Forward-backward SDEs with jumps and classical solutions to nonlocal quasilinear parabolic PDEs
FFORWARD-BACKWARD SDES WITH JUMPS AND CLASSICALSOLUTIONS TO NONLOCAL QUASILINEAR PARABOLIC PDES
EVELINA SHAMAROVA AND RUI S ´A PEREIRA
Abstract.
We obtain an existence and uniqueness theorem for fully coupledforward-backward SDEs (FBSDEs) with jumps via the classical solution to theassociated quasilinear parabolic partial integro-differential equation (PIDE),and provide the explicit form of the FBSDE solution. Moreover, we embed theassociated PIDE into a suitable class of non-local quasilinear parabolic PDEswhich allows us to extend the methodology of Ladyzhenskaya et al [8], origi-nally developed for traditional PDEs, to non-local PDEs of this class. Namely,we obtain the existence and uniqueness of a classical solution to both theCauchy problem and the initial-boundary value problem for non-local quasi-linear parabolic PDEs.
Keywords:
Forward-backward SDEs with jumps, Non-local quasilinear parabolic PDEs,Partial integro-differential equations
1. Introduction
One of the well known tools to solve FBSDEs driven by a Brownian motion istheir link to quasilinear parabolic PDEs which, by means of Itˆo’s formula, allowsto obtain the explicit form of the FBSDE solution via the classical solution of theassociated PDE [12, 14, 15, 3]. However, if we are concerned with FBSDEs withjumps, the associated PDE becomes a PIDE whose coefficients contain non-localdependencies on the solution. To the best of our knowledge, there are no results onthe solvability (in the classical sense) of PIDEs appearing in connection to FBSDEswith jumps.In this work, we obtain the existence and uniqueness of a classical solution for aclass of non-local quasilinear parabolic PDEs, which includes PIDEs of interest, andapply this result to obtain the existence and uniqueness of solution to fully coupledFBSDEs driven by a Brownian motion and a compensated Poisson random measureon an arbitrary time interval [0 , T ]:(1) X t = x + (cid:82) t f ( s, X s , Y s , Z s , ˜ Z s ) ds + (cid:82) t σ ( s, X s , Y s ) dB s + (cid:82) t (cid:82) R l ∗ ϕ ( s, X s − , Y s − , y ) ˜ N ( ds, dy ) ,Y t = h ( X T ) + (cid:82) Tt g ( s, X s , Y s , Z s , ˜ Z s ) ds − (cid:82) Tt Z s dB s − (cid:82) Tt (cid:82) R l ∗ ˜ Z s ( y ) ˜ N ( ds, dy ) . The forward SDE is R n -valued while the backward SDE (BSDE) is R m -valued,and the Brownian motion B t is n -dimensional. The coefficients f ( t, x, u, p, w ), g ( t, x, v, p, w ), σ ( t, x, u ), and ϕ ( t, x, u, y ) are functions of appropriate dimensionswhose argument ( t, x, u, p, w ) belongs to the space [0 , T ] × R n × R m × R m × n × a r X i v : . [ m a t h . P R ] M a y EVELINA SHAMAROVA AND RUI S´A PEREIRA L ( ν, R l ∗ → R m ), where ν is the intensity of the Poisson random measure involvedin (1) and R l ∗ = R l − { } .Our second object of interest is the following R m -valued non-local quasilinearparabolic PDE on [0 , T ] × R n associated to FBSDE (1)(2) − n (cid:88) i,j =1 a ij ( t, x, u ) ∂ x i x j u + n (cid:88) i =1 a i ( t, x, u, ∂ x u, ϑ u ) ∂ x i u + a ( t, x, u, ∂ x u, ϑ u ) + ∂ t u = 0 . The coefficients of (2) are expressed via the coefficients of (1) as follows:(3) a ij ( t, x, u ) = (cid:80) nk =1 σ ik σ jk ( T − t, x, u ) ,a i ( t, x, u, p, w ) = (cid:82) Z ϕ i ( T − t, x, u, y ) ν ( dy ) − f i (cid:0) T − t, x, u, p σ ( T − t, x, u ) , w (cid:1) ,a ( t, x, u, p, w ) = − g (cid:0) T − t, x, u, p σ ( T − t, x, u ) , w (cid:1) − (cid:82) Z w ( y ) ν ( dy ) ,ϑ u ( t, x ) = u ( t, x + ϕ ( T − t, x, u ( t, x ) , · )) − u ( t, x ) , where the support Z of the function y (cid:55)→ ϕ ( t, x, u, y ) is assumed to have a finite ν -measure for each ( t, x, u ) ∈ [0 , T ] × R n × R m . In (2), ∂ x i x j u , ∂ x i u , ∂ t u , u , and ϑ u are evaluated at ( t, x ). Non-local PDE (2) is assumed to be uniformly parabolic ,i.e., for all ξ ∈ R n , it holds that ˆ µ ( | u | ) ξ (cid:54) (cid:80) ni,j =1 a ij ( t, x, u ) ξ i ξ j (cid:54) µ ( | u | ) ξ , where µ and ˆ µ are non-decreasing, and, respectively, non-increasing functions.BSDEs and FBSDEs with jumps have been studied by many authors, e.g., [2, 9,10, 11, 13, 19, 20, 21]. Existence and uniqueness results for fully coupled FBSDEswith jumps were previously obtained in [20], [21], and, on a short time interval, in[11]. The main assumption in [20] and [21] is the so-called monotonicity assumption(see, e.g., [20], p. 436, assumption (H3.2)). This is a rather technical condition thatappears unnatural and requires a bit of effort to find objects satisfying it.We remark that our result on the existence and uniqueness of solution to FB-SDE (1) holds on a time interval of an arbitrary length and without any sort ofmonotonicity assumptions. Our assumptions on the FBSDE coefficients are formu-lated in a way that makes it possible to solve the associated PIDE, which is aparticular case of non-local PDE (2). The assumptions on the coefficients of (2)are, in turn, natural extensions of the assumptions in [8] for traditional quasilinearparabolic PDEs and coincide with the latter if the coefficients of (2) do not dependon ϑ u . It is known that the work of Ladyzhenskaya et al [8] provides assumptionson solvability of quasilinear parabolic PDEs in the most general form, which makesus believe that both problems, FBSDE (1) and the associated PIDE, are solved inquite general assumptions (unlike [20] and [21]).Importantly, we obtain a link between the solution to FBSDE (1) and the so-lution to the associated PIDE. A similar link in the case of FBSDEs driven by aBrownian motion was established by Ma, Protter, and Yong [12], and the relatedresult on solvability of FBSDEs is known as the four step scheme. The main toolto establish this link, and, consequently, to solve Brownian FBSDEs, was the afore-mentioned result of Ladyzhenskaya et al [8] on quasilinear parabolic PDEs. Sincethe consideration of FBSDEs with jumps leads to PDEs of type (2), i.e., containingthe non-local dependence ϑ u , the theory developed in [8] is not applicable anymore.Thus, this article has the following two main contributions. First of all, we definea class of non-local quasilinear parabolic PDEs containing the PIDE associated to BSDES WITH JUMPS AND CLASSICAL SOLUTIONS TO NONLOCAL PDES 3
FBSDE (1) and establish the existence and uniqueness of a classical solution to theCauchy problem and the initial-boundary value problem for PDEs of this class; and,secondly, we prove the existence and uniqueness theorem for fully coupled FBSDEswith jumps (1) and provide the formulas that express the solution to FBSDE (1)via the solution to associated non-local PDE (2) with coefficients and the function ϑ u given by (3). The major difficulty of this work appears in obtaining the first ofthe aforementioned results.The following scheme is used to obtain the existence and uniqueness result fornon-local PDEs. We start with the initial-boundary value problem on an openbounded domain. The maximum principle, the gradient estimate, and the H¨oldernorm estimate are obtained in order to show the existence of solution by means ofthe Leray-Schauder theorem. The uniqueness follows from the maximum principle.Further, the diagonalization argument is employed to prove the existence of solutionto the Cauchy problem. Remark that obtaining the gradient estimate is straighfor-ward and can be obtained from the similar result in [8] by freezing the non-localdependence ϑ u . However, the estimate of H¨older norms cannot be obtained in thesimilar manner, and requires obtaining a bound for the time derivative of the so-lution, which turns out to be the most non-trivial task. Importantly, the H¨oldernorm estimates are crucial for application of the Leray-Schauder theorem and thediaganalization argument.The organization of the article is as follows. Section 2 is dedicated to the exis-tence and uniqueness of solution to abstract multidimensional non-local quasilinearparabolic PDEs of form (2). We consider both the Cauchy problem and the initial-boundary value problem. In Section 3, we show that by means of formulas (3),the PIDE associated to FBSDE (1) is included into the class of non-local PDEsconsidered in Section 2. Then, by means of the existence and uniqueness result forPIDEs, we obtain the existence and uniqueness theorem for FBSDEs with jumpsand provide the formulas connecting the solution to an FBSDE with the solutionto the associated PIDE.
2. Multidimensional non-local quasilinear parabolic PDEs
In this section, we obtain the existence and uniqueness of solution for the initial-boundary value problem and the Cauchy problem for abstract R m -valued non-localquasilinear parabolic PDE (2), where ϑ u ( t, x ) is a function built via u , taking valuesin a normed space E , and satisfying additional assumptions to be specified later.Remark that the function ϑ u considered in this section is not necessary of the formmentioned in (3).Let F ⊂ R n be an open bounded domain with a piecewise-smooth boundary andnon-zero interior angles. For a more detailed description of the forementioned classof domains we refer the reader to [8] (p. 9). Further, in case of the initial-boundaryvalue problem we consider the boundary condition u ( t, x ) = ψ ( t, x ) , ( t, x ) ∈ { (0 , T ) × ∂ F } ∪ (cid:8) { t = 0 } × F (cid:9) , (4)where ψ is the boundary function defined as follows ψ ( t, x ) = (cid:40) ϕ ( x ) , x ∈ { t = 0 } × F , , ( t, x ) ∈ [0 , T ] × ∂ F . (5) EVELINA SHAMAROVA AND RUI S´A PEREIRA
In case of the Cauchy problem, we consider the initial condition u (0 , x ) = ϕ ( x ) , x ∈ R n . (6)Further, in case of the initial-boundary value problem, the coefficients of PDE (2)are defined as follows: a ij : [0 , T ] × F × R m → R , a i : [0 , T ] × F × R m × R m × n × E → R , i, j = 1 , . . . , n , a : [0 , T ] × F × R m × R m × n × E → R m . In case of the Cauchy problem,everywhere in the above definitions, F should be replaced with the entire space R n .We remark that due the presence of the function ϑ u , the existence and uniquenessresults of Ladyzenskaya et al [8] for initial-boundary value problem (2)-(4) andCauchy problem (2)-(6) are not applicable to the present case. Remark . Without loss of generality we assume that { a ij } is a symmetric ma-trix. Indeed, since we are interested in C , -solutions of (2), then for all i, j , ∂ x i x j u = ∂ x j x i u . Therefore, { a ij } can be replaced with ( a ij + a ji ) for non-symmetricmatrices. In this subsection, we briefly explain the main steps to obtaining the existenceand uniqueness theorem for non-local PDE (2). In each step, we mention whetherit is a slight adaptation of the similar result in [8], or the differences are essential.1.
Maximum principle.
As in [8], we start with the initial-boundary value problemfor PDE (2) on a bounded domain. At this step, obtaining the maximum principleis an adaptation of the similar result in [8].2.
Gradient estimate.
To obtain an a priori estimate for the gradient of a classicalsolution, we freeze the function ϑ u in (2). After this, we are able to apply theresult from [8] on the gradient estimate.3. Estimate for the time derivative of the solution.
This is the only step, where thedifference with [8] becomes essential due to the presence of the function ϑ u . Toobtain the time derivative estimate, we study linear-like non-local PDEs writtenin the divergence form (linear w.r.t. u and but not linear w.r.t. ϑ u ), and obtainthe maximum principle for the latter. The maximum principle requires several L -type estimates. In brief, the difficulty arises from the fact that it is not clearhow to estimate an L -type norm of ϑ u , given by (3), via the similar norm of u .4. H¨older norm estimate.
After the previous step is done, the estimate of the H¨oldernorm of the solution can be obtained from the similar result in [8].5.
Existence and uniqueness theorem for an initial-boundary value problem.
Thistheorem is announced in [8] for systems of quasilinear PDEs but not actuallyproved, although one can recover the scheme of the proof (but not the details)from the case of one PDE. The main tools in our proof are a priori estimates ofH¨older norms and the version of the Leray-Schauder theorem from [5]. At thisstep, the presence of the function ϑ u in PDE (2) is not essential.6. Existence theorem for a Cauchy problem.
A Cauchy problem in connection tosystems of quasilinear PDEs is not actually discussed in [8], expect for the caseof one PDE, where the main technique is the diagonalization argument. Thistheorem, therefore, even for traditional systems of PDEs, can be regarded asan additional contribution of this work. The theorem easily extends to non-localPDEs, and the presence of the function ϑ u is not essential. Also, using our results,the original four step scheme ([12]) works in more general assumptions. BSDES WITH JUMPS AND CLASSICAL SOLUTIONS TO NONLOCAL PDES 5 Uniquneness theorem for a Cauchy problem.
To prove the uniqueness, we use theresults on fundamental solutions from the book of Friedman [6]. The presence ofthe function ϑ u adds a little bit more work, but can be treated by Gronwall’sinequality. In this subsection we introduce the necessary notation that will be used through-out this article.
