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Backward Doubly Stochastic Equations with Jumps and Comparison Theorems

In this work we mainly prove the existence and pathwise uniqueness of solutions to general backward doubly stochastic differential equations with jumps appearing in both forward and backward integral parts. Several comparison theorems under some weak conditions are also given. Finally we apply comparison theorems in proving the existence of solution to some special backward doubly stochastic differential equations with drift coefficient increasing linearly.


Introduction
Backward stochastic differential equations (BSDEs) in the linear case were introduced in Kushner (1972), Bismut (1976), Bensoussan (1982) and Haussmann (1986) as adjoint processes in the maximum principle for stochastic control problems and the pricing of options. Since the important work of Pardoux and Peng (1990), the interest in BSDEs has increased considerably in recent years. The significance of BSDEs is not only proved by the considerably important role they are playing in the study of partial differential equations (PDEs); see Peng (1991), Pardoux and Peng (1992) and Darling and Pardoux (1997), but also can be found in many other fields such as mathematical economics, financial mathematics, insurance and stochastic control. Here we just list several important works in every field. Duffie and Epstein (1992a,b) used BSDEs as a powerful tool to study stochastic differential utility. Moreover, in the insurance market BSDEs are used in pricing and hedging insurance equity-linked claims and asset-liability management problems, see El Karoui et al. (1997) and Delong (2013). Peng (1993) studied stochastic optimal control systems, where the state variables are described by a system of ordinary-SDE and BSDEs, and derived a local form of the maximum principle.
As further extensions of BSDEs, backward doubly stochastic differential equations (BDSDEs) contain both forward and backward stochastic integrals. Those equations were first introduced by Pardoux and Peng (1994) in the study of quasi-linear parabolic stochastic partial differential equations (SPDEs). Compared to BSDEs, much less results about BDSDEs can be found in the literature and most of the results established are about BDSDEs driven by Brownian motions. For the details about applications of BDSDEs to SPDEs driven by Brownian motion, one can refer to Zhang and Zhao (2007) which studied the existence and uniqueness of solution to BDSDEs on infinite horizons, and the stationary solutions to SPDEs by virtue of the solutions to BDSDEs on infinite horizons. Moreover, some work about BDSDEs with jumps appearing in the system of ordinary-SDE have been published recently. For instance, Zhu and Shi (2012) studied BDSDEs driven by Brownian motions and Poisson process with non-Lipschitz coefficients on random time interval. Aman (2012), Aman and Owo (2012) and Ren et al. (2009) study a special reflected generalized BDSDEs (driven by Teugel's martingales associated with Lévy process) with means of the penalization method and the fixed-point theorem. Existence and uniqueness of the solution to the BDSDE with jumps in the forward integral are studies in Sow (2011) for the case of non-Lipschitz coefficients. Recently, some results about stochastic control problems of BDSDEs have been obtained by Han et al. (2010) and Bahlali and Gherbal (2010). This work is motivated by Xiong (2013) and He et al. (2014) which mainly studied the distribution function valued process of super-Brownian motions and super-Lévy processes characterized as the pathwise unique solution to a SPDE. For any super-Lévy process with transition semigroup (Q t ) t≥0 defined by (1.4) in He et al. (2014), they proved that its distribution function valued process solves the following stochastic integral equation: for any x ∈ R, where {W (ds, du); t ≥ 0, u > 0} is a Gaussian white noise with intensity dsπ(du), {N (dt, dz, du) : t ≥ 0, u > 0} is a Poisson random measure with intensity dtµ(dz)du and A * is the dual operator of A defined by: for any f (x) ∈ C 2 0 (R), Furthermore, for any fixed T > 0, define the Gaussian white noise {W T (dt, du) : In the proof of the pathwise uniqueness of solutions to (1.1), they established its connection to the following BDSDE: where L r t = L t − L r and {L t : t ≥ 0} is a Lévy process with generator A * . The purpose of this work is extending the above equations into more general BDSDEs with jumps appearing not only in the forward stochastic integral part but also in the backward stochastic integral part; see (2.2) in Section 2. Pathwise uniqueness and existence of their solutions are proved in Section 2 and 3 respectively under Lipschitz conditions. In addition, several comparison theorems for BDSDEs will also be given in Section 4, since they play an important role in both theory and applications; see Shi et al. (2005). Effected by random terms in the backward integrals, classical methods are not applicable, we use another method to get comparison theorems with some reasonable and weak conditions. The main difficulty is to deal with the influence of forward integrals to the drift coefficient and backward integrals. As an applications of comparison theorems, in Section 5 we prove that solutions to a special kind of BDSDEs with drift coefficient increasing linearly exist.

