A Kneser-type theorem for backward doubly stochastic differential equations

A class of backward doubly stochastic differential equations (BDSDEs in short) with continuous coefficients is studied. We give the comparison theorems, the existence of the maximal solution and the structure of solutions for BDSDEs with continuous coefficients. A Kneser-type theorem for BDSDEs is obtained. We show that there is either unique or uncountable solutions for this kind of BDSDEs.


Introduction
Nonlinear backward stochastic differential equations (BSDEs in short) have been independently introduced by Pardoux and Peng [14] and Duffie and Epstein [3]. The comprehensive applications of BSDEs have motivated many efforts to establish the existence and uniqueness of adapted solution under general hypotheses on the coefficients. For instance, for the one-dimensional case, Lepeltier and San Martin [11] proved the existence of a solution to BSDEs under the assumption of continuous coefficient. Recently, Jia and Peng [5] studied the structure of the solutions to BSDEs with continuous coefficients.
The comparison theorem is an important and effective technique in the theory of BSDEs. There are much works concerning the comparison theorems of BSDEs. For instance, for applications to finance, El Karoui, Peng and Quenez [7] have given the comparison theorem of BSDEs with Lipschitz coefficients. And then Liu and Ren [12] have given some comparison theorems of BSDEs with continuous coefficients.
In 1923, Kneser [10] proved that the cardinality of the set of solutions for ordinary differential equations (ODEs in short) with continuous coefficients is either one or of continuum. Many interesting generalizations have followed. For example, in 1956, Alexiewicz and Orlicz [1] got the same theorem for a class of partial differential equations (PDEs in short). In 2008, Jia [6] proved Kneser-type theorem for BSDEs with continuous coefficients. For more information about Kneser-type theorems, one can refer to [8,9,17,18].
A class of backward doubly stochastic differential equations (BDSDEs in short) was introduced by Pardoux and Peng [15] in 1994, in order to provide a probabilistic interpretation for the solutions of a class of semilinear stochastic partial differential equations (SPDEs in short). They have proved the existence and uniqueness of solution for BDSDEs under uniformly Lipschitz conditions. Since then, many efforts have been made to relax the assumption on the coefficients. For instance, Gu [4] proved the existence and uniqueness of solution for BDSDEs under the local Lipschitz conditions. Shi, Gu and Liu [16] have relaxed the Lipschitz assumptions to linear growth conditions by virtue of their comparison theorem of BDSDEs with Lipschitz conditions, and showed the existence of the minimal solution of BDSDEs. Bally and Matoussi [2] have given a probabilistic interpretation of the solutions in Sobolev spaces for parabolic semilinear stochastic PDEs in terms of BDSDEs. Zhang and Zhao [19] have proved the existence and uniqueness of solution for BDSDEs on infinite horizon, and described the stationary solutions of SPDEs by virtue of the solutions of BDSDEs on infinite horizon.
Due to their important significance to SPDEs, it is necessary to give intensive investigation to the theory of BDSDEs. The aim of this paper is to study the structure of the solutions to BDSDEs with continuous and linear growth conditions. We firstly generalize the comparison theorems to the case where the coefficients are continuous. As an application of the comparison theorems, we give the existence of the maximal solution of BDSDEs with continuous coefficient by means of our comparison theorems. Finally we will show that there is either unique or uncountable solutions for this kind of BDSDEs. In fact, our result shows the structure of those solutions, that is, we obtain a Kneser-type theorem for BDSDEs.
The rest of the paper is organized as follows. In Section 2, we present the main assumptions and some preliminary results. In Section 3, we generalize the comparison theorem of BDSDEs in [16] to BDSDEs with continuous coefficients. In Section 4, we prove the existence of the maximal solution of BDSDEs. Finally, in Section 5, we discuss the structure of solutions of BDSDEs with continuous coefficients and linear growth conditions.

Preliminaries
Let (Ω, F, P ) be a probability space, and T > 0 be an arbitrarily fixed constant throughout this paper. Let {W t ; 0 ≤ t ≤ T } and {B t ; 0 ≤ t ≤ T } be two mutually independent standard Brownian Motions with values in R d and R l , respectively, defined on (Ω, F, P ). Let N denote the class of P -null sets of F.
Note that the collection {F t ; t ∈ [0, T ]} is neither increasing nor decreasing, so it does not constitute a filtration. All the equalities and inequalities mentioned in this paper are in the sense of dt × dP almost surely on [0, T ] × Ω.
We use the usual inner product ·, · and Euclidean norm |·| in R k , R k×l and R k×d . All the equalities and inequalities mentioned in this paper are in the sense of dt × dP almost surely on [0, T ] × Ω.
For any k ∈ N , let M 2 (0, T ; R k ) denote the set of (classes of dP ⊗ dt a.e. equal) k-dimensional jointly measurable stochastic processes {ϕ t ; t ∈ [0, T ]} which satisfy: For any t ∈ [0, T ], denote by L 2 (Ω, F t , P ; R k ) the set of k-dimensional random variables ξ, which satisfy: Consider the following BDSDE: We assume that and there exist a constant C > 0 such that for any t and there exist constants C > 0 and 0 < α < 1 such that for any t ∈ [0, T ], Note that the integral with respect to {B t } is a "backward Itô integral", in which the integrand takes values at the right end points of the subintervals in the Riemann type sum (for details refer to [15]), and the integral with respect to {W t } is a standard forward Itô integral. These two types of integrals are particular cases of the Itô-Sokorohod integral (see [13] for details).
This proposition was derived in [15].

