applications, we suppose thatthe initial datum is locally Lipschitz continuous and that the measures µx,t

are of type

dµx,t(z) =m∑

k=1

νk(x, t) δ(ζk(x, t) − z) + ν(x, t; z) dz,

where νk, ν are continuous and nonnegative real functions, ζk are continuousfunctions with values in R

n, and δ stands for the Dirac measure. Eventually,the following regularity preserving property is established.

Theorem 1.2 (Lipschitz continuity). We assume that the initial datum uo

satisfies H’.0 and that the hypotheses H.1–H.3 about the semilinear integro–differential equation (1.9) are strengthened by

nonlinear integro-differential equations 791

H’.1) σij , bj , c ∈ C(R

n× [0, T ])∩ L∞(

0, T ; W 1,∞(Rn));

H’.2) (1.10) is strengthened by

(1.10′)∣∣H(x, t, u, I, p) − H(y, t, u, I, p)

∣∣ ≤ l(1+|u|+|I|+|p|

)|x − y|,

for all x, y, p ∈ Rn, t ∈ [0, T ],u, I ∈ R, and (1.11), (1.12) still hold;

H’.3) νk, ζk ∈ C(R

n×[0, T ])∩L∞(

0, T ; W 1,∞(Rn)), and there exists d′ > 0

such that∫Rn

(1 + en|z|

)|ν(x, t; z) − ν(y, t; z)| dz ≤ d′|x − y|.

Then, the viscosity solution u to the problem (1.9)–(1.2) is locally Lipschitzcontinuous with respect to x, namely e−n|x|u(x, t) ∈ L∞(

0, T ; W 1,∞(Rn)).

2. The financial model

The motivation for our study relies in the problem of derivatives pricingin jump–diffusion markets. This contest, which was already considered byMerton [20] in 1976, allows to account for abrupt price movements caused byexogenous events on information, and so to reduce the discrepancies betweenthe standard Black & Scholes model and empirical evidence. In addition tothe small variation of the trend which may be modeled by a continuousprocess based on Brownian motion, sudden and rare breaks are taken intoaccount: they are described by means of some point process that counts theoccurrences of rare and random events.

In this section, we describe a market model where a finite number of riskyassets are traded in continuous time, evolving according to jump–diffusionprocesses that possibly are correlated. The presence of a large investor isallowed, actually it is assumed that the interest rate of bank deposits dependson the invested wealth. The price of derivatives is characterized by means ofan integro–differential parabolic equation, which is semilinear (of type (1.9))as a consequence of the large investor assumption. Besides it is stronglydegenerate if the diffusion matrix driving the underlying assets is degenerate:it is always the case, for instance, if the market is complete (see later onRemark 2.3).

Let (Ω,F ,P) be a given probability space, endowed with a filtration (Ft)t.Take then a d–dimensional Brownian motion Wt = (W 1

t ,· · ·, W dt )T and an

m–dimensional Poisson process Nt = (N1t ,· · ·, Nm

t )T , with non-homogeneousintensities λ1(t), · · · , λm(t) fulfilling

λk ∈ C[0, T ], λk > 0. (2.1)

792 Anna Lisa Amadori

We assume that the processes Wt and Nt are uncorrelated.Next we consider in this probability setting a market composed by n risky

assets, whose prices will be denoted by S = (S1, · · · , Sn)T , and a moneymarket account, whose value will be denoted by B.

We assume that the vector of risky assets prices evolves according to ajump–diffusion process in R

n+, described by the stochastic differential equa-

tiondSt = St− [αdt + σdWt + γdNt] , (2.2)

where S = diag (S1, · · · , Sn), α = (α1, · · · , αn)T is the drift vector, σ =(σij) i = 1,· · ·, n

j = 1,· · ·, dis the diffusion matrix, and γ = (γik) i = 1,· · ·, n

k = 1,· · ·, mis the jump

matrix. The coefficients αj , σij , γik are assumed to be deterministic and tosatisfy

A.1) αj , σij , γik are continuous functions of S and t on Rn+× [0, T ], with

the regularity property

f ∈ L∞(0, T ; W 1,∞(Rn

+)), S ·Df ∈ L∞(Rn

+× [0, T ]), (2.3)

for f equal to αj , σij , and γik. Moreover, inf γik > −1.

Assumption A.1 leads a wide extension to previously studied jump–diffusionmodels like the one in [16], where all coefficients are assumed to be constant.Notice that the first part of property (2.3) is just needed to guarantee thewell–posedness of (2.2) (see, for instance, [12]). With regard to the secondpart, it is a technical request needed for Theorem 2.8. It is worth pointingout that (2.3) does not bring any restriction to the application to “real”problems. Actually in practice all the coefficients are calibrated on a discretegrid of statistical data, so that one may obtain all the needed regularity bysuitable extending them to the continuum.

The value of the bank deposit evolves according to the differential equation

dB(t) = rB(t)dt, (2.4)

where r stands for the interest rate. We assume that r depends, beneath onthe stocks price S and on the time t, on the wealth invested in bonds, whichwill be denoted by ξ. Usually r is a non-increasing function of ξ; neverthelesswe shall suppose that r(S, t, ξ)ξ is non-decreasing with respect to ξ (as S, tfixed). If not, the agents where boosted to draw back their deposits fromthe bank. In addition, we do not impose that r is continuous with respect toξ at zero, in order to include interesting examples as well as 2.1. Thereforewe assume that

nonlinear integro-differential equations 793

A.2) r is a non–negative continuous function of S, t, ξ on Rn+× [0, T ] ×

R\0. For all ξ ∈ R \ 0 the function (S, t) → r(S, t, ξ) satisfies(2.3), uniformly with respect to ξ, and for all (S, t) ∈ R

n+× [0, T ], the

function ξ → r(S, t, ξ)ξ is non–decreasing and Lipschitz continuouson R, uniformly with respect to (S, t).

This large investor model describes situations not included in the standardtheory.

Example 2.1 (Different interest rates for borrowing or lending).

r(S, t, ξ) =

R(S, t) if ξ ≤ 0, R borrowing rate,ρ(S, t) if ξ > 0, ρ lending rate;

where R, ρ are bounded smooth functions and R(S, t) > ρ(S, t).

Example 2.2 (Large institutional investor). The interest rate decreaseswhen too many wealth is invested in bonds, according to the law r(S, t, ξ) =R(S, t) f(ξ), where R is bounded and smooth, and f is a positive Lipschitzcontinuous function on R, such that f(ξ) = 1 if ξ ≤ ξo, −f(ξ)ξ ≤ f ′(ξ) ≤ 0almost everywhere in (ξo,∞).

