1 Introduction

Abstract

We formulate a new class of stochastic partial differential equations (SPDEs), named high-order vector backward SPDEs (B-SPDEs) with jumps, which allow the high-order integral-partial differential operators into both drift and diffusion coefficients. Under certain type of Lipschitz and linear growth conditions, we develop a method to prove the existence and uniqueness of adapted solution to these B-SPDEs with jumps. Comparing with the existing discussions on conventional backward stochastic (ordinary) differential equations (BSDEs), we need to handle the differentiability of adapted triplet solution to the B-SPDEs with jumps, which is a subtle part in justifying our main results due to the inconsistency of differential orders on two sides of the B-SPDEs and the partial differential operator appeared in the diffusion coefficient. In addition, we also address the issue about the B-SPDEs under certain Markovian random environment and employ a B-SPDE with strongly nonlinear partial differential operator in the drift coefficient to illustrate the usage of our main results in finance.

Key words and phrases: Backward Stochastic Partial Differential Equations with Jumps, High-Order Partial Differential Operator, Vector Partial Differential Equation, Existence and Uniqueness, Random Environment

A New Class of Backward Stochastic Partial Differential Equations with Jumps and Applications

Wanyang Dai

Department of Mathematics

Nanjing University, Nanjing 210093, China

Email: nan5lu8@netra.nju.edu.cn

Date: 5 May 2011

1 Introduction

Motivated from mean-variance hedging (see, e.g., Dai [10]) and utility based optimal portfolio choice (see, e.g., Becherer [3], Musiela and Zariphopoulou [22]) in finance, and multi-channel (or multi-valued) image regularization such as color images in computer vision and network application (see, e.g., Caselles et al. [6], Tschumperlé and Deriche [33, 34, 35], and references therein), we formulate a new class of SPDEs, named high-order vector B-SPDEs with jumps, which allow high-order integral-partial differential operators and into both drift and diffusion coefficients as shown in the following equation (1.1),

 (1.1) V(t,x) = H(x)+∫TtL(s−,x,V,⋅)ds+∫Tt(J(s−,x,V,⋅)−¯V(s−,x))dW(s) −∫Tt∫z>0~V(s−,x,z,⋅)~N(λds,x,dz).

where the operator depends not only on but also on their associated partial derivatives, i.e., for each integer and , and are defined by

 L(s,x,V,⋅) ≡ L(s,x,V(s,x),V(k)(s,x),¯V(s,x),¯V(m)(s,x),~V(s,x),⋅), J(s,x,V,⋅) = (J1(s,x,V,⋅),...,Jd(s,x,V,⋅)), Ji(s,x,V,⋅) ≡ Ji(s,x,V(s,x),V(k)(s,x),⋅),i∈{1,...,d}.

Under certain type of Lipschitz and linear growth conditions, we prove the existence and uniqueness of adapted triplet solution to these B-SPDEs. When the partial differential operator depends only on , and but not on their associated derivatives and , our B-SPDEs with jumps reduce to conventional BSDEs with jumps (see, e.g., Becherer [3], Dai [10], Tang and Li [32]).

BSDEs were first introduced by Bismut [5] and the first result for the existence of an adapted solution to a continuous nonlinear BSDE was obtained by Pardoux and Peng [26]. Since then, numerous extensions along the line have been conducted, such as, Tang and Li [32] get the first adapted solution to a BSDE with Poisson jumps for a fixed terminal time and Situ [31] extended the result to the case where the BSDE is with bounded random stopping time as its terminal time and non-Lipschitz coefficient. Currently, BSDEs are still an active area of research in both theory and applications, see, e.g., Becherer [3], Cohen and Elliott [8], Crépey and Matoussi [9], Dai [10], Lepeltier et al. [18], Yin and Mao [36], and references therein.