T > F ⊂ R n is an open bounded domain with a piecewise-smooth boundary ∂ F andnon-zero interior angles. F T = (0 , T ) × F and F t = (0 , t ) × F , t ∈ (0 , T ).( ∂ F ) T = [0 , T ] × ∂ F and ( ∂ F ) t = [0 , t ] × ∂ F , t ∈ (0 , T ). F T = [0 , T ] × F , where F is the closure of F .Γ t = ( { t = 0 } × F ) ∪ ( ∂ F ) t , t ∈ [0 , T ].( E, (cid:107) · (cid:107) ) is a normed space.For a function φ ( t, x, u, p, w ) : [0 , T ] × F × R m × R m × n × E → R k , where k = 1 , , . . .∂ x φ or φ x denotes the partial gradient with respect to x ∈ R n ; ∂ x i φ or φ x i denotes the partial derivative ∂∂x i φ ; ∂ x i x j φ or φ x i x j denotes the second partial derivative ∂ ∂x i ∂x j φ ; ∂ t φ or φ t denotes the partial derivative ∂∂t φ ; ∂ u φ denotes denotes the partial gradient with respect to u ∈ R m ; ∂ u i φ or φ u i denotes the partial derivative ∂∂u i φ ; ∂ p φ denotes denotes the partial gradient with respect to p ∈ R m × n ; ∂ p i φ or φ p i denotes the partial gradient with respect to the i th column p i of thematrix p ∈ R m × n ; ∂ w φ denotes denotes the partial Gˆateaux derivative of φ with respect to w ∈ E .ˆ µ ( s ), s (cid:62)
0, is a positive non-increasing continuous function. µ ( s ) and ˜ µ ( s ), s (cid:62)
0, are positive non-decreasing continuous functions. P ( s, r, t ) and ε ( s, r ), s, r, t (cid:62)
0, are positive and non-decreasing with respect toeach argument, whenever the other arguments are fixed. ϕ ( x ) is the initial condition. m is the number of equation in the system. M is the a priori bound on F T for the solution u to problem (2)-(4) (as definedin Remark 3). M is the a priori bound for ∂ x u on F T .ˆ M is the a priori bound for (cid:107) ϑ u (cid:107) E on F T . K is the common bound for the partial derivatives and the H¨older constants,mentioned in Assumption (A8), over the region F T × {| u | (cid:54) M } × {(cid:107) w (cid:107) E (cid:54) ˆ M } ×{| p | (cid:54) M } , as defined in Remark 5. K ξ,ζ is the constant defined in Assumption (A10).The H¨older space C β ( F ), β ∈ (0 , (cid:107) φ (cid:107) C β ( F ) = (cid:107) φ (cid:107) C ( F ) + [ φ (cid:48)(cid:48) ] β , where [ ˜ φ ] β = sup x,y ∈ F , < | x − y | < | ˜ φ ( x ) − ˜ φ ( y ) || x − y | β . (7) EVELINA SHAMAROVA AND RUI S´A PEREIRA
For a function ϕ ( x, ξ ) of more than one variable, the H¨older constant with respectto x is defined as [ ϕ ] xβ = sup x,x (cid:48) ∈ F , < | x − x (cid:48) | < | ϕ ( x, ξ ) − ϕ ( x (cid:48) , ξ ) || x − x (cid:48) | β , (8)i.e., it is understood as a function of ξ .The parabolic H¨older space C β , β ( F T ), β ∈ (0 , u ( t, x ) possessing the finite norm(9) (cid:107) u (cid:107) C β , β ( F T ) = (cid:107) u (cid:107) C , ( F T ) + sup t ∈ [0 ,T ] [ ∂ t u ] xβ + sup t ∈ [0 ,T ] [ ∂ xx u ] xβ + sup x ∈ F [ ∂ t u ] t β + sup x ∈ F [ ∂ x u ] t β + sup x ∈ F [ ∂ xx u ] t β . C β ,β ( F T ), β ∈ (0 , u ∈ C( F T ) possessing thefinite norm (cid:107) u (cid:107) C β ,β ( F T ) = (cid:107) u (cid:107) C( F T ) + sup t ∈ [0 ,T ] [ u ] xβ + sup x ∈ F [ u ] t β . C , ( F T ) and C , ( F T ) denotes the space of functions from C , ( F T ) andC , ( F T ), respectively, vanishing on ∂ F . H ( E, R m ) is the Banach space of bounded positively homogeneous maps E → R m with the norm (cid:107) φ (cid:107) H = sup {(cid:107) w (cid:107) E (cid:54) } | φ ( w ) | .The H¨older space C βb ( R n ), β ∈ (0 , (cid:107) φ (cid:107) C βb ( R n ) = (cid:107) φ (cid:107) C b ( R n ) + [ φ (cid:48)(cid:48) ] β , (10)where C b ( R n ) denotes the space of twice continuously differentiable functions on R n with bounded derivatives up to the second order. The second term in (10) isthe H¨older constant which is defined as in (7) but the domain F has to be replacedwith the entire space R n .Similarly, for a function ϕ ( x, ξ ), x ∈ R n , of more than one variable, the H¨olderconstant with respect to x is defined as in (8) but F should be replaced with R n .Further, the parabolic H¨older space C β , βb ([0 , T ] × R n ) is defined as the Ba-nach space of functions u ( t, x ) possessing the finite norm (cid:107) u (cid:107) C β , βb ([0 ,T ] × R n ) = (cid:107) u (cid:107) C , b ([0 ,T ] × R n ) + sup t ∈ [0 ,T ] [ ∂ t u ] xβ + sup t ∈ [0 ,T ] [ ∂ xx u ] xβ + sup x ∈ R n [ ∂ t u ] t β + sup x ∈ R n [ ∂ x u ] t β + sup x ∈ R n [ ∂ xx u ] t β , where C , b ([0 , T ] × R n ) denotes the space of bounded continuous functions whosemixed derivatives up to the second order in x ∈ R n and first order in t ∈ [0 , T ] arebounded and continuous on [0 , T ] × R n .We say that a smooth surface S ⊂ R n (or S ⊂ [0 , T ] × R n ) is of class C γ (resp.C γ ,γ ), where γ, γ , γ > x ∈ S , the surface S is represented as a graph offunction of class C γ (resp. C γ ,γ ). For more details on surfaces of the classes C γ and C γ ,γ , we refer the reader to [8] (pp. 9–10).Furthermore, we say that a piecewise smooth surface S ⊂ R n is of class C γ , γ >
1, if its each smooth components is of this class.
BSDES WITH JUMPS AND CLASSICAL SOLUTIONS TO NONLOCAL PDES 7
The H¨older norm of a function u on Γ T is defined as follows (cid:107) u (cid:107) C β , β (Γ T ) = max (cid:110) (cid:107) u (cid:107) C β ( F ) , (cid:107) u (cid:107) C β , β (( ∂ F ) T ) (cid:111) , where the norm (cid:107) u (cid:107) C β , β (( ∂ F ) T ) is defined in [8] (p. 10). However, since we restrictour consideration only to functions vanishing on the boundary ∂ F , we do not needthe details on the definition of H¨older norms on ( ∂ F ) T , i.e., in our case it alwaysholds that (cid:107) u (cid:107) C β , β (Γ T ) = (cid:107) u (cid:107) C β ( F ) . Remark . Some notation of this article is different than in the book of Ladyzhen-skaya et al. [8]. For reader’s convenience, we provide the correspondence of the mostimportant notation: Ω = F , S = ∂ F , S T = ( ∂ F ) T , Q T = F T , Γ T = Γ T , N = m . In this subsection, we obtain the maximum principle for problem (2)-(4) underAssumptions (A1)–(A4) below. Obtaining an a priori bound for the solution toproblem (2)-(4) is an essential step for obtaining other a priori bounds, as well asproving the existence of solution.We agree that the functions µ ( s ) and ˆ µ ( s ) in the assumptions below are non-decreasing and, respectively, non-increasing, continuous, defined for positive argu-ments, and taking positive values. Further, L E , c , c , c are non-negative constants.We assume the following. (A1) For all ( t, x, u ) ∈ F T × R m and ξ = ( ξ , . . . , ξ n ) ∈ R n ,ˆ µ ( | u | ) | ξ | (cid:54) n (cid:88) i,j =1 a ij ( t, x, u ) ξ i ξ j (cid:54) µ ( | u | ) | ξ | . (A2) The function ϑ u : F T → E is defined for each u ∈ C , ( F T ), and such thatsup F T (cid:107) e − λt ϑ u ( t, x ) (cid:107) E (cid:54) L E sup F T | e − λt u ( t, x ) | for all λ (cid:62) (A3) There exists a function ζ : R × R n → [0 , ∞ ), where R = F T × R m × R m × n × E , such that for all ( t, x, u, p, w ) ∈ R , ζ ( t, x, u, p, w,
0) = 0 and (cid:0) a ( t, x, u, p, w ) , u (cid:1) (cid:62) − c − c | u | − c (cid:107) w (cid:107) E − ζ ( t, x, u, p, w, p (cid:62) u ) . (A4) The function ϕ : F → R m is of class C β ( F ) with β ∈ (0 , Lemma . Assume (A1). If a twice continuously differentiable function ϕ ( x ) achieves a local maximum at x ∈ F , then for any ( t, u ) ∈ [0 , T ] × R m , (cid:88) i,j a ij ( t, x , u ) ϕ x i x j ( x ) (cid:54) . Proof.
For each ( t, u ) ∈ [0 , T ] × R m , we have n (cid:88) i,j =1 a ij ( t, x , u ) ϕ x i x j ( x ) = n (cid:88) i,j,k,l =1 ϕ y k y l ( x ) a ij ( t, x , u ) v ik v jl = n (cid:88) k =1 λ k ϕ y k y k ( x ) , where { v ij } is the matrix whose columns are the vectors of the orthonormal eigenba-sis of { a ij ( t, x , u ) } , ( y , · · · , y n ) are the coordinates with respect to this eigenbasis,and ( λ , · · · , λ n ) are the eigenvalues of { a ij ( t, x , u ) } .Note that by (A1), λ k = (cid:80) ni,j =1 a ij v ik v jk (cid:62) ˆ µ ( | u | ) >
0. Let us show that ϕ y k y k ( x ) (cid:54)
0. Since ϕ ( y , . . . , y n ) has a local maximum at x , then ϕ y k ( x ) = 0 EVELINA SHAMAROVA AND RUI S´A PEREIRA for all k . Suppose for an arbitrary fixed k , ϕ y k y k ( x ) >
0. Then, by the secondderivative test, the function ϕ ( y , . . . , y n ), considered as a function of y k while therest of the variables is fixed, would have a local minimum at x . The latter is notthe case. Therefore, ϕ y k y k ( x ) (cid:54)
0. The lemma is proved. (cid:3)
Lemma 2 below will be useful.
Lemma . For a function ϕ : F T → R , one of the mutually exclusive conditions1)–3) necessarily holds:1) sup F T ϕ ( t, x ) (cid:54) ;2) < sup F T ϕ ( t, x ) = sup Γ T ϕ ( t, x ) ;3) ∃ ( t , x ) ∈ (0 , T ] × F such that φ ( t , x ) = sup F T ϕ ( t, x ) > .Proof. The proof is straightforward. (cid:3)
Theorem . As-sume (A1)–(A4). If u ( t, x ) is a C , ( F T ) -solution to problem (2) - (4) , then sup F T | u ( t, x ) | (cid:54) e λT max (cid:8) sup F | ϕ ( x ) | , √ c (cid:9) , where λ = c + c L E + 1 . (11) Proof.
Let v ( t, x ) = u ( t, x ) e − λt . Then, v satisfies the equation − n (cid:88) i,j =1 a ij ( t, x, u ) v x i x j + e − λt a ( t, x, u, u x , ϑ u )+ a i ( t, x, u, u x , ϑ u ) v x i + λv + v t = 0 . Multiplying the above identity scalarly by v , and noting that ( v x i x j , v ) = ∂ x i x j | v | − ( v x i , v x j ), we obtain(12) − n (cid:88) i,j =1 a ij ( t, x, u ) ∂ x i x j | v | + e − λt ( a ( t, x, u, u x , ϑ u ) , v )+ n (cid:88) i,j =1 a ij ( t, x, u )( v x i , v x j ) + 12 n (cid:88) i =1 a i ( t, x, u, u x , ϑ u ) ∂ x i | v | + λ | v | + 12 ∂ t | v | = 0 , where u and v are evaluated at ( t, x ). If t = 0, then (11) follows trivially. Otherwise,for the function w = | v | , one of the conditions 1)–3) of Lemma 2 necessarily holds.Note that condition 1) is excluded. Furthermore, if 2) holds, thensup F T | u ( t, x ) | (cid:54) e λT sup F T | v ( t, x ) | (cid:54) e λT sup F | ϕ ( x ) | . (13)Suppose now that 3) holds, i.e., the maximum of | v | is achieved at some point( t , x ) ∈ (0 , T ] × F . Then, we have ∂ x w ( t , x ) = 0 and ∂ t w ( t , x ) (cid:62) . (14)By Lemma 1, the first term in (12) is non-negative at ( t , x ). Further, assumption(A1) and identities (14) imply that the third, fours, and the last term on the left-hand side of (12), evaluated at ( t , x ), are non-negative. Consequently, substituting v ( t , x ) = u ( t , x ) e − λt , we obtain e − λt (cid:0) a ( t , x , u x ( t , x ) , ϑ u ( t , x )) , u ( t , x ) (cid:1) + λ | v ( t , x ) | (cid:54) . (15)Further, we remark that v x ( t , x ) (cid:62) v ( t , x ) = w x ( t , x ) = 0. Therefore, by (A2), BSDES WITH JUMPS AND CLASSICAL SOLUTIONS TO NONLOCAL PDES 9 (16) 0 (cid:62) e − λt (cid:0) a ( t , x , u ( t , x ) , u x ( t , x ) , ϑ u ( t , x )) , u ( t , x ) (cid:1) + λ | v ( t , x ) | (cid:62) − c e − λt − c | v ( t , x ) | − c (cid:107) e − λt ϑ u ( t , x ) (cid:107) E − ζ ( . . . ,
0) + λ | v ( t , x ) | (cid:62) − c − c | v ( t , x ) | − c L E | v ( t , x ) | + λ | v ( t , x ) | . Picking λ = c + c L E + 1, we obtain that | v ( t , x ) | (cid:54) c . Therefore,sup F T | u ( t, x ) | (cid:54) √ c e λT . The above inequality together with (13) implies (11). (cid:3)
Corollary . Let assumptions of Theorem 1 hold. If, in (A3), c = 0 , then sup F T | u ( t, x ) | (cid:54) e λT sup F | ϕ ( x ) | with λ = c + c L E + 1 . (17) If c > , then sup F T | u ( t, x ) | (cid:54) e ( λ + c ) T max { sup F | ϕ ( x ) | , } . (18) Remark . Let M denote the biggest of right-hand sides of (11) and (18). ByTheorem 1 and Corollary 1, M is an a priori bound for | u ( t, x ) | on F T . Every-where below throughout the text, by M we understand the quantity defined above.Furthermore, by (A2), L E M is a bound for (cid:107) ϑ u ( t, x ) (cid:107) E , which we denote by ˆ M . Remark . The function ζ was added to the right-hand side of the inequality in(A3) just for the sake of generality (it is not present in the similar assumption in[8]). Indeed, the presence of this function does not give any extra work in the proof. We now formulate assumptions (A5)–(A7), which, together with previously in-troduced assumptions (A1)–(A4), will be necessary for obtaining an a priori boundfor the gradient ∂ x u of the solution u to problem (2)-(4). Obtaining the gradientestimate is crucial for obtaining estimates of H¨older norms, as well as for the proofof existence. Everywhere below, R and R are regions defined as follows R = F T × {| u | (cid:54) M } × R m × n × {(cid:107) w (cid:107) E (cid:54) ˆ M } ; R = F T × {| u | (cid:54) M } . (19)Further, the functions ˜ µ ( s ), η ( s, r ), P ( s, r, t ), and ε ( s, r ) in the assumptions beloware continuous, defined for positive arguments, taking positive values, and non-decreasing with respect to each argument, whenever the other arguments are fixed.Assumptions (A5)–(A7) read: (A5) For all ( t, x, u, p, w ) ∈ R it holds that | a i ( t, x, u, p, w ) | (cid:54) η ( | u | , (cid:107) w (cid:107) E )(1 + | p | ) , i ∈ { , . . . , n } , and | a ( t, x, u, p, w ) | (cid:54) (cid:0) ε ( | u | , (cid:107) w (cid:107) E ) + P ( | u | , | p | , (cid:107) w (cid:107) E ) (cid:1) (1 + | p | ) , where lim r →∞ P ( s, r, q ) = 0 and 2( M + 1) ε ( M, ˆ M ) (cid:54) ˆ µ ( M ). (A6) a ij , a and a i are continuous on R ; ∂ x a ij and ∂ u a ij exist and are continuouson R ; moreover, max (cid:8)(cid:12)(cid:12) ∂ x a ij ( t, x, u ) (cid:12)(cid:12) , (cid:12)(cid:12) ∂ u a ij ( t, x, u ) (cid:12)(cid:12)(cid:9) (cid:54) ˜ µ ( | u | ). (A7) The boundary ∂ F is of class C β . In Theorem 2 below, we obtain the gradient estimate for a C , ( F T )-solution u ( t, x )of problem (2)-(4). The main idea is to freeze ϑ u in the coefficients a i and a andapply the result of Ladyzhenskaya et al [8] on the gradient estimate of a classicalsolution to a system of quasilinear parabolic PDEs. Theorem . (Gradient estimate) Let (A1)–(A7) hold, and let u ( t, x ) be a C , ( F T ) -solution to problem (2) – (4) . Further let M be the a priori bound for | u ( t, x ) | on F T whose existence was established by Theorem 1. Then, there exists a constant M > , depending only on M , ˆ M , sup F | ∂ x ϕ | , µ ( M ) , ˆ µ ( M ) , ˜ µ ( M ) , η ( M, ˆ M ) , sup q (cid:62) P ( M, q, ˆ M ) , and ε ( M, ˆ M ) such that sup F T | ∂ x u | (cid:54) M . (20) Proof.