Large investor models of this kind have been studied by Cvitanic and Ma[7], assuming a pure diffusion process for assets’ price and rankσ = n = d(equivalently, that the market is complete), in the framework of forward–backward stochastic differential equations. Existence of solution to the sto-chastic equation characterizing the price of derivatives is attained throughexistence and regularity of the solution to an associate partial differentialequation. In the non–linear setting caused by the large investor assump-tion, the PDE’s approach is nowadays the only way to price claims, to theauthor’s knowledge.

We recall (see, for instance, [17, Theoremes 2.6, 3.4]) that the marketcomposed by St and Bt is without arbitrage opportunities (respectively,complete) if and only if there exists a probability P∗ (respectively, a uniqueprobability P∗ ), equivalent to the given one, such that the processes Si

t/Bt

are martingales under P∗. In force of the characterization [3, VIII T10] (seealso [21]), the market is without arbitrage opportunities if and only if

there exist θ ∈ Rd and φ ∈ R

m+ such that

α − r1 = (σ, γ)(

θ−λφ

),

(2.5)

where 1 stands for the vector of Rn (1, · · · , 1)T and λ = diag (λ1, · · · , λm).

794 Anna Lisa Amadori

Moreover, the couple θ, φ satisfying (2.5) (usually named market pricefor risks) is in one–to–one correspondence with an equivalent martingalemeasure P∗. Therefore, the market is complete if and only the market pricefor risks is unique.

Remark 2.3. Notice that, in the jump–diffusion setting, if rankσ = n = d,the market is not complete, because for any choice of φ ∈ R

m+, there exists

θ = σ−1 (α − r1 + γλφ) satisfying (2.5).Instead a sufficient condition for completeness is

rank (σ, γ) = n = d + m, (2.6)

which has already been used in [16]. We emphasize that, in this case,rankσ ≤ d < n.

Furthermore, if θ1, θ2 ∈ Rd and φ1, φ2 ∈ R

m (only depending by S and t)are defined by(

θ1

−λφ1

)= (σ, γ)−1α,

(θ2

−λφ2

)= −(σ, γ)−11, (2.7)

the non–arbitrage condition (2.5) becomes φ1 + rφ2 ∈ Rm+. Since in the

large investor case r also depends by the quantity ξ (that is decided by theinvestor), we are forced to strengthen it by asking that

φ1, φ2 ∈ Rm+. (2.8)

In Mathematical Finance completeness is a crucial point, because it al-lows to perfectly hedge the risk endowed in any contingent claim and, as aconsequence, to price them unambiguously. It is worth mentioning that, inthe jump–diffusion setting, one may reduce to a market which satisfies con-dition (2.6), and therefore which is complete, even if dealing with a claim onsome underlying assets S1, · · · , Sn which do not compose a complete mar-ket. This practice, often referred to as “completion of the market”, standsin adding to the natural market composed by S1, · · · , Sn, some new assetsSn+1, · · · , Sd+m until condition (2.6) is fulfilled. It is not a mere theoreticaltrick, since in most applications the role of Sn+1, · · · , Sd+m is played by somecall options on the former assets S1, · · · , Sn which are really marketed; see[23], [18] for more details about this topic.

By now we suppose that the market composed by S and B fulfills (2.6)–(2.7) and we take an European contingent claim with maturity T and payoffG(ST ). We recall that a hedging strategy is a dynamic portfolio whichinvests the wealth ∆t = (∆1

t , · · · , ∆nt ) in the risky stokes and ξt in the bank,

nonlinear integro-differential equations 795

in such a way that

d (∆t1 + ξt) = ∆tS−1t dSt + ξtB

−1t dBt, for all t ∈ (0, T ),

∆T1 + ξT = G(ST ).

By making use of the generalized Ito Lemma (see, for instance, [8, TheoremVIII.27]), the well known ∆–hedging technique (illustrated in [9, Paragraph3.5]) characterizes the replicating strategy in the given probability P as

∆t =(DUSσ, IU

)(σ, γ)−1

ξt = U − Sσθ2 · DU − λφ2 · IU,(2.9)

where U(St, t) is the arbitrage price of the claim at time t < T . Moreover,the deterministic function U solves the modified “Black & Scholes equation”

−∂tU =12tr

[(Sσ)(Sσ)T D2U

]+ S(α − σθ1)·DU + λφ1 ·IU

−r(S, t, U − Sσθ2 ·DU − λφ2 ·IU

)×

(U − Sσθ2 ·DU − λφ2 ·IU

) (2.10)

on (0,∞)n× [0, T ) and satisfies the final condition

U(S, T ) = G(S). (2.11)

Here DU = (∂S1U, · · · , ∂SnU)T is the gradient of U , D2U =(∂2

SiSjU)

isthe Hessian matrix of U , and the nonlocal term IU takes into account theincrements of U due to the jumps of the underlying assets, being

IkU = U(S + Sγk, t) − U(S, t)

(γk standing for the kth row of γ) for k = 1, · · ·m.

Remark 2.4. It is worth noticing that the dependence of equation (2.10) bythe drift vector α is merely seeming, because by making use of the relation(2.7) one gets that α − σθ1 = −γλφ1.

Remark 2.5. Notice that r gives rise to a nonlinear, nonlocal term, becauseof the large investor assumption. Actually r depends on the wealth investedin bonds, i.e., on the process ξt characterized by (2.9).

The equation (2.10) has the same boundary degeneracy of the classicalBlack & Scholes equation, that can be removed by the change of variablesxi = log Si. Now U solves (2.10)–(2.11) if and only if

u(x, t) = U (ex1 , · · · , exn , T − t)

796 Anna Lisa Amadori

solves a semilinear integro–differential Cauchy problem of the type (1.9)–(1.2). Still writing α, σ, γ, θ, φ for the economical parameters composed withthe change of variables, we have

LIu = −12tr

[σσT D2u

]+ b · Du − I1u

with bi = 12σ2

ii − (α − σθ1)i, and

H(x, t, u, I2u, Du) = r(ex1,· · ·, exn, T− t, u − σθ2 · Du − I2u

)×

(u − σθ2 · Du − I2u

).

As = 1, 2 I is a nonlocal term of type (1.3) where µx,t is the discrete

bounded measure

µx,t(z) =

m∑k=1

λkφk δ(ζk(x, t) − z).

Here ζk stands for the vector of Rn (log(1 + γ1k), · · · , log(1 + γnk)), and δ

stands for the Dirac measure.

Remark 2.6. By making use of the relations (2.7), it is possible to rewritethe equation with integral terms of type (1.7) (according to the same mea-sures µ

x,t), by setting

LJ u = −12tr

[σσT D2u

]+ b · Du − J 1u,

H(x, t, u,J 2u, Du) = r(ex1,· · ·, exn, T− t, u − div u − J 2u

)×

(u − div u − J 2u

),

where b1 = 12σ2

ii.