Comparison theorem of BDSDEs with continuous coefficients
In this paper, we only consider one-dimensional BDSDEs, i.e., k = 1. Assume (H3) for fixed ω and t, f (ω, t, ·, ·) is continuous; We consider the following BDSDEs: (0 ≤ t ≤ T ) where The comparison theorem was established by Shi, Gu and Liu [16] for onedimensional BDSDEs, where both the coefficients f 1 and f 2 satisfy Lipschitz conditions. In this section, we firstly generalize the comparison theorem to the case where one of the coefficients f 1 and f 2 is only continuous, another is Lipschitz continuous.  (2) and (3), respectively. If (H5) holds, then Proof. It is easy to see that ( Applying Itô-Tanaka's formula (cf. [16] Since (Y 1 , Z 1 ) and (Y 2 , Z 2 ) are in S 2 (0, T ; R)×M 2 (0, T ; R d ), by virtue of Lemma 1.3 of [15], it follows that From (H1) and Young's inequality, it follows that where c > 0 only depends on the Lipschitz constant C in (H1). By the assumption (H2), we deduce By Gronwall's inequality, it follows that In next section, we will prove the maximal solution of BDSDE with continuous coefficients as an application of Theorem 3.1. Next, let us generalize the comparison theorem to the case where the coefficients are only continuous.  Proof. First, for fixed t, we define, as in Lemma 1 of [11], the sequence f 2 n (t, y, z) associated with f 2 where Q is the rational number set. Then, for n ≥ K, f 2 n are measurable and Lipschitz functions, and f 1 ≥ f 2 ≥ f 2 n . Hence, we know that the following BDSDE has a unique solution (Y 2 n , Z 2 n ) From Theorem 3.1, it follows that Y 1 ≥ Y 2 n and Y 2 ≥ Y 2 n a.s. for all n ≥ K. However, (Y 2 n , Z 2 n ) converges uniformly in t to (Y 2 , Z 2 ) (cf. [16]). Therefore, Y 1 ≥ Y 2 a.s., in particular, Y 1 ≥ Y 2 a.s..
Next, for fixed t, we define the sequence f 1 n (t, y, z) associated with f 1 , then, by virtue of Lemma 4.2 in next section, for n ≥ K, f 1 n are measurable and Lipschitz functions, and f 1 n ≥ f 1 ≥ f 2 . Hence, we know that the following BDSDE has a unique solution (Y 1 n , Z 1 n ) From Theorem 3.1, it follows that Y 1 n ≥ Y 1 and Y 1 n ≥ Y 2 a.s. for all n ≥ K. However, (Y 1 n , Z 1 n ) converges uniformly in t to (Y 1 , Z 1 ) (cf. Lemma 4.2). (2) has a unique solution (Y 1 , Z 1 ), then for any solutions (Y 2 , Z 2 ) of (3) we have
(iii) If the uniqueness of neither (2) nor (3) holds, we were unable to derive a comparison result for any solutions of (2) and (3).
In particular, we easily see that the Lipschitz condition is a special case of our proposed conditions. In other words, Theorem 3.1 and Theorem 3.2 generalize the comparison result in [16].
The second step, it remains to remove the additional condition (H9). Fix N > 0 and choose a Lipschitz continuous mapping b(t, y, z) in y and z such that f 1 (t, y, z) > b(t, y, z) > f 2 (t, y, z) for t ∈ [0, T ], y ∈ R with |y| < N , z ∈ R d . From (H10), there exists a constantε > 0 such that To show the existence of such a mapping b(t, y, z), we writeb(t, y, z) := f 2 (t, y, z) +ε 2 , and J δ (y) = δ −1 J(y/δ), where the constant k satisfies R J(y)dy = 1. Let us smooth outb in y to obtain b δ , i.e., setting whereε and δ are as in (H7). Now any b δ ′ with δ ′ ≤ δ is our candidate. Let (Y, Z) denote a solution of (1) when f is replaced by b. By the above consideration, we observe As a result of that N > 0 is arbitrary, the desired result is obtained.

The existence of the maximal solution of BDSDEs
Under the conditions (H2)-(H4), Shi, Gu and Liu [16] have proved that BDSDE (1) has the minimal solution (Y , Z). Now, we will prove that BDSDE (1) has also the maximal solution (Y , Z). In order to prove Theorem 4.1, we need the following lemma which can be obtained by means of the similar arguments in [11], so we omit its proof.

Kneser-type theorem for BDSDEs
In this section, we shall discuss an interesting question: How many solutions does a one-dimensional BDSDE (1) satisfying (H2)-(H4) have? This is a classical Kneser-type problem. Under some appropriate conditions, we shall prove a Kneser-type theorem for BDSDEs satisfying (H2)-(H4).
It is easy to see that (Y t , Z t ) t∈[0,T ] ∈ S 2 ([0, T ]; R) × M 2 (0, T ; R d ) is a solution of BDSDE (1) with Y T = ξ and Y t 0 = η. By the similar arguments in Theorem 4.1, it is easy to check the closedness of the set of solutions of BDSDE (1) by the continuity of f with respect to y and z.
Corollary 5.2 Indeed, in the case when the solution of BDSDE (1) is not unique, the cardinality of the set of the associated solutions is at least of continuum since we can take η = αY t 0 + (1 − α)Y t 0 for each α ∈ [0, 1]. Thus Theorem 5.1 is a Kneser-type theorem for BDSDEs.