Backward doubly stochastic differential equations with polynomial growth coefficients

In this paper we study the solvability of backward doubly stochastic differential equations (BDSDEs for short) with polynomial growth coefficients and their connections with SPDEs. The corresponding SPDE is in a very general form, which may depend on the derivative of the solution. We use Wiener-Sobolev compactness arguments to derive a strongly convergent subsequence of approximating SPDEs. For this, we prove some new estimates to the solution and its Malliavin derivative of the corresponding approximating BDSDEs. These estimates lead to the verifications of the conditions in the Wiener-Sobolev compactness theorem and the solvability of the BDSDEs and the SPDEs with polynomial growth coefficients.

1. Introduction. In this paper, we use the Malliavin calculus to study the solvability of BDSDEs with polynomial growth coefficients valued in a weighted L 2 (dx) space Here the Brownian motionB could be a Q-Wiener process with values in a separable Hilbert space U and the stochastic integral with respect toB is a backward Itô's integral. But for simplicity, we only consider the finite dimensional Brownian motion valued in R l . The other Brownian motion W is independent ofB and takes values in R d . The coefficients are given functions h : By the polynomially growing coefficient we mean that f in (1) is of a polynomial growth with power p, p ≥ 2, with respect to the solution Y . Specific assumptions on f and g are given in the next section. And X is the solution of the SDE: with b : R d → R d , σ : R d×d → R d . Actually, BDSDE (1) and SDE (2) constitute a forward-backward stochastic differential system, and its connection with the classical solution of parabolic semilinear SPDE was first indicated in Pardoux and Peng [6]. In this paper, we consider the connection between them in the sense of weak solution of the following SPDE: Here the second order differential operator L is given by This work is a further study of [11]. Different from the previous work, the corresponding SPDE we consider here involves the first order derivatives of the solution in both the drift and the diffusion terms. This causes difficulties in applying the Malliavin calculus method. Even in the case that g in (3) does not depend on ∇u, the estimates in [11] are not enough. We need some new estimates on the integrability and the continuity of Z and the Malliavin derivative of Z in the ρ-weighted space L m ρ (dx, dP ), m > 2. This kind of estimates was not given in the estimates derived from BDSDE (1), where only L 2 ρ (dx, dP ) can be given. The idea here is to associate Z with ∇Y , which can actually be the solution of another BDSDE if we differentiate BDSDE (1). We construct a sequence of approximating SPDEs with linear growth drift and deduce the desired estimates of their corresponding sequence of BDSDEs, then the new estimates can be transferred from the BDSDEs to the SPDEs. With these estimates, we verify that the approximating SPDEs satisfy the Wiener-Sobolev compactness theorem. As a consequence, we are able to get a strongly convergent subsequence of the solutions of approximating SPDEs and the approximating BDSDEs.
As we have shown in our previous works [8,9,11], if f and g in (3) are independent of the time variable, we can apply the time reverse transformation to (3) to obtain a SPDE with an initial value where the Brownian motion B is the time reverse ofB. Then we can construct the stationary solution of SPDE (4) after extending the solvability from the finite time horizon to the infinite time horizon. But we don't intend to include this result here and only show the main differences when the finite time horizon BDSDE (1) and SPDE (3).
The rest of this paper is organized as follows. In Sections 2, we introduce some useful definitions and estimates. In section 3, some new estimates to the solutions of approximating BDSDEs are proved. A strongly convergent subsequence of the solutions of corresponding approximating SPDEs as well as approximating BDSDEs is derived. In Section 4, we finally prove the existence and uniqueness of the solution to BDSDE with the polynomial growth coefficient in our setting by weak convergence and strong convergence arguments, and demonstrate the correspondence between the BDSDEs and the SPDEs.