The study on SPDEs receives a great attention recently (see, e.g., Pardoux [25] and Hairer [14]). Particularly, Pardoux and Peng [27] introduces a system of semi-linear parabolic SPDE in a backward manner and establish the existence and uniqueness of adapted solution to the SPDE under smoothness assumptions on the coefficients, and moreover, the authors in [27] also employ backward doubly SDEs (BDSDE) to provide a probabilistic representation for the parabolic SDE. Since then, numerous researches have been conducted in terms of weak solution and stationary solution to the semi-linear SPDE (see, e.g., Bally and Matoussi [2], Zhang and Zhao [38], and references therein). However, our B-SPDEs exhibited in (1.1) are fundamentally different from the SPDEs as introduced in Pardoux and Peng [27] and as studied in most of the existing researches in the following aspects: First, our system formulation is a direct generalization of the conventional BSDEs, i.e., both the drift and diffusion coefficients of our B-SPDEs depend on the triplet and its associated partial derivatives not just on and its associated partial derivatives; Second, our B-SPDEs are based on high-order partial derivatives and are subject to jumps. One special case of our B-SPDEs available in the literature is the one derived in Musiela and Zariphopoulou [22] for the purpose of optimal-utility based portfolio choice, which is strongly nonlinear in the sense that is addressed in Lions and Souganidis [19].

Note that the B-SPDEs presented in (1.1) are vector B-SPDEs with jumps, which are motivated from various aspects such as multi-channel image regularization in computer vision and network application through vector PDEs (see, e.g., Caselles et al. [6], Tschumperlé and Deriche [33, 34, 35], and references therein), coupling and synchronization in random dynamic systems through vector SPDEs (see, e.g., Mueller [21], Chueshov and Schmalfu [7], and references therein).

To show our formulated system well-posed, we develop a method based on a scheme used for conventional BSDEs (see, e.g., Yong and Zhou [37]) to prove the existence and uniqueness of adapted solution to our B-SPDEs with jumps in (1.1) under certain Lipschitz and linear growth conditions. One fundamental issue we need to handle in the method is the differentiability of the triplet solution to our B-SPDEs with jumps, which is a subtle part in the analysis due to the inconsistency of differential orders on two sides of the B-SPDEs and the partial differential operators appeared in the diffusion coefficient. So more involved functional spaces and techniques are required. In addition, although there is no perfect theory in dealing with the strongly nonlinear SPDEs (see, e.g., Pardoux [25]), our discussions about the adapted solution to (1.1) can provide some reasonable interpretation concerning the unique existence of adapted solution before a random bankruptcy time to the strongly nonlinear B-SPDE derived in Musiela and Zariphopoulou [22].

In the paper, we also provide some discussion concerning our B-SPDEs under random environment, e.g., the variable in (1.1) is replaced by a continuous Markovian process . To be convenient for readers, we present a rough graph in Figure 1 with respect to sample surfaces for a solution to a B-SPDE and in terms of sample curves for a solution to the B-SPDE under random environment.

The rest of the paper is organized as follows. In Section 2, we first introduce a class of B-SPDEs with jumps in finite space domain, then we state and prove our main theorem. In Section 3, we extend our discussions in the previous section to the case corresponding to infinite space domain and under random environment. In Section 4, we use an example to illustrate the usage of our main results in finance.

2 A Class of B-SPDEs with Jumps in Finite Space Domain

2.1 Required Probability and Functional Spaces

First of all, we introduce some notations to be used in the paper. Let be a fixed complete probability space on which are defined a standard -dimensional Brownian motion with and dimensional subordinator with and càdlàg sample paths for some fixed (see, e.g., Applebaum [1], Bertoin [4], and Sato [30] for more details about subordinators and Lévy processes), where the prime denotes the corresponding transpose of a matrix or a vector. Moreover, , and their components are assumed to be independent of each other. In addition, each subordinator with can be represented by (see, e.g., Theorem 13.4 and Corollary 13.7 in Kallenberg [17])

 (2.2) Li(t)=ait+∫(0,t]∫zi>0ziNi(ds,dzi),t≥0

where denotes a Poisson random measure with a deterministic, time-homogeneous intensity measure , where is the index function over the set , the constant is taken to be zero, and is the Lévy measure. Related to the probability space , we suppose that there is a filtration with for each , , and .