Pathwise Uniqueness
In this section, we mainly study the pathwise uniqueness of solutions to general backward doublystochastic equations. Suppose that T > 0 is a fixed constant and (Ω, F , P) is a complete probability space endowed with filtration {G (1) t } 0≤t≤T satisfying the usual hypotheses. Let B(s) is a n-dimensional (G t )-Poisson random measure with intensity dtν(du), where ν(du) is a σ-finite Borel measure on the Polish spaces F . Let {G (2) t } 0≤t≤T be another filtration on (Ω, F , P) satisfying the usual hypotheses and independent from {G (1) t } 0≤t≤T . Let {W (ds, du) = (W 1 (ds, du), · · · , W n (ds, du)) T ; 0 ≤ t ≤ T, u ∈ E} be a n-dimensional (G (2) t )-Gaussian white noise constructed with n orthogonal white noises W i (ds, du) on R + × E with intensity dsπ i (du) respectively. Here we denote π(du) = (π 1 (du), · · · , π n (du)) T . Suppose µ 0 (du) is a σ-finite Borel measure on the Polish space U 0 and µ 1 (du) is a finite Borel measure on the Polish space U 1 . Moreover, For each t )-Poisson random measure with intensity dtµ i (du). Obviously, all the random elements introduced above are independent of each other.
t ) are two filtrations satisfying the usual hypotheses. It is easily seen that {B t } is a (G 0 t )-Brownian motion and M (dt, du) is a (G 0 t )-Poisson random measure. Define the Gaussian white noise {W T (dt, du) : 0 ≤ t ≤ T, u ∈ E} by For i = 0, 1, define the Poisson random measures {N T i (dt, du) : 0 ≤ t ≤ T, u ∈ U i } by: Roughly speaking, we can consider W T (dt, du) and N T i (dt, du) as the time reversal of W (dt, du) and N i (dt, du), respectively.
A real process {ξ s } 0≤s≤T is said to be (G r t )-progressive if for any 0 ≤ r ≤ t ≤ T , the mapping (s, ω) → ξ s (ω) restricted to [r, t] × Ω is B([r, t]) × G r t -measurable. A two-parameter real process {ζ s (u)} 0≤s≤T,u∈E is said to be (G r t )-progressive if for any 0 ≤ r ≤ t ≤ T , the restriction of (s, u, ω) → ζ s (u, ω) to [r, t] × E × Ω is B([r, t]) × B(E) × G r t -measurable. Let P denote the σ-algebra on Ω × [0, T ] generated by all real-valued left continuous processes which are (G r t )-progressive. A process (ξ s ) 0≤s≤T is said to be predictable if the mapping (ω, s) → ξ s (ω) is P-measurable. Also a two-parameter process {ζ s (u)} 0≤s≤T,u∈E is said to be predictable if the mapping (ω, s, x) → ζ s (ω, x) is (P × B(E))-measurable. For the theory of timespace stochastic integrals of predictable two parameter processes with respect to point processes or random measures, readers can refer to Section II.3 in Ikeda and Watanabe (1989). The stochastic integrals with respect to martingale measures were discussed in Section 7.3 of Li (2011). We make the convention that the stochastic integral of a progressive process refers to a predictable version of the integrand. The existence of such a version was briefly discussed in Section 2 of He et al. (2014). To simplify the following statements, we introduce several Banach spaces: Before giving the main results, we extend Itô formula to the general case. Let X t be a mdimensional stochastic process defined by: where b(s), a(s, u), γ 0 (s, u), γ 1 (s, u) and ζ s (u) are m-dimensional (G r t )-progressive processes, a(s, u) and Z s are (G r t )-progressive (m × n)-matrix-valued processes. Now let us introduce the backward doubly stochastic integral equation to work with. Suppose that we have the following measurable mappings: Given Y T ∈ G 0 T , we consider the equation:

Condition 2.4
There exist constants C > 0 and 0 < α < 1 such that for any s ∈ [0, T ] and and From Cauchy's inequality, for any a, b > 0 we have Since µ 1 is a finite measure, by Hölder's inequality and Condition 2.4, Here we can choose a, b small enough such thatα := Ca + α + bα < 1. Then By Gronwall's lemma, we have This implies (2.6). Then for any fixed t ∈ [0, T ], the six terms on the right-hand side of (2.7) vanish almost surely. Since each of the six terms is right-continuous or left-continuous, they almost surely vanish for all t ∈ [0, T ]. We have finished the proof.

Existence
In this section, we study the existence of solutions to (2.2). For any 0 ≤ r ≤ t ≤ T we define the natural σ-algebras: where N denotes the totality of P-null sets. For simplicity, we write Obviously, combining this theorem with Theorem 2.5, we have solution to (2.2) exists uniquely Before giving the proof of Theorem 3.1, we introduce a lemma about the solution to some simple backward doubly stochastic equation, which is very important in the proof of this theorem.
Proof. The uniqueness of the solution follows from Theorem 2.5.
Then we can define a Doob's martingale: By the martingale representation theorem, see Lemma 2.3 in Tang and Li (1994), there exist and hence Since M T = Ψ T , we can substitute (3.2) and (3.3) into (3.5) to obtain (3.1). Finally, we need to prove for any 0 where F W N T −r and F W N T −r,T are independent. By (3.4) it is easy to see that Ξ(r) is independent of Then by the uniqueness of the martingale representation, the process (Z t , ζ t (u)) has an (F r t )progressive version. Since each term in (3.1) is right or left continuous, the process (Y t ) is (F r t )progressive.
Proof of Theorem 3.1. We shall use a Picard iteration argument to construct a solution to ( According to Proposition 2.1,, we have By integration by parts, one can see, for any λ > 0, Let a, b be small enough such that aC + bα + α < 1. Then choose λ > 0 large enough such that It follows that Since the right-hand side of the inequality is summable, we see that s (u))} is a Cauchy sequence. By Burkholder-Davis-Gundy Inequality, it is easy to see Y (n) s is also a Cauchy sequence in S 2 F ,T . Then it converges in S 2 F ,T × L 2 F ,T × L 2 F ,T (F ) to some process (Y s , Z s , ζ s (u)), which is clearly a solution to (2.2). Then we have finished the proof.