In (2), we freeze ϑ u in the coefficients a i and a . Non-local PDE (2) is,therefore, reduced to the following quasilinear parabolic PDE with respect to v (21) − n (cid:88) i,j =1 a ij ( t, x, v ) ∂ x i x j v + n (cid:88) i =1 a i ( t, x, v, ∂ x v, ϑ u ( t, x )) ∂ x i v + a ( t, x, v, ∂ x v, ϑ u ( t, x )) + ∂ t v = 0with initial-boundary condition (4). Since ˆ M is an a priori bound for (cid:107) ϑ u ( t, x ) (cid:107) E (see Remark 3), we are in the assumptions of Theorem 6.1 from [8] (p. 592) on thegradient estimate for solutions of PDEs of form (21). Indeed, assumptions (A1) and(A5) are the same as in Theorem 6.1, and (A6) immediately implies the continuityof functions ( t, x, v, p ) → a ( t, x, v, p, ϑ u ( t, x )) and ( t, x, v, p ) → a i ( t, x, v, p, ϑ u ( t, x ))in the region F T × {| v | (cid:54) M } × R m × n . Further, (A5) implies conditions (6.3) on p.588 and inequality (6.7) on p. 590 of [8]. It remains to note that by (A3), (cid:0) a ( t, x, v, p, ϑ u ( t, x )) , v ) (cid:62) − c (cid:48) − c | v | + ζ ( t, x, v, p, ϑ u ( t, x ) , p (cid:62) v ) , where c (cid:48) = c + c ˆ M . Therefore, by Corollary 1, any solution v ( t, x ) of (21) satisfiesthe estimate sup F T | v ( t, x ) | (cid:54) e ( c +1+ c (cid:48) ) T max (cid:8) sup F | ϕ ( x ) | , (cid:9) (cid:54) M .Since v ( t, x ) = u ( t, x ) is a C , ( F T )-solution to (21), then by Theorem 6.1 of [8],estimate (20) holds true. By the same theorem, the constant M only depends on M ,sup F | ∂ x ϕ | , µ ( M ), ˆ µ ( M ), ˜ µ ( M ), η ( M, ˆ M ), sup q (cid:62) P ( M, q, ˆ M ), and ε ( M, ˆ M ). (cid:3) ∂ t u Now we complete the set of assumptions (A1)–(A7) with assumptions (A8)–(A10) below. All together, these assumptions are necessary to obtain an a prioribound for the time derivative ∂ t u which is crucial for proving that any C , ( F T )-solution to problem (2)–(4) belongs to class C β , β ( F T ) and obtaining a boundfor the C β , β ( F T )-norm of this solution. The region R is defined, as before, by(19), and the region R is defined as follows R = F T × {| u | (cid:54) M } × {| p | (cid:54) M } × {(cid:107) w (cid:107) E (cid:54) ˆ M } . (22)Assumptions (A8)–(A10) read: (A8) ∂ t a ij , ∂ uu a ij , ∂ ux a ij , ∂ xt a ij , ∂ ut a ij exist and are continuous on R ; ∂ t a , ∂ u a , ∂ p a , ∂ w a , ∂ t a i , ∂ u a i , ∂ p a i , ∂ w a i exist and are continuous and bounded on R ; a and a i are β -H¨older continuous in x , β ∈ (0 , w with the H¨older and Lipschitz constants bounded over R . BSDES WITH JUMPS AND CLASSICAL SOLUTIONS TO NONLOCAL PDES 11 (A9)
For each u ∈ C , ( F T ), ∂ t ϑ u and ∂ x ϑ u exist and are continuous andbounded; moreover, the bounds for (cid:107) ∂ t ϑ u (cid:107) and (cid:107) ∂ x ϑ u (cid:107) only depend onthe bounds for | ∂ t u ( t, x ) | and | ∂ x u ( t, x ) | in F T . (A10) For all u ∈ C , ( F T ), ( t, x ) ∈ F T , it holds that ϑ u ( t + ∆ t, x ) − ϑ u ( t, x )∆ t = ˆ ϑ v ( t, x ) + ζ u,u x ( t, x ) v ( t, x ) + ξ u,u x ( t, x ) , (23) where v ( t, x ) = (∆ t ) − (cid:0) u ( t +∆ t, x ) − u ( t, x ) (cid:1) , ζ u,u x , ξ u,u x are bounded func-tions with values in L ( R m , E ) and E , respectively, depending non-locally on u and u x (their common bound will be denoted by K ξζ ), and ˆ ϑ v : F T → E ,defined for each v ∈ C , ( F T ), is such that for all α > τ ∈ (0 , T ), (cid:90) F ατ ( | v | ) (cid:107) ˆ ϑ v ( t, x ) (cid:107) E dt dx (cid:54) ˆ L E (cid:16) (cid:90) F ατ ( | v | ) | v ( t, x ) | dtdx + α λ ( F ατ ) (cid:17) , (24) where ˆ L E > (cid:107) u (cid:107) C , ( F T ) , F ατ ( | v | ) = { ( t, x ) ∈ F τ : | v ( t, x ) | > α } , and λ is the Lebesgue measure on R n +1 . Remark . The common bound over R for the partial derivatives and the H¨olderconstants mentioned in assumption (A8) and related to the functions a and a i willbe denoted by K . Remark . According to the results of [18] (p. 484), for locally Lipschitz mappingsin normed spaces, the Gˆateaux and Hadamard directional differentiabilities areequivalent. Moreover, the local Lipschitz constant of a function is the same as theglobal Lipschitz constant of its Gˆateaux derivative. Thus, under (A8), the chainrule holds for the Gˆateaux derivatives ∂ w a and ∂ w a i , which, moreover, are globallyLipschitz and positively homogeneous.The following below maximum principle for non-local linear-like parabolic PDEswritten in the divergence form, is crucial for obtaining the a priori bound for ∂ t u .Consider the following system of non-local PDEs in the divergence form(25) ∂ t u − n (cid:88) i =1 ∂ x i (cid:104) n (cid:88) j =1 ˆ a ij ( t, x ) ∂ x j u + A i ( t, x ) u + f i ( t, x ) (cid:105) + n (cid:88) i =1 B i ( t, x ) ∂ x i u + A ( t, x ) u + C ( t, x ) (cid:0) ˆ ϑ u ( t, x ) (cid:1) + f ( t, x ) = 0 , u (0) = u , where ˆ a ij : F T → R , A i : F T → R m × m , B i : F T → R m × m , f i : F T → R m ,i, j = 1 , . . . , n , A : F T → R m × m , f : F T → R m , and C : F T → H ( E, R m ), where H ( E, R m ) is the Banach space of bounded positively homogeneous maps E → R m with the norm (cid:107) φ (cid:107) H = sup {(cid:107) w (cid:107) E (cid:54) } | φ ( w ) | . In (25), the function u together withits partial derivatives, as usual, is evaluated at ( t, x ) and ˆ ϑ u ( t, x ) is an E -valuedfunction built via u and satisfying inequality (24). Remark that all terms in (25),except the term containing ˆ ϑ u ( t, x ), are linear in u The lemma below, which is a version of the integration-by-parts formula, can befound in [8] (p. 60).
Lemma . Let f and g be real-valued functions from the Sobolev spaces W ,p ( G ) and W ,q ( G ) ( p + q (cid:54) n ) , respectively, where G ⊂ R n is a bounded domain.Assume that the boundary ∂ G is piecewise smooth and that f g = 0 on ∂ G . Then, (cid:90) G f ∂ x i g dx = − (cid:90) G g ∂ x i f dx. Further, for each τ, τ (cid:48) ∈ [0 , T ], τ < τ (cid:48) , we define the squared norm (cid:107) v (cid:107) τ,τ (cid:48) = sup t ∈ [ τ,τ (cid:48) ] (cid:107) v ( t, · ) (cid:107) L ( F ) + (cid:107) ∂ x v (cid:107) L ( F τ,τ (cid:48) ) , (26)where F τ,τ (cid:48) = F × [ τ, τ (cid:48) ]. Furthermore, for an arbitrary real-valued function φ on F T and a number α >
0, we define φ α = ( φ − α ) + and F ατ ( φ ) = { ( t, x ) ∈ F τ : φ > α } ,where τ ∈ (0 , T ]. The following result was obtained in [8] (Theorem 6.1, p. 102). Itwill be used in Lemma 4. Proposition . Let φ ( t, x ) be a real-valued function of class C( F τ ) such that sup ( ∂ F ) τ φ (cid:54) ˆ α, where ˆ α (cid:62) . Assume for all α (cid:62) ˆ α and for a positive constant γ , it holds that (cid:107) φ α (cid:107) ,τ (cid:54) γα (cid:112) λ n +1 ( F ατ ( φ )) , where λ n +1 is the Lebesgue measureon R n +1 . Then, there exists a constant δ > , depending only on n , such that sup F τ φ ( t, x ) (cid:54) α (cid:0) δ γ n τ λ n ( F ) (cid:1) . Remark . We attributed the values r = q = 4, κ = 1 for the space dimensions n = 1 , r = q = 2 + n − , κ = n − for n (cid:62) r , q , and κ appearing in the original version of Theorem 6.1 in [8] (p. 102), since for ourapplication we do not need Theorem 6.1 in the most general form. Also, we remarkthat by our choice of the parameters, 1 + κ (cid:54) n for all space dimensions n . Lemma . Assume the coefficients ˆ a ij , A i , B i , f i , f , A , and C are of class C( F T ) and that (cid:80) ni,j =1 ˆ a ij ( t, x ) ξ i ξ j (cid:62) (cid:37) (cid:107) ξ (cid:107) for all ( t, x ) ∈ F T , ξ ∈ R m , and for someconstant (cid:37) > . Let u ( t, x ) be a generalized solution of problem (25) which is ofclass C , ( F T ) and such that ˆ ϑ u satisfies (24) . Further let v = | u | . Then, thereexist a number τ ∈ (0 , T ] and a constant γ > , where τ depends on the commonbound A over F T for the coefficients A i , B i , f i , f , A , C , and also on ˆ L E , (cid:37) , n , and λ n ( F ) , and γ depends on the same quantities as τ and on sup F | u | , such that (cid:107) v α (cid:107) ,τ (cid:54) γ α (cid:112) λ n +1 ( F ατ ( v )) for all α (cid:62) sup F | u | + 1 . (27) Proof.
