One Kind of Multiple Dimensional Markovian BSDEs with Stochastic Linear Growth Generators

In this article, we deal with a multiple dimensional coupled Markovian BSDEs system with stochastic linear growth generators with respect to volatility processes. An existence result is provided by using approximation techniques.


Introduction
Backward stochastic differential equations (BSDEs) was proposed firstly by Bismut (1973) [3] in linear case to solve the optimal control problems. Later this notion was generalized by Pardoux and Peng (1990) [21] into the general nonlinear form and the existence and uniqueness results were proved under the classical Lipschitz condition. A class of BSDE is also introduced by Duffie and Epstein (1992) [8] in point of view of recursive utility in economics. During the past twenty years, BSDEs theory attracts many researchers' interests and has been fully developed into various directions. Among the abundant literature, we refer readers the florilegium book edited by El-Karoui and Mazliark (1997) [9] for the early works before 1996. Surveys on BSDEs theory also includes [10] which is written by El-Karoui, Hamadène and Matoussi collected in book (2009) [7] (see Chapter 8) and the book by Yong and Zhou (1999) [23] (see Chapter 7). Some applications on optimization problems can be found in [10]. About Other applications such as in field of economics, we refer to El-Kaoui, Peng and Quenez (1997) [11] . Recently, a complete review on BSDEs theory as well as some new results on nonlinear expectation are introduced in a survey paper by Peng (2011) [22].
One possible extension to the pioneer work of [21] is to relax as much as possible the uniform Lipschitz condition on the coefficient. A weaker hypothesis is presented by Mao (1995) [19] which we translate as follows: for all y,ȳ, z,z and t ∈ [0, T ], the generator of the BSDE satisfies |f (t, y, z) − f (t,ȳ,z)| 2 ≤ κ|y −ȳ| 2 + c|y −ȳ| 2 a.s. where c > 0 and κ is a concave non-decreasing function from R + to R + such that κ(0) = 0, κ(u) > 0 for u > 0 and 0 + 1/κ(u)du = ∞. An existence and uniqueness result is proved under such condition in [19]. Hamadène introduced in (1996) [13] a one-dimensional BSDE with local Lipschitz generator. Later Lepeltier and San Martin (1997) [18] provided an existence result of minimum solution for one dimensional BSDE where the generator function f is continuous and of linear growth in terms of (y, z). When f is uniformly continuous in z with respect to (ω, t) and independent of y, a uniqueness result was obtained by Jia [17]. BSDEs with polynomial growth generator is studied by Briand in [4]. The case of 1-dimensional BSDEs with coefficient which is monotonic in y and non-Lipschitz on z is shown in work [5]. About the BSDE with continuous and quadratic growth driver, a classical research should be the one by Kobylanski [12] (2000) which investigated a onedimensional BSDE with driver |f (t, y, z) ≤ C(1 + |y| + |z| 2 ) and bounded terminal value. This result was generated by Briand and Hu into the unbounded terminal value case in [6] (2006).
There are plenty works on one-dimensional BSDE. However, limited results have been obtained about the multi-dimensional case. We refer Hamadène, Lepeltier and Peng [14] for an existence result on BSDEs system of Markovian case where the driver is of linear growth on (y, z) and of polynomial growth on the state process. See Bahlali [1] [2] for high-dimension BSDE with local Lipschitz coefficient.
As the generator of the BSDE, actually, H i is of stochastic linear growth on Z i , or in another word, it is of linear growth ω by ω. Similar situation was considered in [15] in the background of nonzero-sum stochastic differential game problem. However, in [15], the generator H i is independent on (y 1 , ..., y n ). According to our knowledge, this general form of high dimensional coupled BSDEs system with stochastic linear growth generator has not been considered in literature. This is the main motivation of the present work. The rest of this article is organized as follows: in Section 2, we give some notations ans assumptions on the coefficient. The properties of the forward SDE are also provided. The main existence result of BSDEs is proved in Section 3 where a measure domination result plays an important role. This domination result holds true when we assume that the diffusion process of the SDE satisfies the uniform elliptic condition. For the proof of the main result, we adopt an approximation scheme following the well know mollify technique. The irregular coefficients are approximated by a sequence of Lipschiz functions. Then, we obtain the uniform estimates of the sequence of solutions as well as the convergence result in some appropriate spaces. Finally, we verify that the limit of the solutions is exactly the solution to the original BSDE which completes the proof.

Notations and assumptions
In this section, we will give some basic notations, the preliminary assumptions throughout this paper, as well as some useful results. Let (Ω, F , P) be a probability space on which we define a m-dimensional Brownian motion B = (B t ) 0≤t≤T with integer m ≥ 1. Let us denote by F = {F t , 0 ≤ t ≤ T } for fixed T > 0, the natural filtration generated by process B and augmented by N P the P-null sets, i.e. F t = σ{B s , s ≤ t} ∨ N P .
Let P be the σ-algebra on [0, T ] × Ω of F t -progressively measurable sets. Let p ∈ [1, ∞) be real constant and t ∈ [0, T ] be fixed, We then define the following spaces: Hereafter, S p 0,T and H p 0,T are simply denoted by S p T and H p T . The following assumptions are in force throughout this paper. Let σ be the function defined as: which satisfies the following assumption.

is invertible and bounded and its inverse is bounded, i.e., there exits a con-
where I is the identity matrix of dimension m.
Suppose that we have a system whose dynamic is described by a stochastic differential equation as follows: (2. 2) The solution X = (X t,x s ) s≤T exists and is unique under Assumption 2.1. (cf. Karatzas and Shreve 1991 [20], p.289). We recall a well-known result associates to integrability of the solution. For any fixed (t, where the constant C is only depend on the Lipschitz coefficient and the bound of σ. In addition, property (2.3) holds true, as well, for the expectation under the probability which is equivalent to P.
For integer n ≥ 1, we first present the following Borelian function as the terminal coefficient of the n-dimensional BSDE that we considering: which satisfy Assumption 2.2 The function g i , i = 1, 2, ..., n, are of polynomial growth with respect to x, i.e. there exist constants C g and γ ≥ 0 such that x, y 1 , ..., y n , z 1 , ..., z n ), i = 1, 2, ..., n which satisfy the following hypothesis: The BSDE that we concern in this work is the following: 3 Existence of solutions for the multiple dimensional coupled BSDEs system In this section, we will provide an existence result of BSDEs (2.5) when n = 2 as an example. Actually, the case for n > 2 can be dealt with in the same way without any difficulties.

Measure domination
Before we state our main theorem, let us first recall a result related to measure domination.

High dimensional coupled BSDEs system
Our main result in this section is the following theorem.
and Z i is dt-square integrable P-a.s.; The result holds true as well for case i = n > 2 following the same way.
Proof The structure of this proof is as follows. We first use the mollify technique on the generator H i , to construct a sequence of BSDEs with generators which are of Lipschitz continuous. Then, we provide uniform estimates of the solutions, as well as the convergence property. Finally, we verify that the limits of the sequences are exactly the solutions for BSDE (3.1).