Comparison theorems
Comparison theorems are very important in both theory and applications. For instance, if you want to earn more money from a complete capital market in the future time T , you should either invest more money in the market at time 0 or improve your investment policy. This section will mainly introduce several comparison theorems under Condition 2.4. There are two classical ways to prove comparison theorems in the theory of BSDEs; see Situ (2005, p.243-250). One is transforming the BSDE into a summation of a non-negative processes and a martingale under a new probability measure. Then the desired results can be gotten by taking conditional expectation under the new probability measure. Another one is called "a duality method" which mainly by constructing a relative forward SDE (FSDE). Applying Itô formula to the multiplication of the solutions of these two stochastic equations (FBSDE), we will get a new process which is a summation of a non-negative processes and a martingale. Similarly, we get the comparison theorem by taking conditional expectation. Actually, both of these two methods come from the same ideas.
Unfortunately, effected by backward integral parts in (2.2), neither of these two methods works. Here we use another method to get comparison theorems under some conditions which are not really stronger than those in BSDEs. The main difficulty is to deal with the influence of ζ s to the drift coefficient and backward integrals. We divide the influence into several parts and deal with them one by one. Here we only consider the one-dimensional case, comparison theorem for multidimensional case is still an open problem; see Peng (1999). Firstly, we give a simple comparison theorem about the non-positivity of solution to the following one-dimensional BDSDEs, which can be used to derive other results.
Let x → g n (x) be a positive continuous function supported by (α n , α n−1 ) such that Moreover, for any n > 0, define It is easy to see that (a) f n (z) → |z + | 2 increasingly.
(2) both y + g 0 (s, y, u) and y + g 1 (s, y, u) are nondecreasing with respect to y.
(4) there exists a constant K > 0 such that one of the following conditions is satisfied: Proof. Here we just prove this theorem under condition (a) holds. Moreover like the proof of Theorem 4.3, we assume ν(F ) < ∞. Let (Ȳ t ,Z t ,ζ t (u)) = (Y s ). For any n ≥ 0, recall g n (z) defined in the proof of Theorem 4.3, let It is easy to see that (a) f n (z) → z + increasingly. By Proposition 2.1, sinceȲ T ≤ 0 a.s., we have . Taking the expectation, since α < 1, we have By the assumption of this theorem and (c), we have Moreover, from Lemma 3.1 in Li and Pu (2012), Thus by Fatou's lemma, we have as n → ∞, Now let us discuss the integrand of (4.8) denoted by η: From this, (4.8) and conditions in this theorem we have By Gronwall's inequality, we have E Ȳ + t = 0 and P Y (2) t = 1 for any t ∈ [0, T ]. Like the proof of Theorem 2.5, we will get the desired result.

Maximum and Minimum Solutions
In the proof of Theorem 3.1, Picard iteration argument seriously depends on the Lipschitz condition. Actually, sometimes solutions still exist (maybe not unique), even the drift term is linear increasing which is much weaker than Lipschitz condition. As a simple application of comparison theorems, in this section we will prove the existence of solution to (4.5) under some weak conditions. (b) |β(s, y, z, ζ)| ≤ K(1 + |y| + |z| + ζ L 2 (F ) ) and for any y, z where C(s, u) ∈ [0, 1] and sup s∈[0,T ] F |C(s, u)| 2 ν(du) ≤ K a.s.
Then solutions to (4.5) exist. Moreover, there exist two solutions (Y I t , Z I t , ζ I t ) and (Y S t , Z S t , ζ S t ) such that for any solution (Y t , Z t , ζ t ) to (4.1) have Before using comparison theorem to prove this theorem, we need to construct a suitable sequence of BDSDEs with solutions exist and satisfy the conditions of comparison theorems; see the following lemma. Since the proof is easy and similar to Lemma 1 in Lepeltier and Martin (1997), we will omit it.
Since we have showed {Y I n (t)} and {Y S n (t)} converge, it suffices to identify (Z I n , ζ I n ) and (Z S n , ζ S n ) are Cauchy sequences in L 2 F ,T × L 2 F ,T (F ). Since Y S n is a Cauchy sequence, so are Z I n and ζ I n .
Proof of Theorem 5.1. With the preparations of Lemma 5.2, 5.3 and 5.4, like the classical proof in SDE theory, we can get the desired result (omit the details).