Let τ ∈ (0 , T ]. Multiplying PDE (25) scalarly by a W ,p ( F τ )-function η ( t, x )( p >
1) vanishing on ∂ F τ and applying the integration-by-parts formula (Lemma3), we obtain(28) (cid:90) F τ (cid:104) ( u t ( t, x ) , η ( t, x )) + n (cid:88) i =1 (cid:16) n (cid:88) j =1 ˆ a ij ( t, x ) u x j + A i ( t, x ) u + f i ( t, x ) , η x i ( t, x ) (cid:17) + (cid:16) n (cid:88) i =1 B i ( t, x ) u x i + A ( t, x ) u + f ( t, x ) + C ( t, x ) (cid:0) ˆ ϑ u ( t, x ) (cid:1) , η ( t, x ) (cid:17)(cid:105) dtdx = 0 . For simplicity of notation, we write F ατ for F ατ ( v ). Define η ( t, x ) = 2 u ( t, x ) v α ( t, x )and note that v α and its derivatives vanish outside of F ατ . Further, since ( ∂ t u, η ) =2( ∂ t u, u ) v α = ( ∂ t v ) v α = ∂ t ( v α ) v α = ∂ t ( v α ) , we rewrite (28) as follows12 (cid:90) F ( v α ) (cid:12)(cid:12)(cid:12) τ dx + 2 (cid:90) F ατ (cid:104)(cid:16) n (cid:88) i =1 (cid:16) n (cid:88) j =1 ˆ a ij ( t, x ) u x j + A i ( t, x ) u + f i ( t, x ) (cid:17) , ( uv α ) x i (cid:17) + 2 (cid:16) n (cid:88) i =1 B i ( t, x ) u x i + A ( t, x ) u + C ( t, x )( ˆ ϑ u ) + f ( t, x ) , uv α (cid:17)(cid:105) dtdx = 0 . (29) BSDES WITH JUMPS AND CLASSICAL SOLUTIONS TO NONLOCAL PDES 13
Note that the following inequalities hold on F ατ :2 n (cid:88) i,j =1 ˆ a ij ( t, x )( u x j , ( uv α ) x i ) = 2 n (cid:88) i,j =1 ˆ a ij ( t, x )( u x i , u x j ) v α + n (cid:88) i,j =1 ˆ a ij ( t, x ) v x j v αx i (cid:62) (cid:37) | u x | v α + (cid:37) ( v αx ) ;2( A i u, ( uv α ) x i ) (cid:54) | A i | ( | u || u x i | v α + v | v αx i | ) (cid:54) (cid:15) | A i | vv α + 1 (cid:15) | A i | v + (cid:15)v α | u x i | + (cid:15) | v αx i | (cid:54) (cid:15) | A i | v + (cid:15)v α | u x i | + (cid:15) | v αx i | ;2( f i , ( uv α ) x i ) (cid:54) | f i | ( | u x i | v α + | u || v αx i | ) (cid:54) (cid:15) | f i | ( v α + v ) + (cid:15) (cid:2) v α | u x i | + | v αx i | (cid:3) ;2( B i u x i , uv α ) (cid:54) (cid:15) | B i | vv α + (cid:15) | u x i | v α ; 2( Au, uv α ) (cid:54) | A | v ;2( f, uv α ) (cid:54) | f | v (cid:54) | f | (1 + v );2 (cid:90) F ατ ( C ( ˆ ϑ u ) , uv α ) dtdx (cid:54) A (cid:90) F ατ (cid:0) (cid:107) ˆ ϑ u (cid:107) E + v + ( v α ) (cid:1) dtdx (cid:54) ˆ A (cid:104) (cid:90) F ατ v dtdx + α λ ( F ατ ) (cid:105) , where (cid:15) > A beinga constant that depends only on A and the constant ˆ L E from (24). By virtue ofthese inequalities, from (29) we obtain12 (cid:90) F ( v α ( τ, x )) dx + (cid:37) (cid:90) F ατ { | u x | v α + ( v αx ) } dxdt (cid:54) (cid:90) F ( v α (0 , x )) dx + (cid:90) F ατ (cid:0) ˜ A (cid:15) (1 + v ) + 3 (cid:15) | u x | v α + 2 (cid:15) | v αx | (cid:1) dtdx + ˆ A α λ ( F ατ ) , where ˜ A (cid:15) = (cid:15) − sup F τ (cid:16) (cid:80) ni =1 | A i | + (cid:80) ni =1 | f i | + (cid:80) ni =1 | B i | + (cid:15) | A | + (cid:15) | f | (cid:17) . Picking (cid:15) = (cid:37) and defining ˜ (cid:37) = min( , (cid:37) ), we obtain˜ (cid:37) (cid:16) (cid:90) F ( v α ( τ, x )) dx + (cid:90) F ατ ( v αx ) dtdx (cid:17) (cid:54) (cid:90) F ( v α (0 , x )) dx + ˜ A (cid:37) (cid:90) F ατ (1 + v ) dtdx + ˆ A α λ n +1 ( F ατ ) . Introducing ¯ A = 4 ˜ A (cid:37) + 2 ˆ A , by (26), we obtain˜ (cid:37) (cid:107) v α (cid:107) ,τ (cid:54) (cid:107) v α ( x, (cid:107) L ( F ) + 12 ¯ A (cid:0) (1 + α ) λ n +1 ( F ατ ) + (cid:107) v α (cid:107) L ( F ατ ) (cid:1) since (cid:107) v (cid:107) L ( F ατ ) (cid:54) (cid:107) v α (cid:107) L ( F ατ ) + 2 α λ n +1 ( F ατ ). Finally, for all α (cid:62) sup F | u | + 1,it holds that ˜ (cid:37) (cid:107) v α (cid:107) ,τ (cid:54) ¯ A (cid:0) α λ n +1 ( F ατ ) + (cid:107) v α (cid:107) L ( F ατ ) (cid:1) . Further, from inequality(3.7) (p. 76) in [8] it follows that (cid:107) v α (cid:107) L ( F τ ) (cid:54) γλ n +1 ( F ατ ) n +2 (cid:107) v α (cid:107) ,τ , where γ > n . Since, by Fubini’s theorem, λ n +1 ( F ατ ) = (cid:82) τ λ n ( x ∈ F : ( t, x ) ∈ F ατ ) dt , then λ n +1 ( F ατ ) (cid:54) τ λ n ( F ). Picking τ sufficiently small, we obtain that ¯ A γ (cid:0) τ λ n ( F ) (cid:1) n +2 (cid:54) ˜ (cid:37)/
2. This implies (27) with γ = (cid:0) A ˜ (cid:37) − (cid:1) . (cid:3) Theorem . Let assumptions of Lemma 4 be fulfilled. Further let a solution u to problem (25) vanishes on ∂ F . Then sup F T | u | is bounded by a constant depending only on A , ν , n , T , λ n ( F ) , ˆ L E , and linearly depending on sup F | u | .Proof. It follows from Proposition 1 and Lemma 4 that there exist a bound forsup F τ | u | depending only on A , ν , n , λ n ( F ), ˆ L E , and sup F | u | , where τ ∈ (0 , T ] issufficiently small and depends on A , ν , n , λ n ( F ), and ˆ L E . Remark that by Propo-sition 1, the above bound is a multiple of ˆ α = sup F | u | + 1. It is important to em-phasize that τ does not depend on sup F | u | . By making the time change t = t − τ in problem (25), we obtain a bound for sup F τ, τ | u | depending on A , ν , n , λ n ( F ),and sup F | u ( τ, x ) | , where the latter quantity was proved to have a bound which isa multiple of sup F | u | + 1. On the other hand, by Proposition 1, the bound forsup F τ, τ | u | is a multiple of sup F | u ( τ, x ) | + 1. In a finite number of steps, dependingon T , we obtain a bound for | u | in the entire domain F T . This bound will dependlinearly on sup F | u | by Proposition 1. The theorem is proved. (cid:3) Since the maximum principle for systems of non-local PDEs of form (25) isobtained, we can prove the theorem on existence of an a priori bound for ∂ t u . Theorem . Let (A1)–(A10) hold, and let u ( t, x ) be a C , -solution to problem (2) – (4) . Then, there exists a constant M , depending only on M , ˆ M , M , K , K ξ,ζ , T , λ n ( F ) , ˆ L E , (cid:107) ϕ (cid:107) C β ( F ) , such that sup F T | ∂ t u | (cid:54) M . Proof.
Rewrite (2) in the divergence form, i.e., ∂ t u − n (cid:88) i =1 ∂ x i (cid:104) n (cid:88) j =1 a ij ( t, x, u ) u x j (cid:105) + ˆ a ( t, x, u, u x , ϑ u ) = 0 withˆ a ( t, x, u, p, w ) = (cid:80) ni =1 a i ( t, x, u, p, w ) p i + a ( t, x, u, p, w ) + (cid:80) ni,j =1 ∂ x i a ij ( t, x, u ) p j + (cid:80) ni,j =1 ( ∂ u a ij ( t, x, u ) , p i ) p j , where p i is the i th column of the matrix p , and u , u x and ϑ u are evaluated at ( t, x ). Further, we define v ( t, x ) = (∆ t ) − (cid:0) u ( t + ∆ t, x ) − u ( t, x ) (cid:1) and t (cid:48) = t + ∆ t , where ∆ t is fixed. If t = 0, we assume that ∆ t >
0, and if t = T , then ∆ t <
0. The PDE for the function v takes form (25) with ˆ a ij ( t, x ) = a ij ( t (cid:48) , x, u ( t (cid:48) , x )); A i ( t, x ) = (cid:80) nj =1 u x j ( t, x ) (cid:82) dλ ∂ u a ij ( t, x, λu ( t (cid:48) , x ) + (1 − λ ) u ( t, x )) (cid:62) ; f i ( t, x ) = (cid:80) nj =1 (cid:82) dλ ∂ t a ij ( t + λ ∆ t, x, u ( t (cid:48) , x )) u x j ( t, x ); f ( t, x ) = (cid:82) dλ ∂ t ˆ a ( t + λ ∆ t, x, u ( t (cid:48) , x ) , u x ( t (cid:48) , x ) , ϑ u ( t (cid:48) , x ))+ (cid:82) dλ ∂ w ˆ a ( t, x, u ( t, x ) , u x ( t, x ) , λϑ u ( t (cid:48) , x ) + (1 − λ ) ϑ u ( t, x )) (cid:0) ξ u,u x ( t, x ) (cid:1) ; A ( t, x ) = (cid:82) dλ ∂ u ˆ a ( t, x, λu ( t (cid:48) , x ) + (1 − λ ) u ( t, x ) , u x ( t (cid:48) , x ) , ϑ u ( t (cid:48) , x ))+ (cid:82) dλ ∂ w ˆ a ( t, x, u ( t, x ) , u x ( t, x ) , λϑ u ( t (cid:48) , x ) + (1 − λ ) ϑ u ( t, x )) ζ u,u x ( t, x ); B i ( t, x ) = (cid:82) dλ ∂ p i ˆ a ( t, x, u ( t, x ) , λu x ( t (cid:48) , x ) + (1 − λ ) u x ( t, x ) , ϑ u ( t (cid:48) , x )); C ( t, x ) = (cid:82) dλ ∂ w ˆ a ( t, x, u ( t, x ) , u x ( t, x ) , λϑ u ( t (cid:48) , x ) + (1 − λ ) ϑ u ( t, x )) . Above, ξ u,u x and ζ u,u x are bounded functions from representation (23). Remarkthat the above coefficients are bounded by a constant, say A , depending on M ,ˆ M , M , K , and K ξ,ζ (where the latter is the bound for ξ u,u x and ζ u,u x defined in(A10)). By Theorem 3, sup F T | v | is bounded by a constant depending only on A , T , BSDES WITH JUMPS AND CLASSICAL SOLUTIONS TO NONLOCAL PDES 15 λ n ( F ), ˆ L E , and sup F | v (0 , x ) | . Moreover, the dependence on sup F | v (0 , x ) | is linear.Letting ∆ t go to zero, we obtain that the bound for | ∂ t u | on F T depends only on A , T , λ n ( F ), ˆ L E , and sup F | ∂ t u (0 , x ) | . Finally, equation (2) implies that | ∂ t u (0 , x ) | can be estimated via (cid:107) ϕ (cid:107) C ( F ) , and the bounds for the coefficients a ij , a i , and a over R , defined by (22). Further, by virtue of (A1) and (A5), the latter boundscan be estimated by a constant depending only on M , ˆ M , and M . The theorem isproved. (cid:3) In this subsection, we prove that any C , -solution of problem (2)–(4) is, in fact,of class C β , β . Moreover, we obtain a bound for its C β , β -norm. Unlike thebound for the gradient, this bound cannot be obtained directly from the results ofLadyzenskaya et al [8] by freezing ϑ u . Our proof essentially relies on the estimateof the time derivative ∂ t u obtained in the previous subsection. Theorem . (H¨older norm estimate) Let (A1)–(A10) hold, and let u ( t, x ) be a C , ( F T ) -solution to problem (2) – (4) . Further let M and M be the a priori boundsfor u and, respectively, ∂ x u on F T (whose existence was established by Theorems 1and 2). Then, u ( t, x ) is of class C β , β ( F T ) . Moreover, there exists a constant M > depending only on M , ˆ M , M , K , K ξ,ζ , T , λ n ( F ) , ˆ L E , (cid:107) ϕ (cid:107) C β ( F ) , andon the C β -norms of the functions defining the boundary ∂ F , such that (cid:107) u (cid:107) C β , β ( F T ) (cid:54) M . Proof.
Freeze the function ϑ u in the coefficients a i and a , and consider the followingPDE with respect to v − n (cid:88) i,j =1 a ij ( t, x, v ) ∂ x i x j v + ˜ a ( t, x, v, ∂ x v ) + ∂ t v = 0 , (30)where ˜ a ( t, x, v, p ) = a ( t, x, v, p, ϑ u ( t, x ))+ (cid:80) ni =1 a i ( t, x, v, p, ϑ u ( t, x )) p i . Let us provethat the coefficients of (30) satisfy the assumptions of Theorem 5.2 from [8] (p. 587)on the H¨older norm estimate. First we show that the assumptions on the continuityof the partial derivatives ∂ t ˜ a , ∂ u ˜ a , ∂ p ˜ a and on the β -H¨older continuity of ˜ a in x ,mentioned in the formulation of Theorem 5.2 in [8], are fulfilled. Indeed, they followfrom (A8) and (A9). To see this, we first note that a and a i depend on t and x not just via their first two arguments but also via the function ϑ u ( t, x ) (assumedknown a priori) whose differentiability in t and x follows from (A9). Therefore, by(A8) and (A9), a and a i are β -H¨older continuous in x and differentiable in t .Further, Theorem 5.2 of [8] introduces a common bound (denote it by C ) for thepartial derivatives ∂ t ˜ a , ∂ u ˜ a , ∂ p ˜ a and the H¨older constant [˜ a ] xβ which, in case of [8],exists due to the continuity of the above functions on F T × {| u | (cid:54) M } × {| p | (cid:54) M } .However, in our case, the expression for ∂ t ˜ a will contain ∂ t ϑ u , and the expression for[˜ a ] xβ will contain ∂ x ϑ u . Therefore, by (A9), the bound C , required for the applicationof Theorem 5.2, will depend on M and M , i.e., the bounds for ∂ x u and ∂ t u . Thatis why the existence of a bound for ∂ t u is indispensable and must be obtained inadvance.The verification of the rest of the assumptions of Theorem 5.2 in [8] is straight-forward and follows from assumptions (A1), (A4), (A7), and (A8). Since v = u is aC , ( F T )-solution of problem (30)-(4), by aforementioned Theorem 5.2, u belongs to class C β , β ( F T ), and its H¨older norm (cid:107) u (cid:107) C β , β ( F T ) is bounded by a constant M , depending on the constants specified in the formulation of this theorem. (cid:3) The rest of this subsection deals with estimates of other H¨older norms of thesolution u under assumptions that do not require the a priori bound M for ∂ t u .These estimates will be useful for the proof of existence of solution to Cauchyproblem (2)–(6). The need of these bounds comes from the fact that M dependson λ n ( F ), the Lebesgue measure of the domain F . Theorem . Assume (A1)–(A7). Let u ( t, x ) be a generalized C , ( F T ) -solution toequation (2) such that | u | (cid:54) M and | ∂ x u | (cid:54) M on F T . Then, there exists a number α ∈ (0 , β ) and a constant M , both depending only on M , M , ˆ M , β , n , m , and sup F (cid:107) ϕ (cid:107) C β ( F ) such that (cid:107) u (cid:107) C α ,α ( F T ) (cid:54) M . Proof.