Since in the new variables uo(x) = G(ex1,· · ·, exn), the study of the for-mer problem (2.10)–(2.11) with payoff G growing polynomially at infinity isequivalent to the study of the Cauchy problem (1.9)–(1.2) with initial datumuo growing exponentially at infinity.

In principle, one may directly study the problem (2.10)–(2.11), for whichpolynomial growth is sufficient; but in this case one has to face the difficul-ties related to the boundary degeneracy in ∂R

n+ (notice that this boundary

is not of class W 3,∞, which is a standard assumption for the viscosity so-lutions approach) and of the rate of growth of the coefficients of the linearpart (which may play a role when checking Lipschitz continuity). Since thetwo problems are equivalent, we have chosen here to study the one withoutboundary, with bounded coefficient, but with faster increasing datum.

nonlinear integro-differential equations 797

We emphasize that the obtained equation of type (1.9) is always of stronglydegenerate type (i.e., rankσ < n everywhere) in force of the completenesscondition (2.6).

In the linear case corresponding to a small investor, integro–differentialequations arising in finance have been studied by Pham (see [21] and thereferences therein) for n = d = 1, and by Mastroeni and Matzeu (see [19]and the references therein) for n = d arbitrary, under the non-degeneracyassumption rankσ = n. Mastroeni and Matzeu studied also a semilinearcase, allowing nonlinearities with respect to DU (but not with respect toIU). Their variational technique deals with weak degeneracy: namely, σσT

is positive, defined at all (S, t), except a set of zero measure in which itvanishes. Hence it may not be applied to problems of type (1.sl)–(2), whenthe completeness condition (2.6) holds true.

Let us state the main results that may be obtained as easy consequencesof Theorems 1.1 and 1.2, respectively.

Theorem 2.7 (Well posedness). We assume that the market parameterssatisfy A.1, A.2, and that (2.6)–(2.8) hold true with inf det(σ, γ) > 0. Thenthe final value problem (2.10)–(2.11) is well posed in the class of continuousfunctions with polynomial growth for large |S|. In particular, for all payoffsG ∈ C

(R

n+

)with 0 ≤ G(S) ≤ bo(1 + |S|no), the viscosity solution satisfies

0 ≤ U(S, t) ≤ b(1 + |S|no).

Theorem 2.8 (First order estimate). Under the same assumptions of The-orem 2.7, and for all payoffs G ∈ W 1,∞

loc

(R

n+

)with

|G(S)| + |DG(S)| ≤ bo(1 + |S|n),

the viscosity solution of (2.10)–(2.11) is locally Lipschitz continuous withrespect to S, namely

|U(S, t)| + |DU(S, t)| ≤ b(1 + |S|n)

for all t ∈ [0, T ].

The regularity property stated by Theorem 2.8 is quite relevant. Indeed,the trader expects that, in order to hedge from the risk of his contingentclaim, he need to place a quantity of money which grows up exactly inthe same way of the expected payoff, as the prices S increase. Since thegradient of U is connected with the hedging portfolio by the relation (2.9),such property is just established by Theorem 2.8.

798 Anna Lisa Amadori

3. Existence and uniqueness of viscosity solution

In this section, we show that the Perron’s method, introduced by Ishii [13]for viscosity solutions, can be used also for the integro–differential problem(1.1)–(1.2); then, by a comparison principle, we obtain existence and unique-ness of a continuous solution. We refer to [4] for a general presentation ofthe theory of viscosity solutions in the pure differential case and to Alvarezand Tourin [1] for a simpler nonlocal situation.

We begin by defining viscosity solutions for integro–differential problemswith non local term Iu of type (1.3), under the structural assumptions (1.4),(1.5).

At this step we may deal with the wide class of integro–differential prob-lems (1.1), (1.2) under the assumptions F.0–F.3.

In order to define viscosity solutions to nonlocal equations, it is needed toface two main difficulties: the first one is that the meaning of “viscosity in-equality” changes by using the local notion of semijets, or the global one; thesecond one is that, in general, the integral term I does not preserves semi-continuity. In force of assumption F.3, they may be overcame by following[1].

Definition 3.1. Given a function u and (x, t) ∈ Rn × [0, T ), we shall say

that ∂tu+F (x, t, u, Iu, Du, D2u) ≤ 0 (respectively, ≥ 0) in viscosity senseat (x, t) if one of the following equivalent items holds:

(1) τ + F (x, t, u(x, t), Iu(x, t), p, X) ≤ 0 (resp., ≥ 0) for all (τ, p, X) inP+u(x, t) (respectively, P−u(x, t));

(2) ∂tϕ(x, t) + F(x, t, u(x, t), Iu(x, t), Dϕ(x, t), D2ϕ(x, t)

)≤ 0 (resp.,

≥ 0) in classical sense, for each smooth function ϕ on Rn× [0, T )

such that u − ϕ has a local maximum (resp., a minimum) at (x, t);(3) ∂tϕ + F

(x, t, ϕ, Iϕ, Dϕ, D2ϕ

)≤ 0 (resp., ≥ 0) in classical sense, for

all smooth functions ϕ on Rn× [0, T ) such that u − ϕ has a global

strict maximum (resp., a minimum) at (x, t).

We recall that the parabolic semijets P± are defined as

P±u(x, t) =

(τ, p, X) ∈ R × Rn × Sn : u(y, s) ≤ ( resp. ≥)

u(x, t) + τ(t − s) + p · (x − y) +12X(x − y) · (x − y)

+o(|t − s| + |x − y|2

)as (y, s) → (x, t)

,

Next we define sub/supersolutions to the problem (1.1)–(1.2) as follows.

nonlinear integro-differential equations 799

Definition 3.2. An upper (resp., lower) semicontinuous, locally boundedfunction u on R

n× [0, T ) is a subsolution (resp., a supersolution) to theproblem (1.1)–(1.2) if ∂tu+F (x, t, u, Iu, Du, D2u) ≤ 0 (resp., ≥) in viscositysense at all (x, t) ∈ R

n× (0, T ], and u(x, 0) ≤ uo(x) (resp., ≥) at all x ∈ Rn.

A solution is any function which is is both a sub/supersolution.