2.
Preliminaries. As we know, when we consider such kind of BDSDEs and SPDEs with a general form, we usually construct a sequence of approximating BDSDEs and SPDEs. To get the strongly convergent subsequence is a key step. The Wiener-Sobolev compactness theorem, which is a natural but not trivial extension of Rellich-Kondrachov compactness theorem to stochastic case, is the method we use to get the strongly convergent subsequence. Here the Malliavin derivative plays a key role in the assumptions. We begin our preliminaries with Malliavin derivatives. For a smooth random variable F such that is the set of infinitely differentiable functions whose differentials of any order all grow in a polynomial way. Let K be the class of smooth random variables F . The derivative operator of F denoted by the stochastic process {D t F, t ∈ [0, T ]} is then defined by (cf. [4]) We denote the domain of the operator D in L 2 (Ω) by D 1,2 with the norm below . We first recall a version of a Wiener-Sobolev compactness theorem in the space L 2 (Ω × [0, T ] × O; R 1 ) used in this paper. The theorem was proved by Bally and Saussereau [1]. See Da Prato, Malliavin and Nualart [2] and Peszat [7] for an earlier version of time and space independent case and Feng and Zhao [3] for a relative compactness result in the space C([0, T ], L 2 (Ω × O; R 1 ).
The conditions of Theorem 2.1 are not easy to verify, without the exception to our case when we apply the theorem to the approximating SPDEs of (3). We will utilize the correspondence between BDSDE and SPDE for the corresponding approximating BDSDEs to verify the conditions of Theorem 2.1.
Since we will consider the solution of BDSDE (1) in a weighted L 2 (dx) space which connects the weak solution of corresponding SPDE (3), a necessary equivalence between the norms of the BDSDE and SPDE solution spaces is needed. For this, we utilize the property of stochastic flow and always assume that Here C k l,b , k ≥ 0 denotes the set of C k -functions for which the partial derivatives from the order 1 to k are bounded, but the functions themselves may not be bounded, and C k b denotes the set of C k -functions for which the partial derivatives from the order 0 to k are bounded.

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The inequality (5) implies the equivalence of norm between BDSDEs and SPDEs in their respective solution spaces when Ψ is regarded as the weak solution of SPDE. This equivalence of norm principle will be more clear after we clarify the definitions for the solution spaces of BDSDEs and SPDEs in the following. Both the solutions of BDSDEs and SPDEs are valued in a ρ −1 -weighted L q , q ≥ 2, space, denoted by L q ρ , where the weight function ρ(x) = (1 + |x|) q , q > d + 32p.
For t ≤ s ≤ T and q ≥ 2, we denote by s. continuous and Correspondingly, we give the definition for weak solution of SPDE (3) in the sense of C ∞ c test function.
We then assume the conditions to BDSDE (1). Given constantsL ≥ 0 and 0 ≤ α < √ 2 2 , for any s, s 1 , f is locally Lipschitz on x and globally Lipschitz on z as follows: and there exists a constant µ ∈ R 1 s.t. , (H4): g is globally Lipschitz as follows: and the derivatives ∂ y g, ∂ z g exist and satisfy Remark 1. As indicated in the literatures (e.g. [5,11]), the monotonic constant µ in (H2) can be assumed, without losing any generality, to be 0. To simplify the calculation, we always take µ = 0 in the rest of the paper.
Then we construct a sequence {f n } n∈N which converges to f a.s. For this, first set f n (s, x, y, z) = f s, x, y, z when |y| ≤ n and f n (s, x, y, z) = f s, x, n+1 |y| y, z + ∂ y f (s, x, n+1 |y| y, z)(y − n+1 |y| y) when |y| ≥ n + 1. Then we use the standard partition and unity method to define f n such that f n has smooth enough connections and is monotone on the interval [n, n + 1] and [−n − 1, −n]. For any s ∈ [0, T ], x, x 1 , x 2 , z, z 1 , z 2 ∈ R d , y, y 1 , y 2 ∈ R 1 , f n satisfies the following conditions with the constant L depending onL: (H2) : for the same f 0 in (H2), f n is locally Lipschitz on x and globally Lipschitz on z as follows: and there exists a constant µ ∈ R 1 s.t. , .
Note that f n is of linear growth in y for each n ∈ N. We consider a sequence of BDSDEs of the linearly growing drift f n : For BDSDEs with linear growth coefficients, we have some estimates for which the reader can refer to [9,11] for detailed proofs.
Here and in the rest of this paper C p is a generic constant depending only on given parameters. Moreover, 3. The compactness of solutions to approximating equations. The main task in this section is to prove the relative compactness of the sequence of solutions to SPDEs (7) and BDSDEs (6). We need some preparations. First, as we indicate before, the Malliavin derivatives are used to obtain the strongly convergent subsequence of solutions of approximating equations. By (H3) , (H4) and the results of [1] or [6], the Malliavin derivative of (Y t,x,n s , Z t,x,n s ) in BDSDE (6) Furthermore, we need the following estimates for the Malliavin derivatives.
However, different from the case that f and g are independent of z in [11], the estimate for sup  So we only need to estimate ∇Y t,x,n s instead.