Freeze the functions u , ∂ x u , and ϑ u inside the coefficients a ij , a i , and a , andconsider the linear PDE with respect to v∂ t v − n (cid:88) i,j =1 ˜ a ij ( t, x ) ∂ x i x j v + n (cid:88) i =1 ˜ a i ( t, x ) ∂ x i v + ˜ a ( t, x ) = 0 with(31)˜ a i ( t, x ) = a i ( t, x, u, ∂ x u, ϑ u ) , ˜ a ( t, x ) = a ( t, x, u, ∂ x u, ϑ u ) , ˜ a ij ( t, x ) = a ij ( t, x, u ) , where v , u , ∂ x u , and ϑ u are evaluated at ( t, x ). Note that by (A1), (A5), and (A6), a ij , ∂ x a ij , ∂ u a ij , a i , and a are bounded in the region R , defined by (22), and thecommon bound depends on M , M , and ˆ M . The existence of the bound M followsnow from Theorem 3.1 of [8] (p. 582). (cid:3) Theorem . Assume (A1)–(A7). Further, assume the following conditions aresatisfied in the region R , defined by (22) :(i) a ij , a i , a are H¨older continuous in t, x, u, p , with exponents β , β, β, β , re-spectively, and, moreover, locally Lipschitz and Gatˆeaux differentiable w ;all H¨older and Lipschitz constants are bounded (say, by a constant M );(ii) For any C , ( F T ) -solution u ( t, x ) to problem (2) – (4) and for some β (cid:48) ∈ (0 , β ) , the bound for [ ϑ u ] t β (cid:48) is determined only by the bound for [ u ] t β (cid:48) and M ; and the bound for [ ϑ u ] xβ (cid:48) is determined only by M .Let u ( t, x ) be a C , ( F T ) -solution to equation (2) such that | u | (cid:54) M and | ∂ x u | (cid:54) M on F T , and let G ⊂ F be a strictly interior open domain. Then, there exista number α ∈ (0 , β ∧ β (cid:48) ) and a constant M , both depending only on M , M , ˆ M , M , (cid:107) ϕ (cid:107) C β ( F ) , and the distance between G and ∂ F , such that u is of class C α , α ( G T ) , and (cid:107) u (cid:107) C α , α ( G T ) (cid:54) M . Proof.
Freeze the function ϑ u in the coefficients a i , and a , and consider PDE (30)with respect to v . Let α be the smallest of β (cid:48) and the exponent whose existencewas established by Theorem 6. Assumptions (i) and (ii) imply that the coefficient˜ a in PDE (30) is H¨older continuous in t, x, u , and p with exponents α , α, α , and α ,respectively. Moreover, the H¨older constants are bounded and their common bounddepends on M , M , and M . The constant M , in turn, depends on M , M , ˆ M , BSDES WITH JUMPS AND CLASSICAL SOLUTIONS TO NONLOCAL PDES 17 β , and sup F (cid:107) ϕ (cid:107) C β ( F ) . Thus, by Theorem 5.1 of [8] (p. 586), the solution u is ofclass C α , α ( G T ) and the bound for the norm (cid:107) u (cid:107) C α , α ([0 ,T ] × G ) depends onlyon M , M , ˆ M , M , sup F (cid:107) ϕ (cid:107) C α ( F ) , and the distance between G and ( ∂ F ) T . Thetheorem is proved. (cid:3) To obtain the existence and uniqueness result for problem (2)–(4), we need thetwo additional assumptions below: (A11)
The following compatibility condition holds for x ∈ ∂ F : − n (cid:88) i,j =1 a ij (0 , x, ∂ x i x j ϕ ( x ) + n (cid:88) i =1 a i (0 , x, , ∂ x ϕ ( x ) , ϑ ϕ (0 , x )) ∂ x i ϕ ( x )+ a (0 , x, , ∂ x ϕ ( x ) , ϑ ϕ (0 , x )) = 0 . (A12) For any u, u (cid:48) ∈ C , ( F T ), it holds that ϑ u ( t, x ) − ϑ u (cid:48) ( t, x ) = ˜ ϑ u − u (cid:48) ( t, x ) + ς u,u (cid:48) ,u x ,u (cid:48) x ( t, x )( u ( t, x ) − u (cid:48) ( t, x )) , (32) where ς u,u (cid:48) ,u x ,u (cid:48) x : F T → L ( R m , E ) is bounded and may depend non-locallyon u , u (cid:48) , u x , and u (cid:48) x ; ˜ ϑ v : F T → E is defined for each v ∈ C , ( F T ) andsatisfies (A2) (in the place of ϑ u ).Lemma 5 below is a version of the maximum principle for non-local linear-likeparabolic PDEs which will be used to prove the uniqueness. Lemma . Let u ( t, x ) be a C , ( F T ) -solution to the following non-local initial-boundary value problem ∂ t u − n (cid:80) i,j =1 ˜ a ij ( t, x ) ∂ x i x j u + n (cid:80) i =1 B i ( t, x ) ∂ x i u + A ( t, x ) u + C ( t, x ) (cid:0) ˜ ϑ u (cid:1) = f ( t, x ) ,u (0 , x ) = u ( x ) , x ∈ F , u ( t, x ) = 0 ( t, x ) ∈ ( ∂ F ) T , (33) where ˜ a ij : F T → R , B i : F T → R m × m , A : F T → R m × m , f : F T → R m , and C : F T → H ( E, R m ) are of class C( F T ) , and (cid:80) ni,j =1 ˜ a ij ( t, x ) ξ i ξ j (cid:62) ρ (cid:107) ξ (cid:107) for all ( t, x ) ∈ F T , ξ ∈ R m , and for some ρ > . Further, assume that (A2) is fulfilled for ˜ ϑ u : F T → E , Then, sup F T | u ( t, x ) | (cid:54) e λT max { sup F | u ( x ) | ; sup F T (cid:112) | f ( t, x ) |} , (34) where λ = (2 + L E ) D + 1 with D being the maximum of sup F T | A ( t, x ) | , sup F T | f ( t, x ) | , and sup { ( t,x ) ∈ F T } (cid:107) C ( t, x ) (cid:107) H ( E, R m ) .Proof. It is immediate to verify that (A3) is fulfilled for PDE (33) with ζ = 0, c = sup F T | f ( t, x ) | , c = 2 D , and c = D . The statement of the lemma is thenimplied by Theorem 1. (cid:3) The main tool in the proof of existence for initial-boundary value problem (2)-(4) is the following version of the Leray-Schauder theorem proved in [5] (Theorem11.6, p. 286). First, we recall that a map is called completely continuous if it takesbounded sets into relatively compact sets.
Theorem . (Leray-Schauder theorem) Let X be a Banach space, and let Φ be acompletely continuous map [0 , × X → X such that for all x ∈ X , Φ(0 , x ) = c ∈ X .Assume there exists a constant K > such that for all ( τ, x ) ∈ [0 , × X solvingthe equation Φ( τ, x ) = x , it holds that (cid:107) x (cid:107) X < K . Then, the map Φ ( x ) = Φ(1 , x ) has a fixed point. Remark . Theorem 11.6 in [5] is, in fact, proved for the case c = 0. However, letus observe that the assumptions of Theorem 11.6 are fulfilled for the map ˜Φ( τ, x ) =Φ( τ, x + c ) − c , whenever Φ satisfies the assumptions of Theorem 8. To see this, wefirst check that ˜Φ is completely continuous. Let B ⊂ [0 , × X be a bounded set, then B (cid:48) = { ( τ, x + c ) s.t. ( τ, x ) ∈ B } is also a bounded set with the property ˜Φ( B ) =Φ( B (cid:48) ) − c . Therefore, ˜Φ is completely continuous if and only if Φ is completelycontinuous. Next, it holds that ˜Φ(0 , x ) = 0 for all x ∈ X . It remains to note that x is a fixed point of the map ˜Φ( τ, · ) if and only if x + c is a fixed point of the mapΦ( τ, · ).Now we are ready to prove the main result of Section 2 which is the existenceand uniqueness theorem for non-local initial-boundary value problem (2)-(4). Theorem . Let(A1)–(A11) hold. Then, there exists a C β , β ( F T ) -solution to non-local initial-baundary value problem (2) - (4) . If, in addition, (A12) holds, then this solution isunique.Proof. Existence. For each τ ∈ [0 , ∂ t u − (cid:80) ni,j =1 ( τ a ij ( t, x, u ) + (1 − τ ) δ ij ) ∂ x i x j u + (1 − τ )∆ ϕ + τ (cid:80) ni =1 a i ( t, x, u, ∂ x u, ϑ u ) ∂ x i u + τ a ( t, x, u, ∂ x u, ϑ u ) = 0 ,u (0 , x ) = ϕ ( x ) , u ( t, x ) (cid:12)(cid:12) ( ∂ F ) T = 0 , where u , u x , and ϑ u are evaluated at ( t, x ). In the above equation, we freeze u ∈ C , ( F T ) whenever it is in the arguments of the coefficients a ij ( t, x, u ), a i ( t, x, u, ∂ x u, ϑ u ), a ( t, x, u, ∂ x u, ϑ u ), and consider the following linear initial-boundary value problem with respect to v :(36) ∂ t v k − (cid:80) ni,j =1 (cid:0) τ a ij ( t, x, u ) + (1 − τ ) δ ij (cid:1) ∂ x i x j v k + (1 − τ )∆ ϕ k + τ (cid:80) ni =1 a i ( t, x, u, ∂ x u, ϑ u ) ∂ x i v k + τ a k ( t, x, u, ∂ x u, ϑ u ) = 0 ,v k (0 , x ) = ϕ k ( x ) , v k ( t, x ) (cid:12)(cid:12) ( ∂ F ) T = 0 , where v k , ϕ k , and a k are the k th components of v , ϕ , and a , respectively. Remarkthat the assumptions of Theorem 5.2, Chapter IV in [8] (p. 320) on the existenceand uniqueness of solution for linear parabolic PDEs are fulfilled for problem (36).Indeed, the assumptions of Theorem 5.2 in [8] require that the coefficients of (36)are of class C β ,β ( F T ) for some β ∈ (0 , ∂ F and the boundary function ψ is fulfilled by(A4) and (A7). Finally, the compatibility condition on the boundary ∂ F , requiredby Theorem 5.2, follows from (A11). Therefore, by Theorem 5.2 (p. 320) in [8], weconclude that there exists a unique solution v k ( t, x ) to problem (36) which belongsto class C β , β ( F T ). Clearly, v k is also of class C , ( F T ), and, therefore, for each τ ∈ [0 , , ( F T ) → C , ( F T ), Φ( τ, u ) = v . Note that, fixed BSDES WITH JUMPS AND CLASSICAL SOLUTIONS TO NONLOCAL PDES 19 points of the map Φ( τ, · ), if any, are solutions of (35). In particular, fixed pointsof Φ(1 , · ) are solutions to original problem (2)-(4).To prove the existence of fixed points of the map Φ(1 , · ), we apply the Leray-Schauder theorem (Theorem 8). Let us verify its conditions. First we note that if τ = 0, then the PDE in (36) takes the form ∂ t v k − ∆ v k +∆ ϕ k = 0. Therefore, it holdsthat Φ(0 , u ) = ϕ for all u ∈ C , ( F T ). Let us prove that Φ is completely continuous.Suppose B ⊂ [0 , × C , ( F T ) is a bounded set, i.e., for all ( τ, u ) ∈ B , it holdsthat (cid:107) u (cid:107) C , ( F T ) (cid:54) γ B for some constant γ B depending on B . By aforementionedTheorem 5.2 from [8] (p. 320), the solution v τ,u ( t, x ) = { v kτ,u ( t, x ) } mk =1 to problem(36), corresponding to the pair ( τ, u ) ∈ B , satisfies the estimate (cid:107) v τ,u (cid:107) C β , β ( F T ) (cid:54) γ (cid:0) (cid:107) a ( t, x, u ( t, x ) , ∂ x u ( t, x ) , ϑ u ( t, x )) (cid:107) C β ,β ( F T ) + (cid:107) ϕ (cid:107) C β ( F T ) (cid:1) , where the first term on the right-hand side is bounded by (A8), (A9), and by theboundedness of (cid:107) u (cid:107) C , ( F T ) for all ( τ, u ) ∈ B . Moreover, the bound for this termdepends only on γ B and K (where K is the constant defined in Remark 5). Thisimplies that (cid:107) v τ,u (cid:107) C β , β ( F T ) is bounded by a constant that depends only on K , γ B , γ , and (cid:107) ϕ (cid:107) C β ( F T ) . By the definition of the norm in C β , β ( F T ) (see(9)), the family v τ,u , ( τ, u ) ∈ B , is uniformly bounded and uniformly continuous inC , ( F T ). By the Arzel´a-Ascoli theorem, Φ( B ) is relatively compact, and, therefore,the map Φ is completely continuous.It remains to prove that there exists a constant K > τ ∈ [0 , , ( F T )-solution u τ to problem (35), it holds that (cid:107) u τ (cid:107) C , ( F T ) (cid:54) K .Remark that the coefficients of problem (35) satisfy (A1)–(A10). Hence, by Theorem5, the H¨older norm (cid:107) u τ (cid:107) C β , β ( F T ) , and, therefore, the C , ( F T )-norm of u τ , isbounded by a constant depending only on M , ˆ M , M , K , K ξ,ζ , T , λ n ( F ), ˆ L E , (cid:107) ϕ (cid:107) C β ( F ) , and on the C β -norms of the functions defining the boundary ∂ F .Thus, the conditions of Theorem 8 are fulfilled. This implies the existence of afixed point of the map Φ(1 , · ), and, hence, the existence of a C , ( F T )-solution toproblem (2)-(4). Further, by Theorem 5, any C , ( F T )-solution to problem (2)-(4)is of class C β , β ( F T ). Uniqueness.