It is worst mentioning that, in general, Iu is not upper (respectively,lower) semicontinuous for any subsolution (respectively, supersolution). Onthe other hand [1, Lemma 1] establishes that Iu is upper (resp., lower)semicontinuous at a point (x, t) ∈ R

n× [0, T ] if u satisfies the additionalproperty

there exists Φ ∈ Bx,t such that for all y, s near at x, t and z ∈ Rn

M(u(y + z, s), u(y, s)) ≤ Φ(z) (resp., ≥ −Φ(z)). (3.1)

Here the set Bx,t is defined by

Bx,t =

Φ ∈ C(R

n)∩ L1

(R

n; µx,t

): Φ ≥ 0,

lim(y,s)→(x,t)

∫Rn

Φ(z) dµy,s(z) =∫

RnΦ(z) dµx,t(z)

.

Notice that, since µx,t are finite measures, such property is implied by asuitable rate of growth at infinity. Actually, the proof of next result isimmediate.

Lemma 3.1. Assume that there exists a subadditive, nonnegative g∈ C[0,∞)such that the function (x, t) →

∫[1 + g(|z|)] dµx,t(z) is continuous on R

n ×[0, T ). Then all upper (resp., lower) semicontinuous functions such thatu(x, t)/g(|x|) is bounded satisfy (3.1).

In particular, if µx,t has compact support for any fixed x, t, all semicon-tinuous and locally bounded functions satisfy (3.1).

It is useful to observe that the initial condition can be relaxed, in the samespirit of [2, Proposition 5] for the purely differential case. In the integro-differential case, the relaxation of the initial condition has to be balanced byimposing (3.1) at t = 0.

Definition 3.3. u is a subsolution (respectively, a supersolution) of (1.1)–(1.2) with generalized initial condition, GIC, if the initial conditionu(x, 0) ≤ uo(x) (respectively, ≥) is replaced by

min∂tu + F (x, 0, u, Iu, Du, D2u), u − uo

≤ 0

(resp., max∂tu + F (x, 0, u, Iu, Du, D2u), u − uo

≥ 0 )

800 Anna Lisa Amadori

in viscosity sense and property (3.1) is satisfied at t = 0.

In the next Lemma we show that a sub/supersolution with generalizedinitial condition is a sub/supersolution in the sense of Definition 3.2, indeed.

Lemma 3.2. If u is a subsolution (resp., a supersolution) to (1.1)–(1.2) withGIC, then u(x, 0) ≤ uo(x) (resp. ≥) for all x ∈ R

n.

Proof. Assume by contradiction that u(xo, 0) > uo(xo); by continuity,u(xo, 0) > uo(x) for x near xo. Let V be the neighborhood of (xo, 0) in whichproperty (3.1) is fulfilled and u(xo, 0) > uo(x), in addition. We consider thefamily of test functions

ϕn(x, t) =n

2|x−xo|2 + cnt,

where cn > max(n,− inf

((x,t),v)∈(V,u(V ))F

(x, t, v,

∫Φµx,t, n(x−xo), nI

)).

By upper semicontinuity, u − ϕn has a maximum point (xn, tn) in thecompact set V and, up to an extracted sequence, we may assume that (xn, tn)converges to (x, t) ∈ V . Because

u(xo, 0) ≤ limn→∞

[u(xn, tn)−ϕn(xn, tn)] ≤ u(x, t)− limn→∞

n(1

2|xn−xo|2 + tn

),

we have that (x, t) = (xo, 0). Therefore, for large n, (xn, tn) ∈ V is a localmaximum point for u − ϕn on R

n× [0, T ] and Definition 3.1.2 gives

cn ≤ −F (xn, tn, u(xn, tn), Iu(xn, tn), n(xn−xo), nI).

Next, since Iu(xn, tn) ≤∫

Φdµxntn by (3.1), hypothesis F.3 implies thecontradiction

cn ≤ −F(xn, tn, u(xn, tn),

∫Φdµxntn , n(xn−xo), nI

).

3.1. Perron’s method. In [1], Alvarez and Tourin showed that Perron’smethod can be used for integro–differential Cauchy problem of the kind(1.8)–(1.2), under assumptions F.0–F.2. Here we show that the monotonicityassumption F.3 is sufficient to apply Perron’s method to the wider class(1.1)–(1.2).

Theorem 3.3 (Perron Method). Assume that h, k ∈ C(Rn× [0, T ]) are re-spectively a viscosity sub/supersolution of (1.1)–(1.2) such that:

(i) h ≤ k,

nonlinear integro-differential equations 801

(ii) for every (x, t) ∈ Rn× [0, T ], there exist Φ ∈ Bx,t and a neighborhood

V of (x, t) such that

−Φ(z) ≤ M (h(y + z, s), w) ≤ M (k(y + z, s), w) ≤ Φ(z)

for all (y, s) ∈ V , z ∈ Rn, and w in the segment line [h(y, s), k(y, s)].

We denote by S the set of subsolutions v of (1.1)–(1.2) such that h ≤ v ≤ kon R

n× [0, T ], byu(x, t) = sup

v(x, t) : v ∈ S

, (3.2)

and by u∗, u∗ the upper and lower semicontinuous envelope of u, respectively.Then u∗ and u∗ are respectively a sub/supersolution to (1.1)–(1.2).

Proof. We first observe that, since h and k are continuous and M is non–decreasing with respect to its first argument, hypothesis (ii) guarantees thatu∗ and u∗ satisfy the property (3.1) at all points of R

n× [0, T ].Therefore, thanks to Lemma 3.2, we may prove that u∗ is a subsolution

to (1.1)–(1.2) by checking that it is a subsolution with GIC. Take (x, t) ∈R

n× [0, T ], with u∗(x, 0) > uo(x) if t = 0 (otherwise there is nothing toprove): our goal is to show that ∂tϕ + F

(x, t, u∗, Iu∗, Dϕ, D2ϕ

)≤ 0 for

every smooth function ϕ that u∗ − ϕ has a strict maximum at (x, t). Bystandard arguments, one may find two sequences vn ∈ S and (xn, tn) ∈R

n×[0, T ] such that (xn, tn, v(xn, tn)) converges to (x, t, u∗(x, t)), and (xn, tn)are points of local maximum for vn − ϕ. Up to an extracted sequence, wemay suppose that tn > 0 for all n: otherwise, we should have u∗(x, 0) =lim vn(xn, 0) ≤ lim uo(xn) = uo(x) because of the continuity of uo. Thus∂tϕ + F

(xn, tn, vn, Ivn, Dϕ, D2ϕ

)≤ 0. By passing to the limit as n goes to

infinity and by taking advantage of assumption F.3, we have

0 ≥ lim infn→∞

[∂tϕ(xn, tn) + F

(xn, tn, vn, Ivn, Dϕ, D2ϕ

)]= ∂tϕ(x, t) + F

(x, t, u∗, lim sup

n→∞Ivn(xn, tn), Dϕ, D2ϕ

).

Hence, the proof may be completed by showing that

lim supn→∞

Ivn(xn, tn) ≤ Iu∗(x, t),

but this last inequality is an easy consequence of [1, Lemma 1] and hypothesis(ii).