BDSDES WITH POLYNOMIAL GROWTH COEFFICIENTS 9
Noticing the smooth conditions (H1), (H3) , (H4) and the form of SPDE (7), we have By the standard correspondence of BDSDE and SPDE (see e.g. [1] for details), we know that (Ỹ t,x,n r ,Z t,x,n r ) t≤r≤T is the solution of the following BDSDE: Proof. We rewrite BDSDE (9) as Taking integrations of (9) over R d and carrying out similar calculations as (A.6) in where ε can be taken sufficiently small.
Note that Therefore, On the other hand, by B-D-G inequality and (10),

QI ZHANG AND HUAIZHONG ZHAO
Noticing (11), we further have Since σ(X t,x s )Z t,x,n s = (σσ * )(X t,x s )Ỹ t,x,n s ≥ εỸ t,x,n s , by Proposition 1, Thus by (11) and (12), we get for a sufficiently small δ > 0, Noticing that δ only depends on the given parameters, we can extend the above estimate to the interval [t, T ]. Thus Proposition 4 follows.
From Proposition 4, it follows immediately that for 2 ≤ m ≤ 16, With (13), we can give the proof of Proposition 3.
Proof of Proposition 3. For 2 ≤ m ≤ 16, applying Itô's formula to e Kr |D θ Y t,x,n r | m , we have From Conditions (H2) and (H4), we know that for any s ∈ [0, T ], y ∈ R 1 , x, z ∈ R d , and |∂ y g(s, x, y, z)| ≤ L. Therefore, we get from (14) that Then, it follows from (H4), Proposition 1 and (13) that Using the B-D-G inequality in (15), by the above formula we can further prove The proof is completed.
Proof. First note that by a standard estimate, for 2 ≤ m ≤ 16p we have Since Obviously, (Ỹ ,Z) satisfies the following equation: +Z t,x,n r (j t+h (r, x) − j t (r, x)) + e t+h (r, x)Ỹ r + j t+h (r, x)Z r d †B r − T sZ r dW r .

QI ZHANG AND HUAIZHONG ZHAO
By a similar computation as in the proof of Proposition 4, we have Before we estimate each term on the right hand side of above inequality, we need do some calculations. Firstly, Next for the continuity dependence on b, The continuity dependence on c is shown below: We are ready to estimate each term of (18) with the help of (19)-(20). For the first term, For the second term, using the estimates in Proposition 4 we have For the third term, E[ The estimate for the forth term is similar to the second one, and by the Lipschitz conditions on ∂ x g, ∂ y g, ∂ z g we have Eventually, we have Then Lemma 3.1 follows immediately from (17) and (21).
Denote the Malliavin derivative of (Ỹ t,x,n where C p depends on sup Then we prove that a subsequence of u n (s, x) in SPDE (7) is relatively compact by Theorem 2.1. Proof. We verify that u n satisfies Conditions (1)-(4) in Theorem 2.1.
Step 1. It is not difficult to see that Condition (1) is satisfied. Actually, by Lemma 2.2, Propositions 1 and 2, it yields that Step 2. We now check Condition (2). Note that D θ u ϕ n (s) = O D θ u n (s, x)ϕ(x)dx. By Proposition 3, D θ u n (s, x) = D θ Y s,x,n s exists. We further prove u ϕ n (s) ∈ D 1,2 . Computing as Propositions 1 and 4, we have u ϕ n (s) 2 The right hand side of the above inequality is bounded and independent of s and n, so sup n T 0 u ϕ n (s) 2 D 1,2 ds < ∞.
Step 4. We finally check Condition (4). For (4i), since by the equivalence of norm principle and (16) it turns out that We need to estimate each term in the above formula. From (2), we have