Let us prove the uniqueness under (A12). Rewrite (2) in the form − n (cid:88) i,j =1 a ij ( t, x, u ) ∂ x i x j u + ˜ a ( t, x, u, ∂ x u, ϑ u ) + ∂ t u = 0 , (37)where ˜ a ( t, x, u, p, w ) = a ( t, x, u, p, w ) + (cid:80) ni =1 a i ( t, x, u, p, w ) p i with p i being the i thcolumn of the matrix p . As before, u , ∂ x u , ∂ t u , and ϑ u are evaluated at ( t, x ). Suppose now u and u (cid:48) are two solutions to (2)-(4) of class C , ( F T ). Define v = u − u (cid:48) . The PDE for the function v takes form (33) with(38) ˜ a ij ( t, x ) = a ij ( t, x, u ( t, x )) ,A ( t, x ) = − (cid:80) ni,j =1 ∂ x i x j u (cid:48) ( t, x ) (cid:82) dλ ∂ u a ij ( t, x, λu (cid:48) ( t, x ) + (1 − λ ) u ( t, x )) (cid:62) + (cid:82) dλ ∂ u ˜ a ( t, x, λu (cid:48) ( t, x ) + (1 − λ ) u ( t, x ) , ∂ x u ( t, x ) , ϑ u ( t, x ))+ (cid:82) dλ ∂ w ˜ a ( t, x, u (cid:48) ( t, x ) , ∂ x u (cid:48) ( t, x ) , λϑ u (cid:48) ( t, x ) + (1 − λ ) ϑ u ( t, x )) ◦ ς u,u (cid:48) ,u x ,u (cid:48) x ( t, x ) ,B i ( t, x ) = (cid:82) dλ ∂ p i ˜ a ( t, x, u (cid:48) ( t, x ) , λ∂ x u (cid:48) ( t, x ) + (1 − λ ) ∂ x u ( t, x ) , ϑ u ( t, x )) ,C ( t, x ) = (cid:82) dλ ∂ w ˜ a ( t, x, u (cid:48) ( t, x ) , ∂ x u (cid:48) ( t, x ) , λϑ u (cid:48) ( t, x ) + (1 − λ ) ϑ u ( t, x )) ,f ( t, x ) = 0 , v ( x ) = 0 . By Lemma 5, v ( t, x ) = 0 on F T . The theorem is proved. (cid:3) In this subsection, we consider Cauchy problem (2)–(6). The results of the pre-vious subsection will be used to prove the existence theorem for problem (2)–(6).Below, we formulate assumptions (A1’)–(A12’) needed for the existence anduniqueness theorem. Assumptions (A1’)–(A3’) are the same as (A1)–(A3) but F should be replaced with R n , and C , ( F T ) with C , b ([0 , T ] × R n ). Also, ϑ u is de-fined for all u ∈ C , b ([0 , T ] × R n ).As before, the functions µ ( s ), ˆ µ ( s ), ˜ µ ( s ), η ( s, r ), P ( s, r, t ), ε ( s, r ) are continuous,defined for positive arguments, taking positive values, and non-decreasing (exceptˆ µ ( s )) with respect to each argument, whenever the other arguments are fixed; thefunction ˆ µ ( s ) is non-increasing. Further, ˜ R , ˜ R , ˜ R , and ˜ R are defined as follows˜ R = [0 , T ] × R n × R m × R m × n × E ; ˜ R = [0 , T ] × R n × R m ;˜ R = [0 , T ] × R n × {| u | (cid:54) C } × {| p | (cid:54) C } × {(cid:107) w (cid:107) E (cid:54) C } ;˜ R = [0 , T ] × {| x | (cid:54) C } × {| u | (cid:54) C } × {| p | (cid:54) C } × {(cid:107) w (cid:107) E (cid:54) C } , where C , C , C , C > (A4 ’ ) The initial condition ϕ : R n → R m is of class C βb ( R n ), β ∈ (0 , (A5 ’ ) For all ( s, x, u, p, w ) ∈ ˜ R , | a i ( t, x, u, p, w ) | (cid:54) η ( | u | , (cid:107) w (cid:107) E )(1 + | p | ) | a ( s, x, u, p, w ) | (cid:54) (cid:0) ε ( | u | , (cid:107) w (cid:107) E ) + P ( | u | , (cid:107) w (cid:107) E , | p | ) (cid:1) (1 + | p | ) , where lim r →∞ P ( s, r, q ) = 0 and 2( s + 1) ε ( s, r ) (cid:54) ˆ µ ( s ). (A6 ’ ) ∂ x a ij , ∂ u a ij , ∂ t a ij , ∂ uu a ij , ∂ ux a ij , ∂ xt a ij , and ∂ ut a ij exist and are contin-uous on ˜ R ; moreover, max (cid:8)(cid:12)(cid:12) ∂ x a ij ( t, x, u ) (cid:12)(cid:12) , (cid:12)(cid:12) ∂ u a ij ( t, x, u ) (cid:12)(cid:12)(cid:9) (cid:54) ˜ µ ( | u | ). (A7 ’ ) ∂ t a , ∂ u a , ∂ p a , ∂ w a , ∂ t a i , ∂ u a i , ∂ p a i , and ∂ w a i exist and are bounded andcontinuous on regions of form ˜ R ; a and a i are β -H¨older continuous in x and locally Lipschitz in w with the H¨older and Lipschitz constants boundedin regions of form ˜ R . (A8 ’ ) The same as (A9) but valid for any bounded domain F . (A9 ’ ) The same as (A10) but valid for any bounded domain F . (A10 ’ ) For any bounded domain F ⊂ R n , on F T , the bound for [ ϑ u ] t α , for some α ∈ (0 , β ), is determined only by the bounds for [ u ] tα and ∂ x u , and thebound for [ ϑ u ] xα is determined only by the bound for ∂ x u . BSDES WITH JUMPS AND CLASSICAL SOLUTIONS TO NONLOCAL PDES 21 (A11 ’ ) For all u, u (cid:48) ∈ C , b ([0 , T ] × R n ), representation (32) holds with ˜ ϑ v : [0 , T ] × R n → E , defined for each v ∈ C , b ( R n × [0 , T ]) and satisfying the inequalitysup [0 ,t ] × R n (cid:107) ˜ ϑ v (cid:107) E (cid:54) L E sup [0 ,t ] × R n | v | for all t ∈ (0 , T ]; ς u,u (cid:48) ,u x ,u (cid:48) x ( t, x ) in(32) are bounded, continuous, and β -H¨older continuous in x . (A12 ’ ) ∂ xx a ij , ∂ xu a ij , ∂ uu a ij , ∂ t a ij , ∂ x a i , ∂ u a i , ∂ p a i , ∂ w a i , ∂ u a , ∂ p a , ∂ w a , ∂ px a , ∂ pu a , ∂ pp a , ∂ pw a , ∂ px a i , ∂ pu a i , ∂ pp a i , ∂ pw a i exist and are bounded andcontinuous on regions of form ˜ R , and, moreover, α -H¨older continuous in x , u , p , w for some α ∈ (0 , ∂ p a and ∂ p a i are locally Lipschitz in w .Furthermore, all the Lipschitz constants are bounded over regions of form˜ R , and all the H¨older constants are bounded over regions of form ˜ R .Assumptions (A11’)–(A12’) are required only for the proof of uniqueness. Unlikeinitial-boundary value problems, we do not prove a maximum principle for Cauchyproblems. The uniqueness result for problem (2)-(6) follows from the possibility tosolve linear parabolic systems via fundamental solutions (see [6]).Theorem 10 below is one of our main results. Theorem
10 (Existence and uniqueness for the Cauchy problem) . Let (A1’)–(A10’) hold. Then, there exists a C , b ([0 , T ] × R n ) -solution to non-local Cauchyproblem (2) - (6) which, moreover, belongs to class C α , αb ([0 , T ] × R n ) for some α ∈ (0 , β ) . If, additionally, (A11’) and (A12’) hold, then this solution is unique.Proof. Existence. We employ the diagonalization argument similar to the one pre-sented in [8] (p. 493) for the case of one equation. Consider PDE (2) on the ball B r of radius r > ψ ( t, x ) = (cid:40) ϕ ( x ) ξ ( x ) , x ∈ { t = 0 } × B r , , ( t, x ) ∈ [0 , T ] × ∂B r , (39)where ξ ( x ) is a smooth function such that ξ ( x ) = 1 if x ∈ B r − , ξ ( x ) = 0 if x / ∈ B r ; further, ξ ( x ) decays from 1 to 0 along the radius on B r (cid:31) B r − in a waythat ξ ( l ) ( x ), l = 1 , ,
3, does not depend on r and are zero on ∂B r . Let u r ( t, x ) bethe C β , β ( B r +1 )-solution to problem (2)-(39) in the ball B r +1 whose existencewas established by Theorem 9. Remark, that since u r is zero on ∂B r +1 , it canbe extended by zero to the entire space R n , and, therefore, ϑ u r is well defined.Moreover, by Theorem 1, on B r +1 , the solution u r is bounded by a constant M that only depends on T , L E , sup R n | ϕ | , and the constants c , c , c from (A3’).Next, by Theorem 2, the gradient ∂ x u r possesses a bound M on B r +1 which onlydepends on M , ˆ M , sup R n | ∂ x ϕ | , µ ( M ), ˆ µ ( M ), ˜ µ ( M ), η ( M, ˆ M ), sup q (cid:62) P ( M, q, ˆ M ),and ε ( M, ˆ M ). Thus, both bounds M and M do not depend on r .Remark that the partial derivatives and H¨older constants mentioned in assump-tion (A7’) are bounded in the region [0 , T ] × R n ×{| u | (cid:54) M }×{| p | (cid:54) M }×{(cid:107) w (cid:107) E (cid:54) ˆ M } . Let K be their common bound.Fix a ball B R for some R . By Theorem 7, there exists α ∈ (0 , β ), and a con-stant C >
0, both depend only on M , M , ˆ M , K , and (cid:107) ϕ (cid:107) C β ( R n ) , such that (cid:107) u r (cid:107) C α , α ([0 ,T ] × B r ) (cid:54) C (remark that the distance distance between B r and ∂B r +1 equals to one). Therefore, (cid:107) u r (cid:107) C α , α ([0 ,T ] × B R ) (cid:54) C for all r > R . It is im-portant to mention that the constant C does not depend on r . By the Arzel`a-Ascolitheorem, the family of functions u r ( t, x ), parametrized by r , is relatively compact in C , ([0 , T ] × B R ). Hence, the family { u r } contains a sequence { u (0) r k } ∞ k =1 whichconverges in C , ([0 , T ] × B R ). Further, we can choose a subsequence { u (1) r k } ∞ k =1 of { u (0) r k } ∞ k =1 with r k > R +1 that converges in C , ([0 , T ] × B R +1 ). Proceeding this waywe find a subsequence { u ( l ) r k } with r k > R + l that converges in C , ([0 , T ] × B R + l ).The diagonal sequence { u ( k ) r k } ∞ k =1 converges pointwise on [0 , T ] × R n to a func-tion u ( t, x ), while its derivatives ∂ t u ( k ) r k , ∂ x u ( k ) r k , and ∂ xx u ( k ) r k converge pointwise on[0 , T ] × R n to the corresponding derivatives of u ( t, x ). Therefore, u ( t, x ) is a C , b -solution of problem (2)-(6).Let us prove that u ∈ C α , αb ([0 , T ] × R n ). Note that | u | (cid:54) M and | ∂ x u | (cid:54) M ,where M and M are bounds for | u r | and, respectively | ∂ x u r | , that are independentof r . By Theorem 7, (cid:107) u (cid:107) C α , α ([0 ,T ] × B R ) (cid:54) C , where the constants α and C arethe same that for u r . Moreover, the above estimate holds for any ball B R . Therefore, (cid:107) u (cid:107) C α , αb ([0 ,T ] × R n ) (cid:54) C . Uniqueness.