Next, we shall prove that u∗ is a supersolution to (1.1)–(1.2). By Lemma3.2, it is sufficient to show that it is a supersolution with GIC. Supposeby contradiction that u∗ is not a supersolution with GIC; then there exist

802 Anna Lisa Amadori

(xo, to) ∈ Rn×[0, T ] (with u∗(xo, 0) < uo(xo) if to = 0) and a smooth function

ϕ such that (u∗ − ϕ) (xo, to) = 0 is a global strict minimum and

∂tϕ + F(xo, to, ϕ, Iϕ, Dϕ, D2ϕ

)< 0.

We carry out the proof by constructing a subsolution in S strictly biggerthan u∗ at one point. We begin by setting

Br =(x, t) ∈ R

n× [0, T ) : |x − xo|2 + (t − to)2 < r

,

ϕδ(x, t) = max (ϕ(x, t) + δ, h(x, t)) .

Notice that, by construction, ϕδ ≡ ϕ(x, t) + δ near at (xo, to). So, the samearguments of [1, Proposition 1] yield that, for suitable δ and r > 0, ϕδ ≤ kand Iϕδ is lower semicontinuous near at (xo, to). Therefore F.3 implies that

∂tϕ + F(x, t, ϕδ, Iϕδ, Dϕ, D2ϕ

)≤ 0, on Br,

after possibly choosing a smaller r. Moreover, as u∗−ϕ has a strict minimumat (xo, to) and u∗ ≥ h, we may suppose without loss of generality thatu∗ ≥ ϕδ on R

n× [0, T ] \ B r2. Next we set

w(x, t) =

u∗(x, t), (x, t) ∈ Rn× [0, T ] \ Br,

max(u∗(x, t) , ϕδ(x, t)

), (x, t) ∈ Br.

Hypothesis (ii) guarantees that w satisfies property (3.1), so that it easilyfollows by construction that w is a subsolution to (1.1)–(1.2) with generalizedinitial condition. On the other hand, standard semicontinuity argumentsshow that there is at least one point near (xo, to) where w > u∗.

In general, the function u produced in (3.2) does not have any regular-ity property, but, whenever its continuity is achieved, then u is a viscositysolution. Actually, the only availability of a comparison principle amongsub/supersolutions of (1.1)–(1.2) is sufficient to get the existence of one so-lution, via the Perron’s method.

3.2. The Comparison Principle. It is well understood that, in order toestablish a comparison principle for the Cauchy problem, it is necessary toselect a rate of growth at infinity for sub/supersolutions. In view of thefinancial applications, we are interested in dealing with exponential growth,namely we shall assume that the initial datum satisfies, for a suitable no,

(H.0) e−no|x|uo(x) ∈ C(Rn) ∩ L∞(Rn).

As a consequence, a mere extension to the integro–differential setting ofthe classical results is not sufficient, because they only allow linear growth

nonlinear integro-differential equations 803

(see, for instance [5], [11], or [14]). However, the equation of financial inter-est has some nice properties: first of all, the coefficients of the linear partrespect the growth conditions of the classical Phragmen-Lindelof principle(see for instance [22, Theorem 10]); the nonlinear term is globally Lipschitz–continuous, and finally the nonlocal term I has a natural growth restrictionby virtue of the boundedness of the economical parameters. Therefore, wefocus here our interest into the semilinear integro–differential equation

(1.9) ∂tu + LIu + H(x, t, u, Iu, Du) = 0, (x, t) ∈ Rn× (0, T ),

assuming that H.1–H.3 hold.Let us mention that assumptions H.1 and H.2-(1.10) are standard (see, for

instance, [15, 4]). In addition, the Lipschitz–continuity property H.2-(1.11)and the condition on the rate of growth of the source term H.2-(1.12) areimposed in order to deal with faster increasing solutions, in the spirit of [6].

With respect to hypothesis H.3, we stress that some compatibility amongthe space L1

(R

n; µx,t

)and the rate of growth at infinity of the initial datum

and of the source term is necessary, in view of uniqueness of solutions, alsoin the linear case. We refer to [1, Remark 4], which shows lack of uniquenessin the class of bounded functions, when the constant function 1 does notbelong to L1

(R

n; µx,t

).

The main result of this subsection is the following.

Theorem 3.4 (Comparison Principle). Let u and u, satisfying

e−n|x|u, e−n|x|u ∈ L∞(0, T ; L∞(Rn)

), (3.3)

be respectively sub/supersolution for the Cauchy problem (1.9)–(1.2).Then u ≤ u in R

n× [0, T ].

Notice that the double sided estimate (3.3) is imposed to the rate of growthof sub/supersolutions, while a one sided estimate is sufficient in the purelydifferential setting. Indeed, the additional estimate is related to the request(3.1), which is needed in order to estimate the integral term Iu. Actuallyhypothesis H.3 and the growth condition (3.3) imply that u and u satisfy(3.1), via Lemma 3.1.

The idea for handling faster increasing sub/supersolutions is the same osthe classical Phragmen–Lindelof principle, namely to multiply u and u for asuitable weight function in such a way that their difference is bounded fromabove and take maximum in R

n× [0, T ]. As we shall show during the proof,this technique may be extended to nonlinear equations under assumptionH.2-(1.11)

804 Anna Lisa Amadori

Before entering into details of the proof, we recall for the sake of complete-ness a Lemma previously used by Ishii and Kobayasi [14], which establishesthe well known Osgood condition in the framework of viscosity solutions.

Lemma 3.5 ([14, Theorem 3, page 918]). Let g ∈ C([0, T ]) be such that

g(0) = 0,

∫ 1

0

dr

g(r)= +∞.

Let then f be an upper semicontinuous function on [0, T ), nonnegative,bounded from above and suppose that

min(f ′(t) − g(f(t)) , f(t)

)≤ 0, for all t ∈ [0, T ), (3.4)

in viscosity sense, then f ≡ 0.

Proof of Theorem 3.4. Without loss of generality, we may replace thenorm |x| with (x) =

√1 + |x|2, with the advantage that is of class

C2. By hypothesis v(x, t) = e−n(x)u(x, t) and v(y, t) = e−n(y)u(y, t) arebounded. Besides, v and v are respectively a sub/supersolution for themodified integro–differential Cauchy problem

(1.9) ∂tv + LIv + H

(x, t, v, Iv, Dv

)= 0, (x, t) ∈ R

n×(0, T ],

(1.2) v(x, 0) = e−n(x)uo(x) ∈ C(Rn) ∩ L∞(Rn), x ∈ Rn.