Secondly, let and be a close connected domain in for a given . Then we can use for each to denote the Banach space of all functions having continuous derivatives up to the order with the following uniform norm,

 (2.3) ∥f∥Ck(D,q)=maxc∈{1,...,k}maxj∈{1,...,r(c)}supx∈D∣∣f(c)j(x)∣∣

for each , where for each is the total number of the following partial derivatives of the order

 (2.4) f(c)r,(i1...ip)(x)=∂cfr(x)∂xi11...∂xipp

with , , , and . Moreover, for the late purpose, we let

 (2.5) f(c)(i1,...,ip) ≡ (f(c)1,(i1,...,ip),...,f(c)q,(i1,...,ip)), (2.6) f(c)(x) ≡ (f(c)1(x),...,f(c)r(c)(x)),

where each corresponds to a -tuple and a . In addition, let denote the following Banach space, i.e.,

 (2.7) C∞(D,Rq)≡{f∈∞⋂k=1Ck(D,Rq),∥f∥C∞(D,q)<∞}

where

 (2.8) ∥f∥2C∞(D,q)=∞∑k=1ξ(k)∥f∥2Ck(D,q)

for some discrete function with respect to , which is fast decaying in . For convenience, we take .

Thirdly, we introduce some measurable spaces to be used in the sequel. Let denote the set of all -valued measurable stochastic processes adapted to for each , which are in for each fixed ), such that

 (2.9) E[∫T0∥Z(t)∥2C∞(D,q)dt]<∞

and let denote the corresponding set of predictable processes (see, e.g., Definition 5.2 and Definition 1.1 respectively in pages 21 and 45 of Ikeda and Watanabe [15]). Moreover, let denote the set of all -valued, -measurable random variables for each , where satisfies

 (2.10) E[∥ξ∥2C∞(D,q)]<∞.

In addition, let be the set of all -valued predictable processes for each and , satisfying

 (2.11) E[h∑i=1∫T0∫zi>0∥∥~Vi(t−,z)∥∥2C∞(D,q)νi(dzi)dt]<∞

and let

 (2.12) L2ν,c(D×Rh+,Rq×h)≡{~v:D×Rh+→Rq×h,h∑i=1∫zi>0∥~vi(zi)∥2Cc(D,q)νi(dzi)<∞}

with the associated norm for any and as follows,

 (2.13) ∥~v∥ν,c≡(h∑i=1∫zi>0∥~vi(zi)∥2Cc(D,q)λiνi(dzi))12.

In the end, we define

 (2.14) Q2F([0,T])≡L2F([0,T],Rq)×L2F,p([0,T],Rqd)×L2p([0,T],Rq×h).

2.2 The B-SPDEs

First of all, we introduce a class of -dimensional B-SPDEs with jumps and terminal random variable for each as presented in (1.1), where for each and ,

 ¯V(s,⋅) = (¯V1(s,⋅),...,¯Vd(s,⋅))∈C∞(D,Rq×d), ~V(s,⋅,z) = (~V1(s,⋅,z1),...,~Vh(s,⋅,zh))∈C∞(D,Rq×h), ~N(λds,x,dz) = (~N1(λ1ds,x,dz1),...,~Nh(λhds,x,dzh))′.

Moreover, in (1.1), is a -dimensional integral-partial differential operator satisfying, a.s.,

 (2.15) ∥∥ΔL(c)(s,x,u,v)∥∥≤KD(∥u−v∥Ck+c(D,q)+∥¯u−¯v∥Cm+c(D,qd)+∥~u−~v∥ν,c)

for any , with , where depending on the domain is a nonnegative constant, is the largest absolute value of entries (or components) of the given matrix (or vector) , and

 (2.16) ΔL(c)(s,x,u,v)≡L(c)(s,x,u,⋅)−L(c)(s,x,v,⋅).

Similarly, is a -dimensional partial differential operator satisfying, a.s.,

 (2.17) ∥ΔJ(c)(s,x,u,v)∥≤KD(∥u−v∥Cm+c(D,q)).