As in the proof of uniqueness for the initial-boundary value problem(2)-(4), we rewrite PDE (2) in form (37).Suppose we have two C , b -solutions u and u (cid:48) to Cauchy problem (37)–(6). Then v = u − u (cid:48) is a solution to (33) on [0 , T ] × R n with the coefficients defined by(38). Assumptions (A1’), (A6’), (A7’), and (A10’)–(A12’) imply the conditions ofTheorems 3 and 6 in [6] (Chapter 9, pp. 256 and 260) on the existence and unique-ness of solution to a system of linear parabolic PDEs via the fundamental solution G ( t, x ; τ, z ). Namely, the forementioned Theorems 3 and 6 imply that the function v satisfies the equation v ( t, x ) = (cid:90) t (cid:90) R n G ( t, x ; τ, z ) C ( τ, z ) (cid:0) ˜ ϑ v ( τ, z ) (cid:1) dτ dz. Further, (A11’) and (A12’) imply the boundedness of C ( t, z ) and ˜ ϑ v ( τ, z ). Finally,taking into account the estimate sup [0 ,t ] × R n (cid:107) ˜ ϑ v (cid:107) E (cid:54) L E sup [0 ,t ] × R n sup | v | , as wellas Theorem 2 in [6] (Chapter 9, p. 251) which provides an estimate for the funda-mental solution via a Gaussian-density-type function, by Gronwall’s inequality, weobtain that v ( t, x ) = 0. Therefore, a C , b -solution to (2)-(6) is unique. (cid:3)
3. Fully-coupled FBSDEs with jumps
In this section, we obtain an existence and uniqueness theorem for FBSDEs withjumps by means of the results of Section 2.Let (Ω , F , F t , P ) be a filtered probability space with the augmented filtra-tion F t satisfying the usual conditions. Further, let B t be a d -dimensional stan-dard F t -Brownian motion, N ( t, A ) be an F t -adapted Poisson random measure on R + × B ( R l ∗ ) (where R l ∗ = R l − { } and B ( R l ∗ ) is the σ -algebra of Borel sets), and˜ N ( t, A ) = N ( t, A ) − tν ( A ) be the associated compensated Poisson random measureon R + × B ( R l ∗ ) with the intensity ν ( A ) which is assumed to be a L´evy measure.Fix an arbitrary T > solution to (1) we under-stand an F t -adapted quadruple ( X t , Y t , Z t , ˜ Z t ) with values in R n × R m × R m × d × L ( ν, R l ∗ → R m ), satisfying (1) a.s. and such that the pair ( X t , Y t ) is c`adl`ag. BSDES WITH JUMPS AND CLASSICAL SOLUTIONS TO NONLOCAL PDES 23
Together with FBSDE (1), we consider the associated final value problem for thefollowing partial integro-differential equation:(40) ∂ x θ (cid:8) f (cid:0) t, x, θ, ∂ x θ σ ( t, x, θ ) , ϑ θ ( t, x ) (cid:1) − (cid:82) R l ∗ ϕ ( t, x, θ, y ) ν ( dy ) (cid:9) + tr (cid:0) ∂ xx θ σ ( t, x, θ ) σ ( t, x, θ ) (cid:62) (cid:1) + g (cid:0) t, x, θ, θ x σ ( t, x, θ ) , ϑ θ ( t, x ) (cid:1) + (cid:82) R l ∗ ϑ θ ( t, x )( y ) ν ( dy ) + ∂ t θ = 0; θ ( T, x ) = h ( x ) . In (40), x ∈ R n , and the equation is R m -valued. Further, θ , ∂ x θ , ∂ t θ , and ∂ xx θ are everywhere evaluated at ( t, x ) (we omit the arguments ( t, x ) to simplify theequation). As before, ∂ x θ is understood as a matrix whose ( ij )th component is ∂ x j θ i ,and the first term in (40) is understood as the multiplication of the matrix ∂ x θ by thevector-valued function following after it. Furthermore, tr( ∂ xx θ σ ( t, x, θ ) σ ( t, x, θ ) (cid:62) )is the vector whose i th component is the trace of the matrix ∂ xx θ i σσ (cid:62) . Finally, forany v ∈ C b ([0 , T ] × R n ), we define the function ϑ v ( t, x ) = v ( t, x + ϕ ( t, x, v ( t, x ) , · )) − v ( t, x ) . (41)By introducing the time-changed function u ( t, x ) = θ ( T − t, x ), we transform prob-lem (40) to the following Cauchy problem:(42) ∂ x u (cid:8) (cid:82) R l ∗ ˆ ϕ ( t, x, u, y ) ν ( dy ) − ˆ f ( t, x, u, ∂ x u ˆ σ ( t, x, u ) , ϑ u ( t, x )) (cid:9) − tr (cid:0) ∂ xx u ˆ σ ( t, x, u )ˆ σ ( t, x, u ) (cid:62) (cid:1) − ˆ g ( t, x, u, ∂ x u ˆ σ ( t, x, u ) , ϑ u ( t, x )) − (cid:82) R l ∗ ϑ u ( t, x )( y ) ν ( dy ) + ∂ t u = 0; u (0 , x ) = h ( x ) . In (42), ˆ f ( t, x, u, p, w ) = f ( T − t, x, u, p, w ), and the functions ˆ σ , ˆ ϕ , and ˆ g are definedvia σ , ϕ , and, respectively, g in the similar manner. Furthermore, the function ϑ u is defined by (41) via the function ˆ ϕ (but we use the same character ϑ ).Let us observe that problem (42) is, in fact, non-local Cauchy problem (2)-(6) ifwe define the coefficients a ij , a i , a , and the function ϑ u by formulas (3), and assumethat the normed space E is L ( ν, R l ∗ → R m ). In other words, formulas (3) embedthe PIDE in (42) into the class of non-local PDEs considered in the previous section.Further, it will be shown that assumptions (B1)–(B8) below imply (A1’)–(A12’).As before, µ ( s ), ˆ µ ( s ), ˜ µ ( s ), P ( s, r, t ), ς ( r ), and ε ( s, r ) are continuous functions,defined for positive arguments, taking positive values, and non-decreasing (exceptˆ µ ( s )) with respect to each argument, whenever the other arguments are fixed; thefunction ˆ µ ( s ) is non-increasing. Further, ˜ R , ˜ R , ˜ R , and ˜ R are regions defined asin the previous section with E = L ( ν, R l ∗ → R m ):˜ R = [0 , T ] × R n × R m × R m × n × L ( ν, R l ∗ → R m ); ˜ R = [0 , T ] × R n × R m ;˜ R = [0 , T ] × R n × {| u | (cid:54) C } × {| p | (cid:54) C } × {(cid:107) w (cid:107) ν (cid:54) C } ;˜ R = F T × {| u | (cid:54) C } × {| p | (cid:54) C } × {(cid:107) w (cid:107) ν (cid:54) C } , where, C , C , C are constants, and (cid:107) · (cid:107) ν is the norm in L ( ν, R l ∗ → R m ).We assume: (B1) ˆ µ ( | u | ) I (cid:54) σ ( t, x, u ) σ ( t, x, u ) (cid:62) (cid:54) µ ( | u | ) I for all ( t, x, u ) ∈ ˜ R . (B2) ( t, x, u ) (cid:55)→ ϕ ( t, x, u, · ) is a map ˜ R → L ( ν, Z → R n ), where Z ⊂ R l isa common support of the L -functions y (cid:55)→ ϕ ( t, x, u, y ), which is assumedto be of finite ν -measure. Further, ∂ x ϕ and ∂ u ϕ exist for ν -almost each y ; ∂ t ϕ , ∂ ux ϕ , and ∂ uu ϕ exist w.r.t. the L ( ν, Z → R n )-norm. Moreover, allthe mentioned derivatives are bounded as maps ˜ R → L ( ν, Z → R n ). (B3) There exist constants c , c , c > ζ : ˜ R × R n → (0 , + ∞ )such that for all ( t, x, u, p, w ) ∈ ˜ R , ζ ( t, x, u, p, w,
0) = 0 and (cid:0) g ( t, x, u, p, w ) , u (cid:1) (cid:54) c + c | u | + c (cid:107) w (cid:107) ν + ζ ( t, x, u, p, w, p (cid:62) u ) . (B4) The final condition h : R n → R m is of class C βb ( R n ), β ∈ (0 , (B5) For all ( t, x, u, p, w ) ∈ ˜ R , (cid:12)(cid:12)(cid:12) (cid:90) Z ϕ ( t, x, u, y ) ν ( dy ) (cid:12)(cid:12)(cid:12) (cid:54) ς ( | u | ); | f ( t, x, u, p, w ) | (cid:54) η ( | u | , (cid:107) w (cid:107) ν )(1 + | p | ); | g ( t, x, u, p, w ) | (cid:54) (cid:0) ε ( | u | , (cid:107) w (cid:107) ν ) + P ( | u | , | p | , (cid:107) w (cid:107) ν ) (cid:1) (1 + | p | ) , where lim r →∞ P ( s, r, q ) = 0 and 4(1 + s )(1 + µ ( s )) ε ( s, r ) < ˆ µ ( s ). (B6) There exist continuous derivatives ∂ x σ and ∂ u σ such thatmax (cid:8)(cid:12)(cid:12) ∂ x σ ( t, x, u ) (cid:12)(cid:12) , (cid:12)(cid:12) ∂ u σ ( t, x, u ) (cid:12)(cid:12)(cid:9) (cid:54) ˜ µ ( | u | ) . (B7) For any bounded domain F ⊂ R n and for any u ∈ C , ( F T ), it holds that(1) D ( t, x, y ) > λ n +1 × ν )-almost all ( t, x, y ) ∈ F T × Z and (2) (cid:82) Z D − ( t, x, y ) ν ( dy ) < Λ, where Λ is a constant depending on u and F and D ( t, x, y ) = | det { I + ∂ x ϕ ( t, x, u ( t, x ) , y ) + ∂ u ϕ ( t, x, u ( t, x ) , y ) ∂ x u ( t, x ) }| . (B8) The functions (a) ∂ t f , ∂ t g , [ g ] xβ , ∂ t σ , ∂ xt σ , ∂ ut σ , and (b) ∂ xx σ , ∂ xu σ , ∂ uu σ , ∂ x f , ∂ u f , ∂ p f , ∂ w f , ∂ u g , ∂ p g , ∂ w g , ∂ px f , ∂ pu f , ∂ pp f , ∂ pw f , ∂ px g , ∂ pu g , ∂ pp g , ∂ pw g exist and are bounded and continuous in regions of form ˜ R ;the derivatives of group (b) are α -H¨older continuous in x , u , p , w for some α ∈ (0 , R .Further, f , g , ∂ p f and ∂ p g are locally Lipschitz in w , and all the Lipschitzconstants are bounded over regions of form ˜ R .Theorem 11 below is the existence and uniqueness result for final value problem(40) which involves a PIDE. It can be regarded as a particular case of Theorem 10and is the main tool to show the existence and uniqueness for FBSDEs with jumps.In particular, it is shown that assumptions (A8’)–(A12’), including decompositions(23), (32), and inequality (24), are fulfilled when ϑ θ is given by (41). Theorem . Let (B1)–(B8) hold. Then, final value problem (40) has a unique C , b ([0 , T ] × R n ) -solution.Proof. Since problem (40) is equivalent to problem (42), it suffices to prove the exis-tence and uniqueness for the latter. As we already mentioned, introducing functions(3), letting the normed space E be L ( ν, R l ∗ → R m ), and defining ϑ u by (41), werewrite Cauchy problem (42) in form (2)-(6).Let us prove that (A1’)–(A12’) are implied by (B1)–(B8). Indeed, (B1) implies(A1’). Next, we note that by (B2), the function ϕ ( t, x, u, · ) is supported in Z and ν ( Z ) < ∞ . This implies (A2’) since for any λ (cid:62) u ∈ C b ([0 , T ] × R n ), (cid:107) e − λt ϑ u ( t, x ) (cid:107) ν (cid:54) ν ( Z ) sup [0 ,T ] × R n | e − λt u ( t, x ) | . Further, (A3’) follows from (B3) and (3), since for any u ∈ R m , (cid:82) Z ( w ( y ) , u ) ν ( dy ) (cid:54) (cid:107) w (cid:107) ν + ν ( Z )2 | u | . Next, by (B5) and (B1), (cid:12)(cid:12) ˆ f ( t, x, u, p ˆ σ ( t, x, u ) , w ) (cid:12)(cid:12) (cid:54) η ( | u | , (cid:107) w (cid:107) ν ) (cid:0) | p | | ˆ σ ( t, x, u ) | (cid:1) (cid:54) η ( | u | , (cid:107) w (cid:107) ν ) (cid:0) (cid:112) µ ( | u | ) (cid:1) (1 + | p | ) , BSDES WITH JUMPS AND CLASSICAL SOLUTIONS TO NONLOCAL PDES 25 which, together with the inequality for ϕ in (B5), implies the first inequality in(A5’). The second inequality in (A5’) follows, again, from (B5) and (B1) by virtueof the following estimates (cid:12)(cid:12) ˆ g ( t, x, u, p ˆ σ ( t, x, u ) , w ) (cid:12)(cid:12) (cid:54) (cid:0) ε ( | u | , (cid:107) w (cid:107) ν ) + P (cid:0) | u | , (cid:107) w (cid:107) ν , | p | (cid:112) µ ( | u | ) (cid:1)(cid:1)(cid:0) | p | (cid:112) µ ( | u | ) (cid:1) (cid:54) (cid:0) ˜ ε ( | u | , (cid:107) w (cid:107) ν ) + ˜ P ( | u | , (cid:107) w (cid:107) ν , | p | ) (cid:1) (1 + | p | ) , and (cid:12)(cid:12)(cid:12) (cid:90) Z w ( y ) ν ( dy ) (cid:12)(cid:12)(cid:12) (cid:54) ˆ P ( (cid:107) w (cid:107) ν , | p | )(1 + | p | ) , where ˜ ε ( s, r ) = 2 ε ( s, r )(1 + µ ( s )), ˜ P ( s, r, q ) = 2 P ( s, r, p (cid:112) µ ( s ))(1 + µ ( s )), andˆ P ( s, r ) = ν ( Z ) s (1 + r ) − . Further, (B6) and (B8) imply (A6’), (A7’), and(A12’). Remark, that (A7’) is implied, in particular, by the fact that the function L ( ν, Z → R m ) → R m , w (cid:55)→ (cid:82) Z w ( y ) ν ( dy ) is Gˆateaux-differentiable and Lipschitz.It remains to verify assumptions (A8’)–(A11’). Let us start with (A8’). Firstremark that if u ∈ C , ( F T ), where F is a bounded domain, then it can be extendedby 0 outside of F defining a bounded continuous function on [0 , T ] × R n . Therefore, ϑ u ( t, x ) is well defined on [0 , T ] × R n for any function u which is zero on ∂ F .Further, note that by (B2), ϑ u ( t, x ) takes values in L ( ν, Z → R m ) for any u ∈ C , b ([0 , T ] × R n ). Furthermore, (B2) implies that ∂ t ϑ u ( t, x ) and ∂ x ϑ u ( t, x ) exist in L ( ν, Z → R m ) and are expressed via ∂ t u , ∂ x u , ∂ t ϕ , ∂ x ϕ , and ∂ u ϕ . Hence, (A8’) isfulfilled.Let us verify (A9’). Recall that (A9’) is assumption (A10) from subsection 4 validfor any bounded domain F . Let u ∈ C , ( F T ) and v ( t, x ) = (∆ t ) − ( u ( t (cid:48) , x ) − u ( t, x ))with t (cid:48) = t + ∆ t . The immediate computation implies decomposition (23) with ˆ ϑ v = v ( t, x + ˆ ϕ ( t, x, u ( t, x ) , · )) − v ( t, x ) ,ζ u,u x = (cid:82) dλ ∂ x u ( t (cid:48) , x + λ ∆ ˆ ϕ ) (cid:82) d ¯ λ ∂ u ˆ ϕ ( t, x, ¯ λu ( t (cid:48) , x ) + (1 − ¯ λ ) u ( t, x ) , · ) ,ξ u,u x = (cid:82) dλ ∂ x u ( t (cid:48) , x + λ ∆ ˆ ϕ ) (cid:82) d ¯ λ ∂ t ˆ ϕ ( t + ¯ λ ∆ t, x, u ( t (cid:48) , x ) , · ) , where ∆ ˆ ϕ = ˆ ϕ ( t (cid:48) , x, u ( t (cid:48) , x ) , · ) − ˆ ϕ ( t, x, u ( t, x ) , · ). Further, inequality (24) fol-lows from (B8). Indeed, define the functions Φ t,y ( x ) = x + ˆ ϕ ( t, x, u ( t, x ) , y )and ˜ v ( t, x, y ) = v ( t, Φ t,y ( x )) on [0 , T ] × R n × Z . By the definition (see (B8)), D ( t, x, y ) = | det ∂ x Φ t,y ( x ) | . We have(43) (cid:90) F ατ (cid:16) (cid:90) Z | ˜ v | ( t, x, y ) ν ( dy ) (cid:17) dt dx (cid:54) (cid:90) F ατ (cid:16) (cid:90) { y : | ˜ v | (cid:54) | v |} | ˜ v | ( t, x, y ) ν ( dy ) (cid:17) dt dx + (cid:90) F ατ (cid:16) (cid:90) { y : | ˜ v | > | v |} | ˜ v | ( t, x, y ) (cid:112) D ( t, x, y ) (cid:112) D − ( t, x, y ) ν ( dy ) (cid:17) dt dx (cid:54) ν ( Z ) (cid:90) F ατ | v | dx dt + Λ (cid:90) Z ν ( dy ) (cid:90) τ dt (cid:90) { x : | ˜ v | >α ; D> } | ˜ v | D ( t, x, y ) dx (cid:54) ( ν ( Z ) + Λ ν ( Z )) (cid:90) F ατ | v | dt dx, where Λ is the constant from (B8), depending on F and u . The second integral inthe third line is estimated as follows. First we remark that since D ( t, x, y ) >
0, byTheorem 1.2 in [7] (p. 190), the map Φ t,y : R n → R n is invertible. Therefore, wecan transform this integral by the change of variable x = Φ t,y ( x ). Thus, inequality(43) implies (24), and (A9’) is verified. Further, (A10’) is verified immediately by (41). To verify (A11’), we note thatdecomposition (32) holds with (cid:40) ˜ ϑ v = v ( t, x + ˆ ϕ ( t, x, u ( t, x ) , · )) − v ( t, x ) ,ς u,u x ,u (cid:48) ,u (cid:48) x = (cid:82) dλ ∂ x u (cid:48) ( t, x + λδ ˆ ϕ ) (cid:82) d ¯ λ ∂ u ˆ ϕ ( t, x, ¯ λu ( t, x ) + (1 − ¯ λ ) u (cid:48) ( t, x ) , · ) , where v = u − u (cid:48) and δ ˆ ϕ = ˆ ϕ ( t, x, u ( t, x ) , · ) − ˆ ϕ ( t, x, u (cid:48) ( t, x ) , · ). By (B2), ς u,u x ,u (cid:48) ,u (cid:48) x is bounded, continuous, and has a bounded derivative in x . This verifies (A11’).Thus, we conclude, by Theorem 10, that there exists a unique C , b ([0 , T ] × R n )-solution to problem (42). (cid:3) Remark . As in Theorem 10, Assumptions (B1)–(B7) imply the existence of aC , b -solution to problem (40), and (B8) is required only for the proof of uniqueness. Remark . We formulate Theorem 11 just as a result sufficient to the applicationto FBSDEs. However, we note that, by Theorem 10, the C , b -solution to problem(40) also belongs to class C α , αb ([0 , T ] × R n ) for some α ∈ (0 , β ).Before we prove our main result (Theorem 12 below), which is the existence anduniqueness theorem for FBSDE (1), we state a version of Itˆo’s formula (Lemma 6)used in the proof of Theorem 12. We give the proof of the lemma since we do notknow a reference for the time-dependent case. Lemma . Let X t be an R n -valued semimartingale with c`adl`ag paths of the form X t = x + (cid:90) t F s ds + (cid:90) t G s dB s + (cid:90) t (cid:90) Z Φ s ( y ) ˜ N ( ds dy ) , where the d -dimensional Brownian motion B t and the compensated Poisson randommeasure ˜ N are defined as above. Further, let Z ⊂ R l ∗ be such that ν ( Z ) < ∞ , and F t , G t , and Φ t be stochastic processes with values in R n , R n × d , and L ( ν, Z → R n ) ,respectively. Then, for a real-valued function φ ( t, x ) of class C , b ([0 , T ] × R n ) , a.s.,it holds that φ ( t, X t ) = φ (0 , x ) + (cid:90) t ∂ s φ ( s, X s ) ds + (cid:90) t ( ∂ x φ ( s, X s ) , F s ) ds + (cid:90) t ( ∂ x φ ( s, X s ) , G s dB s )+ 12 (cid:90) t tr (cid:0) ∂ xx φ ( s, X s ) G s G (cid:62) s (cid:1) ds + (cid:90) t (cid:90) Z (cid:2) φ (cid:0) s, X s − + Φ s ( y ) (cid:1) − φ ( s, X s − ) (cid:3) ˜ N ( ds dy )+ (cid:90) t (cid:90) Z (cid:2) φ (cid:0) s, X s − + Φ s ( y ) (cid:1) − φ ( s, X s − ) − ( ∂ x φ ( s, X s − ) , Φ s ( y )) (cid:3) ν ( dy ) ds. (44) Remark . In the above lemma, we agree that X − = X = x . Proof of Lemma 6.
Let us first assume that the function φ does not depend on t .Applying Itˆo’s formula (see Theorem 33 in [17], p. 74), we obtain(45) φ ( X t ) − φ ( x ) = (cid:90) t ( ∂ x φ ( X s ) , F s ) ds + (cid:90) t ( ∂ x φ ( X s − ) , dX s )+ 12 (cid:90) t tr (cid:0) ∂ xx φ ( X s ) G s G (cid:62) s (cid:1) ds + (cid:88)
Now take a partition of the interval [0 , t ]. Then, for each pair of successive points,(46) φ ( t n +1 , X t n +1 ) − φ ( t n , X t n ) = (cid:2) φ ( t n +1 , X t n )) − φ ( t n , X t n ) (cid:3) + (cid:2) φ ( t n +1 , X t n +1 ) − φ ( t n +1 , X t n )) (cid:3) . The first difference on the right-hand side equals to (cid:82) t n +1 t n ∂ s φ ( s, X t n ) ds , while thesecond difference is computed by formula (45). Assume, the mesh of the partitiongoes to zero as n → ∞ . Then, summing identities (46) and letting n → ∞ , we arriveat formula (44). Indeed, the convergence of the stochastic integrals holds in L (Ω)by Lebesgue’s dominated convergence theorem, implying the convergence almostsurely for a subsequence. Further, in the term containing the time derivative ∂ s φ ,we have to take into account that X t has c`adl`ag paths. (cid:3) Let S denote the class of processes ( x t , y t , z t , ˜ z t ) with values in R n , R m , R m × n ,and L ( ν, R l ∗ → R m ), respectively, such thatsup t ∈ [0 ,T ] (cid:8) E | x t | + E | y t | (cid:9) + (cid:90) T (cid:0) E | z t | + E (cid:107) ˜ z t (cid:107) ν (cid:1) dt < ∞ . (47)The main result of this work is the following. Theorem . Assume (B1)–(B8). Then, there exists a solution ( X t , Y t , Z t , ˜ Z t ) toFBSDE (1) , such that X t is a c`adl`ag solution to (48) X t = x + (cid:90) t f (cid:0) s, X s , θ ( s, X s ) , ∂ x θ ( s, X s ) σ ( s, X s , θ ( s, X s )) , ϑ θ ( s, X s ) (cid:1) ds + (cid:90) t σ ( s, X s , θ ( s, X s )) dB s + (cid:90) t (cid:90) R l ∗ ϕ ( s, X s − , θ ( s, X s − ) , y ) ˜ N ( ds dy ) , where θ ( t, x ) is the unique C , b ([0 , T ] , R n ) -solution to problem (40) , whose existencewas established by Theorem 11, and ϑ θ is given by (41) . Furthermore, Y t , Z t , and ˜ Z t are explicitly expressed via θ by the formulas Y t = θ ( t, X t ) , Z t = ∂ x θ ( t, X t ) σ ( t, X t , θ ( t, X t )) , and ˜ Z t = ϑ θ ( t, X t − ) . (49) Moreover, the solution ( X t , Y t , Z t , ˜ Z t ) is unique in the class S , and the pair ( X t , Y t ) is pathwise unique. Remark . Remark that ( X t , Y t , Z t ) is a c`adl`ag process, and ˜ Z t is left-continuouswith right limits. Proof of Theorem 12. Existence.
First we prove that SDE (48) has a unique c`adl`agsolution. Define ˜ f ( t, x ) = f ( t, x, θ ( t, x ) , ∂ x θ ( t, x ) σ ( t, x, θ ( t, x )) , ϑ θ ( t, x )), ˜ σ ( t, x ) = σ ( t, x, θ ( t, x )), and ˜ ϕ ( t, x, y ) = ϕ ( t, x, θ ( t, x ) , y ). With this notation, SDE (48) be-comes X t = x + (cid:90) t ˜ f ( t, X s ) ds + (cid:90) t ˜ σ ( s, X s ) dB s + (cid:90) t (cid:90) R l ∗ ˜ ϕ ( s, X s − , y ) ˜ N ( ds dy ) . (50)Note that since θ is of class C , b ([0 , T ] × R n ), (B1) and (B5) imply that ˜ f ( t, x ),˜ σ ( t, x ), (cid:82) Z ˜ ϕ ( t, x, y ) ν ( dy ) are bounded. Further, (B6) implies the boundedness of ∂ x ˜ σ ( t, x ), while (B1), (B2), (B6), and (B8) imply the boundedness of ∂ x ˜ f ( t, x ).Furthermore, (B2) implies the boundedness of ∂ x (cid:82) R l ∗ ˜ ϕ ( t, x, y ) ν ( dy ). Therefore, theLipschitz condition and the linear growth conditions, required for the existence and uniqueness of a c`adl`ag adapted solution to (50) (see [1], Theorem 2.6.9, p. 374),are fulfilled. By Theorem 2.6.9 in [1] (more precisely, by its time-dependent versionconsidered in Exercise 2.6.10, p. 375), there exists a unique c`adl`ag solution X t toSDE (50).Further, define Y t , Z t , and ˜ Z t by formulas (49). Applying Itˆo’s formula (Lemma6) to θ ( t, X t ), we obtain(51) θ ( t, X t ) = θ ( T, X T ) − (cid:90) Tt ∂ x θ ( s, X s − ) σ ( s, X s − , θ ( s, X s − )) dB s − (cid:90) Tt (cid:110) ∂ x θ ( s, X s ) f (cid:0) s, X s , θ ( s, X s ) , ∂ x θ ( s, X s ) σ ( s, X s , θ ( s, X s )) , ϑ θ ( s, X s ) (cid:1) + ∂ x θ ( s, X s ) (cid:90) R l ∗ ϕ ( s, X s , θ ( s, X s ) , y ) ν ( dy ) + ∂ s θ ( s, X s )+ 12 tr (cid:2) ∂ xx θ ( s, X s ) σ ( s, X s , θ ( s, X s )) σ ( s, X s , θ ( s, X s )) (cid:62) (cid:3) + (cid:90) R l ∗ ϑ θ ( s, X s )( y ) ν ( dy ) (cid:111) ds − (cid:90) Tt (cid:90) R l ∗ ϑ θ ( s, X s − )( y ) ˜ N ( ds, dy ) . Since θ ( t, x ) is a solution to PIDE (40), then Y t , Z t , and ˜ Z t , defined by (49), aresolution processes for the BSDE in (1). Furthermore, Y t and Z t are c`adl`ag, and ˜ Z t is left-continuous with right limits. Uniqueness.
Assume ( X (cid:48) t , Y (cid:48) t , Z (cid:48) t , ˜ Z (cid:48) t ) is another solution to FBSDE (1) satisfy-ing (47). As before, θ ( t, x ) is the unique C , b ([0 , T ] , R n )-solution to PIDE (40).Define ( Y (cid:48)(cid:48) t , Z (cid:48)(cid:48) t , ˜ Z (cid:48)(cid:48) t ) by formulas (49) via θ ( t, x ) and X (cid:48) t . Therefore, ( Y (cid:48) t , Z (cid:48) t , ˜ Z (cid:48) t )and ( Y (cid:48)(cid:48) t , Z (cid:48)(cid:48) t , ˜ Z (cid:48)(cid:48) t ) are two solutions to the BSDE in (1) with the process X (cid:48) t being fixed. By the results of [19] (Lemma 2.4, p.1455), the solution to theBSDE in (1) is unique in the class of processes ( Y t , Z t , ˜ Z t ) whose squared normsup t ∈ [0 ,T ] E | Y t | + (cid:82) T (cid:0) E | Z t | + E (cid:107) ˜ Z t (cid:107) ν (cid:1) dt is finite. Without loss of generality, we canassume that Y (cid:48) t is c`adl`ag by considering, if necessary, its c`adl`ag modification. Sinceboth Y (cid:48) t and Y (cid:48)(cid:48) t are c`adl`ag, then Y (cid:48) t and θ ( t, X (cid:48) t ) coincide pathwise a.s. Further, Z (cid:48) t = Z (cid:48)(cid:48) t and ˜ Z (cid:48) t = ˜ Z (cid:48)(cid:48) t as elements of L ([0 , T ] × Ω → R m ) and L ([0 , T ] × Ω → E ),respectively, where E = L ( ν, R l ∗ → R m ). Therefore, a.s., Z (cid:48) t and Z (cid:48)(cid:48) t (and also,˜ Z (cid:48) t and ˜ Z (cid:48)(cid:48) t ) differ only at a countable number of points t ∈ [0 , T ]. Hence, X (cid:48) t is asolution to SDE (48). Since a c`adl`ag solution is pathwise unique, then X t = X (cid:48) t pathwise a.s. Thus, the pair ( X t , Y t ) is pathwise unique. On the other hand, it isproved that the quadruple ( X t , Y t , Z t , ˜ Z t ) is unique in the class S . (cid:3) References [1] D. Applebaum.
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Departamento de Matem´atica, Universidade Federal da Para´ıba, Jo˜ao Pessoa, Brazil
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