Here, we have used the notations

LIv = −1

2tr

[σσT D2v

]+ b · Dv + cv − Iv,

where b = b− 12n DσσT and c = c+nb ·D− 1

2n tr[σσT D2

]− 1

2n2 |Dσ|2,

H(x, t, v, Iv, Dv

)= e−n(x)H

(x, t, en(x)

(v, Iv, nvD(x) + Dv

)),

Iv(x, t) = e−n(x)

∫Rn

Mv(x, t, z) µxt(dz),

Mv(x, t, z) = e−n(x)M(en(x+z)v(x+z, t), en(x)v(x, t)

).

Notice that the coefficients of LI still satisfies H.1, and that the new Hamil-

tonian H still fulfills (1.10) (for a new modulus of continuity ω) and (1.11);in addition the source term H(·, 0, 0, 0) is bounded. Theorem 3.4 is equiv-alent to the comparison principle among bounded sub/supersolutions to(1.9)–(1.2). We set θ(t) = sup

(v(x, t) − v(x, t))+ : x ∈ R

n, and we

nonlinear integro-differential equations 805

prove the thesis by checking that θ ≡ 0. By invoking Lemma 3.5, it sufficesto show that its upper semicontinuous envelope θ∗ fulfills (3.4) for

g(θ∗) =(− inf c + md + (1 + md)l′

)+θ∗,

where d =∥∥∫ (

1 + en|z|) dµx,t(z)∥∥∞.

To this aim we fix to ∈ [0, T ], with θ∗(to) > 0 (if not, there is nothing toprove) and we take a smooth function ϕ(t) such that θ∗(to) − ϕ(to) = 0 is astrict global maximum.

For any fixed δ > 0, the function

Ψδ(x, t) = v(x, t) − v(x, t) − δ

2|x|2 − ϕ(t)

has a global maximum point (xδ, tδ) on Rn×[0, T ]. We claim that it is possible

to choose an infinitesimal sequence of parameters (that we still denote by δ)in such a way that

δ

2|xδ|2 → 0, tδ → to, v(xδ, tδ) − v(xδ, tδ) → θ∗(to), (3.5)

as δ → 0. As a consequence

θ(tδ) → θ∗(to) as δ → 0. (3.6)

Moreover, we may suppose that tδ > 0 for all δ, even if to = 0: otherwise weshould have 0 < θ∗(to) = limδ→0 [v(xδ, 0) − v(xδ, 0)] ≤ 0 in force of (1.2).

Afterward we double variables by considering (xαδ, yαδ, tαδ) a maximumpoint for the function

Ψαδ(x, y, t) = v(x, t) − v(y, t) − α

2|x − y|2 − δ

2|x|2 − ϕ(t)

on Rn× R

n× [0, T ]. It is an easy exercise to check that, for any fixed δ,α

2|xαδ − yαδ|2 → 0, tαδ → tδ,

v(xαδ, tδ) − v(yαδ, tαδ) → v(xδ, tδ) − v(xδ, tδ),(3.7)

as α → ∞. Hence, in view of (3.5), we are allowed to assume that |xαδ|,|yαδ| ≤ R(δ), tαδ > 0, v(xαδ, tδ) − v(yαδ, tαδ) > 0 for large α.

We defer to check (3.5) and (3.7) later on. For the time being, we wantto take advantage from the fact that v and v are sub/supersolutions ofthe integro–differential equation (1.9) to achieve an estimate from above ofϕ′(to). To this end, we apply [4, Theorem 8.3] with ε = 1/α to v1(x, t) =v(x, t)− δ

2 |x|2, v2(y, t) = −v(y, t), φ(x, y, t) = α2 |x−y|2 +ϕ(t). It yields that

806 Anna Lisa Amadori

there are τ1, τ2 ∈ R and two symmetrical matrices X, Y (depending on α, δ)such that

τ1 − τ2 = ϕ′(tαδ),(

X 00 −Y

)≤ 3α

(I −I−I I

),

andτ1 + F

(x, t, v, Iv, α(x−y)+δx, X+δI

)≤ 0,

τ2 + F (y, t, v, Iv, α(x−y), Y

)≥ 0,

where we have omitted the dependence on α, δ and we have used the abbre-viated notation

F (x, t, v, I, p, X) = −12tr

[σσT X

]+ b · p + cv − I + H(x, t, v, I, p) .

Subtracting this two inequalities yields

ϕ′(t) ≤ F (y, t, v, Iv, α(x−y), Y

)− F

(x, t, v, Iv, α(x−y)+δx, X+δI

).

By making use of (3.7) and repeating the computations of [4, Example 3.6],[15, Theorem II.1], passing to the limit as α → ∞ yields

ϕ′(tδ) ≤ − inf cθ(tδ) + lim supα→∞

∣∣∣Iv(xαδ, tαδ) − Iv(xαδ, tαδ)∣∣∣

+ lim supα→∞

ωR(δ)

((1 + ‖v‖∞ +

∥∥∥Iv∥∥∥∞

+ α|xαδ−yαδ|)|xαδ−yαδ|

)

+l′(θ(tδ) + lim sup

α→∞

∣∣∣Iv(xαδ, tαδ) − Iv(xαδ, tαδ)∣∣∣).

Since v and Iv are bounded by construction (recalling H.3), the third termon the right–hand side is zero, via (3.7). Moreover, thanks to hypothesis H.3,Iv, Iv are continuous, via Lemma 3.1 and [1, Lemma 1]. In particular, forall fixed δ we have

lim supα→∞

∣∣∣Iv(xαδ, tαδ) − Iv(xαδ, tαδ)∣∣∣

≤∫Rn

∣∣∣Mv(xδ, tδ; z) −Mv(xδ, tδ; z)∣∣∣ dµxδ ,tδ(dz) ≤ mdθ(tδ).

Summing up, we have obtained ϕ′(tδ) ≤ g(θ(tδ)).Eventually passing to the limit as δ → 0 and taking advantage of (3.6)

yields the desired inequality ϕ′(to) ≤ g(θ∗(to)), because ϕ′ and g are contin-uous.

Lastly we sketch how (3.5) and (3.7) can be checked, for the sake ofcompleteness.

nonlinear integro-differential equations 807

With respect to (3.5), we set Mδ = max ψδ = ψδ(xδ, tδ) . By constructionMδ ≤ 0. Besides, by definition there is a sequence (xn, tn) such that tn → t0and v(xn, tn)− v(xn, tn) → θ∗(t0). Hence by choosing δn in such a way thatδn2 |xn|2 → 0 we obtain Mδn → 0 and therefore δn

2 |xδn |2 → 0 and

limn→0

[v(xδn , tδn) − v(xδn , tδn)] = limn→0

ϕ(tδn). (3.8)

Up to a subsequence, we may suppose that tδn converges to some to ∈ [0, T ].So (3.8) implies θ∗(to) ≥ ϕ(to) and therefore to = to, because to is a strictmaximum point for θ∗−ϕ, . Lastly, recalling that ϕ is continuous, the same(3.8) yields that v(xδn , tδn) − v(xδn , tδn) → ϕ(to) = θ∗(to), so that (3.5) isestablished.