Moreover, we suppose that

 (2.18) ∥∥L(c)(s,x,u,⋅)∥∥ ≤ KD(∥u∥Ck+c(D,q)+∥¯v∥Cm+c(D,qd)+∥~v∥ν,c), (2.19) ∥∥J(c)(s,x,u,⋅)∥∥ ≤ KD∥u∥Cm+c(D,q).
Example 2.1

The following conventional linear partial differential operators satisfy the conditions as stated in (2.15)-(2.19),

 (Lu)(t,x) = p∑i,j=1aij(x)∂2u(t,x)∂xi∂xj+d∑j=1bj(x)∂u(t,x)∂xj+c(x)u(t,x) +d∑i,j=1¯aij(x)∂2¯u(t,x)∂xi∂xj+d∑j=1¯bj(x)∂¯u(t,x)∂xj+¯c(x)¯u(t,x) (Ju)(t,x) = p∑i,j=1aij(x)∂2u(t,x)∂xi∂xj+d∑j=1bj(x)∂u(t,x)∂xj+c(x)u(t,x),

where , , , , , and are uniformly bounded over all and and .

Theorem 2.1

Under conditions of (2.15)-(2.19), if and are -adapted for each fixed and any given with

 (2.20) L(⋅,x,0,⋅),J(⋅,x,0,⋅)∈L2F([0,T],Rq),

then the B-SPDE (1.1) has a unique adapted solution satisfying, for each and ,

 (2.21) (V(⋅,x),¯V(⋅,x),~V(⋅,x,z))∈Q2F([0,T])

where is a càdlàg process and the uniqueness is in the sense: if there exists another solution as required, we have

 E[∫T0(∥U(t)−V(t)∥2C∞(D,q)+∥¯U(t)−¯V(t)∥2C∞(D,qd)+∥~U(t)−~V(t)∥2ν,∞)dt]=0.

We divide the proof of Theorem 2.1 into the following three lemmas.

Lemma 2.1

Under the conditions of Theorem 2.1, for each fixed , , and a triplet

 (2.22) (U(⋅,x),¯U(⋅,x),~U(⋅,x,z))∈Q2F([0,T]),

there exists another triplet such that

 (2.23) V(t,x) = H(x)+∫TtL(s−,x,U,⋅)ds+∫Tt(J(s−,x,U,⋅)−¯V(s−,x))dW(s) −∫Tt∫z>0~V(s−,x,z)~N(λds,x,dz),

where is a -adapted càdlàg process, and are the corresponding predictable processes, and for each ,

 (2.24) E[∫T0∥V(t,x)∥2dt]<∞, (2.25) E[∫T0∥¯V(t,x)∥2dt]<∞, (2.26) E[h∑i=1∫T0∫zi>0∥∥~Vi(t−,x,z)∥∥2νi(dzi)dt]<∞.

Proof. First of all, for each fixed , , and a triplet as stated in (2.22), it follows from conditions (2.15)-(2.20) that

 (2.27) L(⋅,x,U,⋅)∈L2F([0,T],Rq),J(⋅,x,U,⋅)∈L2F([0,T],Rq×d).

Now consider and in (2.27) as two new starting and , then it follows from the Martingale representation theorem (see, e.g., Lemma 2.3 in Tang and Li [32]) that there exists a unique pair of predictable processes which are square-integrable for each in the senses of (2.25)-(2.26) such that

 (2.28) ^V(t,x) ≡ E[H(x)+∫T0L(s−,x,U,⋅)ds+∫T0J(s−,x,U,⋅)dW(s)∣∣∣Ft] = ^V(0,x)+∫t0¯V(s−,x)dW(s)+∫t0∫z>0~V(s−,x,z)~N(λds,x,dz)

which implies that

 (2.29) ^V(0,x) = H(x)+∫T0L(s−,x,U,⋅)ds+∫T0J(s−,x,U,⋅)dW(s) −∫T0¯V(s−,x)dW(s)−∫T0∫z>0~V(s−,x,z)~N(λds,x,dz).

Moreover, due to the Corollary in page 8 of Protter [29], can be taken as a càdlàg process. Now we define a process as follows,

 (2.30) V(t,x) ≡ E[H(x)+∫TtL(s−,x,U,⋅)ds+∫TtJ(s−,x,U,⋅)dW(s)∣∣∣Ft]

Then by simple calculation, we know that is square-integrable in the sense of (2.24), and moreover, it follows from (2.28)-(2.30) that

 (2.31) V(t,x) = ^V(t,x)−∫t0L(s−,x,U,⋅)ds−∫t0J(s−,x,U,⋅)dW(s)

which indicates that is a càdlàg process. Furthermore, for a given triplet , , it follows from (2.28)-(2.29) and (2.31) that the corresponding triplet satisfies the equation (2.23) as stated in the lemma, which also implies that

 (2.32) V(t,x) ≡ Missing or unrecognized delimiter for \left +∫t0∫z>0~V(s−,x,z)~N(λds,x,dz).

