Infinite Horizon Multi-Dimensional Reflected Backward Stochastic Differential Equations And The Comparison Theorem

In this paper,we study the infinite horizon multi-dimensional reflected backward stochastic differential equations (BSDE in short). The existence and uniqueness result of the solution for this kind of equation was proved, and we also give one kind of multi-dimensional comparison theorem for the reflected BSDE .It is well known that BSDE has become a field of increasing activity.It is becoming an important tool in study of financial mathematics,stochastic optimal control problems and partial differential equation. The following nonlinear BSDEwas first introduced by Pardoux and Peng in1990(see[1]).After that, Peng prove the comparison theorem for one-dimensional BSDE in 1992(see[2]).Zhou proved one kind of comparison theorem for multi-dimensional BSDE in1999(see[3]),and then use the comparison theorem as the tool to prove one existence result for multi-dimensional BSDE where the coefficient is continuous and has the linear growth.El.Karoui er al studied one-dimensional reflected BSDE with one barrier in 1997(see[5]),and then proved the existence a nd uniqueness result and comparison theorem of the solution for this kind of equation.we divide this paper into three chapters.The first chapter is an introduction.In Chapter 2,we give the following model for multi-dimensional reflected BSDE. We first give the comparison definition for two vectors in Rⁿ: and then assume:(H2.3) There are two positive functions u₁(t),u₂(t), satisfying where t ≥ 0, y,y' ∈ Rⁿ, z, z' ∈ H^n×d and U₁(t)dt < ∞, u₂²(t)dt < ∞. and we give one n-dimensional obstacle{5(f),t ≥ 0} ∈ Rⁿ satisfying (H2.4) S(t) is a continuous progressively measurable Rⁿ-valued process satisfying(f, ξ,S) is called one group of standard parameter for n-dimensional reflected BSDE, if it is satisfies (H2.1)-(H2.4).And then we call {(Y(t),Z(t),K(t)),t ≥ 0} to be the solution for infinite horizon n-dimensional reflected BSDE if it satisfies(H2.8) K(t) is continuous and increase process satisfying K(0) = 0 and (Y(t)-S(t))dK(t) = 0.Theorem 2.1: We assume (f,ξ,S) satisfies (H2.1)-(H2.4), then there exists a group of solution (Y,Z,K) for n-dimensional reflected BSDE satisfying (H2.5)-(H2.8) , which is also unique.After we proved the existence and uniqueness of the solution, we have also proved the solution is continuous about the parameter.In Chapter 3,we get Theorem 3.1: Let (f¹ ξ¹, S¹) and(f², ξ², S²) be two standard parameters of the n-dimensional reflected BSDE satisfying(H2.1)-(H2.4), and suppose in addition the followingthen Y¹ ≤ Y² .Then we think about whether it is possible to change to the weaker assumption:From the counterexample 3.1, we know that the comparison theorem does not hold under the assumption (ii').