In order to check (3.7) we set Nαδ = max Ψαδ(x, y, t) = Ψαδ(xαδ, yαδ, tαδ).By construction Nαδ ≥ Mδ for all α, so α

2 |xαδ − yαδ|2 is equibounded withrespect to α and then |xαδ − yαδ| → 0 as α → ∞. If by contradiction|xαδ| where not bounded with respect to α, then Nαδ → −∞. Therefore,up to a subsequence, we may suppose that (xαδ, yαδ, tαδ) converges to somepoint (xδ, xδ, tδ) as α → ∞. So standard semicontinuity arguments yieldthat α

2 |xαδ − yαδ|2 → 0 and Ψα,δ(xαδ, yαδ, tαδ) → Ψδ(xδ, tδ) = Mδ, up toan extracted sequence. In particular (xδ, tδ) is a maximum point for Ψδ, sothat we may suppose without loss of generality that it is equal to (xδ, tδ)considered before. Eventually, recalling that ϕ is continuous, we obtain thesecond and third items of (3.7).

Now we are ready to prove, as an immediate corollary of Theorems 3.3and 3.4, the well posedness result stated by Theorem 1.1.Proof of Theorem 1.1. It is an easy exercise to show that there are threepositive constants λ, a, b such that h(x, t) = −eλt

[at + bemax(no,n1)(x)

]and k(x, t) = eλt

[at + bemax(no,n1)(x)

]are respectively classical sub/super-

solution. So, in view of Lemma 3.1, we may apply Theorem 3.3 and we obtainthat u, defined in (3.2), is such that u∗ and u∗ are respectively a sub/super-solution. Next Theorem 3.4 yields that u∗ ≤ u∗ on R

n× [0, T ]. Therefore,since the inverse inequality holds by construction, u ∈ C(Rn× [0, T )) is aviscosity solution, indeed.

Uniqueness follows immediately from comparison principle 3.4. We conclude this section by applying our results to the pricing of deriva-

tives.

Proof of Theorem 2.7. The assumptions of Theorem 2.7 grant that therelated Cauchy problem (1.9)–(1.2) fulfills H.0–H.3; in particular, condition

808 Anna Lisa Amadori

H.3 holds for all n and the source term H(x, t, 0, 0, 0) is zero. Hence, The-orem 1.1 establishes the existence of a continuous viscosity solution u, withe−no|x|u(x, t) bounded, which is unique in the class

v ∈ C(R

n× (0, T ))

: |v(x, t)| = O(exp n|x|) as |x| → ∞, for any n

.

In terms of the economical variable S, we have established the well posednessof final value problem (2.10)–(2.11) in the class of continuous functions withpolynomial growth, that where stated in the first part of Theorem 2.7.

Moreover, if 0 ≤ G(S) ≤ bo(1 + |S|no), one may easily check that 0and eλt[at + beno(x+

1 ,··· ,x+n )] are respectively sub/supersolution for the re-

lated Cauchy problem (1.9)–(1.2). Hence, the comparison principle statedin Theorem 3.4 guarantees that 0 ≤ U(S, t) ≤ b1(1 + |S|no).

4. Regularity result

Lastly, we investigate the regularity of the solution with respect to x:we expect that the solution of an integro–differential Cauchy problem is assmooth as its initial datum. In the pure differential setting, a regularityresult with respect to the x–variable may be stated by refining the proofof comparison principle; actually, the solution of the Cauchy problem u(·, t)has the same kind of modulus of continuity of its initial datum (see [4] andthe references therein).

We now prove analogous result for the integro–differential problem (1.9)–(1.2), under the assumptions H’.0–H’.3.Proof of Theorem 1.2. We shall prove directly that

v = e−nu ∈ L∞(0, T ; W 1,∞(Rn)

),

by taking advantage from the fact that v is the viscosity solution of theCauchy problem (1.9)–(1.2) assigned in the proof of Theorem 3.4. Noticethat the coefficient of the linear operator L still satisfies H’.1 and thatH satisfies, beneath (1.11) (with |u − v| replaced by (1+n)|u − v|), anestimate of type (1.10’) for a new constant l (after replacing |u| by (1+n)|u|).In addition v is continuous and bounded by virtue of Theorem 1.1, andv(·, 0) = e−nuo ∈ W 1,∞(Rn) by hypothesis H’.0. Thus, the thesis may beachieved by finding a suitable λ such that the function

ϑ(t) = sup

v(x, t) − v(y, t) − boeλt|x − y| : x, y ∈ R

n

is equal to 0. Here bo = ‖D(e−nuo

)‖∞. By invoking Lemma 3.5, it is

sufficient to establish that its upper semicontinuous envelope ϑ∗ fulfills (3.4)

nonlinear integro-differential equations 809

for

g(ϑ∗) =((1+n)l′ − inf c + (l′ − 1)md

)+ϑ∗,

where

d =m∑

k=1

∥∥∥νk(1 + enζk

) ∥∥∥∞

+∥∥∥

∫ (1 + en|z|

)ν(·; z)dz

∥∥∥∞

.

To this purpose, we fix to ∈ [0, T ] such that ϑ∗(to) > 0 and a smooth functionϕ such that (ϑ∗ − ϕ)(to) = 0 is a strict maximum. It is straightforward tocheck that there exists a sequence (xδ, yδ, tδ) ∈ R

n× Rn× [0, T ] of points of

local maximum for the function

Rn× R

n× [0, T ] (x, y, t) → v(x, t) − v(y, t) − δ

2|x|2 − boe

λt|x − y| − ϕ(t),

such thatδ

2|xδ|2 → 0, tδ → to,

v(xδ, tδ) − v(yδ, tδ) − boeλtδ |xδ − yδ| → ϑ∗(to) > 0,

(4.1)

as δ → 0. Therefore we may suppose without loss of generality that xδ = yδ

and that tδ > 0 for all δ > 0, because v(xδ, 0)− v(yδ, 0)− bo|xδ − yδ| ≤ 0 byconstruction; moreover

ϑ(tδ) → ϑ∗(to), as δ → 0. (4.2)