Hence we complete the proof of Lemma 2.1.

Lemma 2.2

Under the conditions of Theorem 2.1, for each fixed , , and a triplet as in (2.22), we define through (2.23). Then , , for each exists a.s. and satisfies a.s.

 (2.33) V(c)(i1...ip)(t,x) = H(c)(i1...ip)(x)+∫TtL(c)(i1...ip)(s−,x,U,⋅)ds +∫Tt(J(c)(i1...ip)(s−,x,U,⋅)−¯V(c)(i1...ip)(s−,x))dWi(s) −∫Tt∫z>0~V(c)(i1...ip)(s−,x,z)~N(λds,x,dz),

where and with . Moreover, for each is a -adapted càdlàg process, and are the corresponding predictable processes, which are square-integrable in the senses of (2.24)-(2.26).

Proof. First of all, we show that the claim in the lemma is true for . To do so, for each given and as in the lemma, let

 (2.34) (V(1)(l)(t,x),¯V(1)(l)(t,x),~V(1)(l)(t,x,z))

be defined through (2.23) where and are replaced by their first-order partial derivatives and in terms of with . Then we can show that the triplet defined in (2.34) for each is indeed the required first-order partial derivative of that is defined through (2.23) for the given .

As a matter of fact, for each , small enough positive constant , and , define

 (2.35) f(l),δ(t,x)≡f(t,x+δel),

where is the unit vector whose th component is one and others are zero. Moreover, let

 (2.36) Δf(1)(l),δ(t,x)=f(l),δ(t,x)−f(t,x)δ−f(1)(l)(t,x)

for each . In addition, let

 (2.37) ΔI(1)(l),δ(s,x,U) = 1δ(I(s,x+δel,U(s,x+δel),⋅)−I(s,x,U(s,x),⋅)) −I(1)(l)(s,x,U(s,x),⋅)

for each . Then, by applying the Ito’s formula (see, e.g., Theorem 1.14 and Theorem 1.16 in pages 6-9 of ksendal and Sulem [24]) to the function

 ζ(ΔV(1)(l),δ(t,x))≡Tr(ΔV(1)(l),δ(t,x))e2γt

for some , where Tr denotes the trace of the matrix for a given matrix , we have

 (2.38) ζ(ΔV(1)(l),δ(t,x))+∫TtTr(ΔJ(1)(l),δ(s,x,U)−Δ¯V(1)(l),δ(s,x))e2γsds +∫Tt∫zj>0Tr(Δ~V(1)(l),δ(s−,x,z))e2γs~N(λds,x,dz) = 2∫Tt(−γTr(ΔV(1)(l),δ(s,x))+(ΔV(1)(l),δ(s,x))′(ΔL(1)(l),δ(s,x,U)))e2γsds−M(t) ≤ (−2γ+3K2D^γ)∫TtTr(ΔV(1)(l),δ(s,x))e2γsds+^γ∫Tt∥∥ΔL(1)(l),δ(s,x,U)∥∥2e2γsds−M(t) = ^γ∫Tt∥∥ΔL(1)(l),δ(s,x,U)∥∥2e2γsds−Mδ(t)

if, in the last equality, we take

 (2.39) ^γ=3K2D2γ>0,

where is a martingale of the following form,

 2d∑j=1∫Tt(ΔV(1)(l),δ(s−,x))′(Δ(Jj)(1)(l),δ(s−,x,U)−Δ(¯Vj)(1)(l),δ(s−,x))e2γsdWj(s) −2h∑j=1∫Tt∫zj>0(ΔV(1)(l),δ(s−,x))′((1/δ)~Vj(s−,x,zj)+~V(1)j(s−,x,zj))e2</