On the other hand, by applying [4, Theorem 8.3] with ε = boeλtδ |xδ−yδ| > 0

to v1(x, t) = v(x, t)+ δ2 |x|2, v2(y, t) = −v(y, t), φ(x, y, t) = boe

λt|x−y|+ϕ(t),and proceeding as in the proof of Theorem 3.4, we have

ϕ′(t) + λboeλt|x−y| ≤ F

(y, t, v(y, t), Iv(y, t),boe

λt x−y

|x−y| , Y)

−F (x, t, v(x, t), Iv(x, t),boe

λt x−y

|x−y| + δx, X + δI),

(4.3)

where we have omitted the dependence from δ for simplicity of notations.With respect to the nonlocal term Iv, we notice that, since |v(ξ, t) −

v(η, t)| ≤ ϑ(t)+boeλt|ξ−η| for all ξ, η, hypothesis H’.3 provides the following

estimate: ∣∣∣Iv(x, t) − Iv(y, t)∣∣∣ ≤ mdϑ(t) + c1boe

λt|x − y|,

where the parameter c1 depends, besides on m and on ‖v‖∞, on the W 1,∞–norms of νk, ζk, and on d′. Hence, by making use of the assumptions H’.1,

810 Anna Lisa Amadori

H’.2, after computations one gets

F (y, t, v(y, t), Iv(y, t),boe

λt x − y

|x − y| , Y)

−F (x, t, v(x, t), Iv(x, t),boe

λt x − y

|x − y| + δx, X + δI)≤

≤ g(ϑ(t)) +[c1 + l′(c1 + c2)

]boe

λt|x − y| + o(1)

as δ goes to zero. Here, the parameter c2 only depends, besides on ‖v‖∞,on the W 1,∞–norms of σij , b

i, c

, H(x, t, 0, 0, 0), and on l.Plugging this estimate in (4.3) and choosing λ = c1 + l′(c1 + c2) yields

ϕ′(tδ) ≤ g(ϑ(tδ)) + o(1). By taking advantage of the continuity of ϕ′, g andof (4.2), passing to the limit with respect to δ gives ϕ′(to) ≤ g(ϑ∗(to)), aswanted.

We lastly illustrate, for the sake of completeness, how the Lipschitz regu-larity of the derivatives’ price function follows by Theorem 1.2.Proof of Theorem 2.8. The same assumptions of Theorem 2.7 grant thatthe related Cauchy problem (1.9)–(1.2) fulfills H’.0–H’.3. Hence, Theorem1.2 may be applied and the belonging of e−nu to L∞(

0, T ; W 1,∞(Rn))

isattained. After coming back to the economical variable S, one exactly getthe statement of Theorem 2.8.

References

[1] O. Alvarez and A. Tourin, Viscosity solutions of nonlinear integro–differential equa-tions, Ann. Inst. H. Poincare Anal. Non Lineaire, 13 (1996), 293–317.

[2] G. Barles and B Perthame, Comparison principle for Dirichlet–type Hamilton–Jacobiequations and singular perturbation of degenerated elliptic equations, Appl. Math. Op-tim., 21 (1990), 21–44.

[3] P. Bremaud, “Point Processes and Queues. Martingales Dynamics,” Springer Verlag,1981

[4] M.G. Crandall, H. Ishii, and P.L. Lions, Users’ guide to viscosity solutions of secondorder partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1–67.

[5] M.G. Crandall and P.L. Lions, Quadratic growth of solutions of fully nonlinear secondorder equations in R

n, Diff. Int. Eq., 3 (1990), 601–616.[6] M.G. Crandall, R.T.II Newcomb, and Y. Tomita, Existence and uniqueness for vis-

cosity solutions of degenerate quasilinear elliptic equations in RN , Appl. Anal., 34(1989), 1–23.

[7] J. Cvitanic and J. Ma., Hedging options for a large investor and forward–backwardS. D. E ’s, Ann. Appl. Prob., 6 (1996), 370–398.

[8] C. Dellacherie and P.A. Meyer, Probabilites et potentiel: theorie des martingales,Hermann, Paris, 1980.

[9] J. Dewinne, S. Howison, and P. Wilmott, The mathematics of financial derivatives. Astudent introduction, Cambridge University Press, Cambridge, 1995.

nonlinear integro-differential equations 811

[10] M.G. Garroni and J.L. Menaldi, Green functions for second order parabolic integro-–differential problems, Pitman Res. Notes Math. Ser., 275, Longman Scientific &Technical, Harlow, 1992.

[11] Y. Giga, S. Goto, H. Ishii, and M.H. Sato, Comparison principle and convexity pre-serving properties for singular degenerated parabolic equations on unbounded domains,Indiana Univ. J., Vol. 40 (1991), 443–470.

[12] I. I. Gıhman and A. V. Skorohod, Stochastic differential equations, Ergebnisse derMathematik und ihrer Grenzgebiete, Band 72, Springer-Verlag, New York, 1972.

[13] H. Ishii, Perron’s method for Hamilton-Jacobi equations, Duke Math. J., Vol. 55(1987), 369–384.

[14] H. Ishii and K. Kobayasi, On the uniqueness and existence of solutions of fully non-linear parabolic PDEs under the Osgood type condition, Diff. Int. Eq., Vol. 7 (1994),909–920.

[15] H. Ishii and P.L. Lions, Viscosity solutions of fully nonlinear second-order ellipticpartial differential equations, J. Differ. Equations, Vol. 83 (1990), 26–78.

[16] M. Jeanblanc-Pique and M. Pointier,Optimal portfolio for a small investor in a marketmodel with discontinuous prices, Appl. Math. Opt., Vol. 22 (1990), 287–310.

[17] D. Lamberton and B. Lapeyre, Introduction au calcul stochastique applique a la fi-nance. second edition, Ellipses, 1998.

[18] C. Mancini, The European Options hedge “any” claim perfectly in a Poisson–Gaussianmodel for the index–linked bond market, Quaderni del Dipartimento di Matematicaper le Decisioni Economiche, Universita degli Studi di Firenze, n.2/2000.

[19] L. Mastroeni and M. Matzeu, Nonlinear variational inequalities for jump–diffusionprocess and irregular obstacles with a financial application, Nonlin. Anal., Vol. 34(1998), 889–905.

[20] R.C. Merton, Option pricing when the underlying stocks returns are discontinuous,Journ. Financ. Econ., Vol. 5 (1976), 125–144.

[21] H. Pham, Optimal stopping of controlled jump–diffusion processes,a viscosity solutionapproach, J. Math. Syst. Extim. Contr., Vol. 8 (1998), 127–130.

[22] M.H. Protter and H.F. Weinberger, Maximum principles in differential equations,Prentice–Hall, 1967.

[23] W.J. Runggaldier, Jump Diffusion Models, to appear in: Handbook of Heavy TailedDistributions in Finance (S.T. Rachev, ed.), North Holland Handbooks in Finance.