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

In this paper, we study the 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 and then use the comparison theorem as the tool to prove one existence result for multi-dimensional reflected BSDE where the coefficient is continuous and has the linear growth.It is well known that BSDE has become a field of increasing activity. It is becoming an important tool in study of fanancial mathematics, stochastic optimal control problems and partial defferential equation. The following nonlinear BSDEwas first introduced by Pardoux and Peng in 1990 (see [1]). After that, Peng proved the comparison theorem for one-dimensional BSDE in 1992 (see [2]). Zhou proved one kind of comparison theorem for mul-dimensional BSDE in 1999 (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 et al studied one-dimensional reflected BSDE with one barrier in 1997 (see [5]), and then proved the existence and uniqueness result and comparison theorem of the solution for this kind of equation.We divide this paper into four 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:and we give one n-dimensional obstacle {S(t),0 ≤ t ≤ T} ∈ Rⁿ satisfying (H2.4) {S{t),0 ≤ t ≤ T} ∈ Rⁿ is a continuous progressively measurable Rⁿ-valued process satisfying E( sup |S+(t)|²) < +∞,S{T) ≤ ξ.Here S+(t) is a Rⁿ vector , the jth element is (S_j⁺(t)).(f, ξ, S) is called one group of standard parameter for n-dimensional reflected BSDE, if it satisfies (H2.1) - (H2.4) .And then we call {(Y(t), Z(t), K(t)),0 ≤ t ≤ T} to be the solution for n-dimensional reflected BSDE if it satisfiesincrease process satisfying K_j(0) = 0 and ∫₀^T (Y_j(t) — S_j(t)) dK_j(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).Theorem 2.2. We assume (f, ξ,S) satisfies (H2.1) - (H2.4), then there exists at most a group of solution (Y, Z, K) for n-dimensional reflected BSDE satisfying (H2.5) —(H2.8).In chapter 3, we getTheorem 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 followingLet (Y¹,Z¹,K¹) and (Y²,Z²,K²) be the solution respective to (f¹,ξ¹,S¹) and (f²,ξ²,S²), then Y¹(t) ≤ Y²{t).We notice that the condition (ii) of Theorem 3.1 , i.e. the quasi-monotonously in-creasing assumption , is different to the common monotonous assumption for the generator of the reflected BSDE . Our question is whether it is possible to change to the weaker assumption :(n')fj(t,y,zl) Rn be a measurable mapping and satisfy the following(H4.1) the j-th line fj of / only contains the j-th element of z,i.e.3g{t,ui,y,j) : [0,T] x ft x Rn x Rd —> Rn,s.t.fj(t,u,y,z) = gj(t,u,y,Zj),Vt€ [0,T],u; € U,y € Rn,z e Rnxd; (H4.2) linear growth : 3 Co > 0, si.\f{t,u,y,z)\ < C0(l + \y\ + \z\),V t € [0,T),u 6 Q,y?nxd(H4.3) for fixed t,oj,f(t,u,-,-) is continuous .(H4.4) for fixed t,u,fj(t,u,-,z) is quasi-monotonously increasing, that is for j = l,2,...,n,fj{t,u,y\z) < fj{t,u,y2,z),Vy1,y2 G Rn, y) = y],y} < y2,1 ± j-And then we haveTheorem 4.1. Assume / satisfies (H4.1)-(H4.4) and ï¿¡ € Lï¿¡, S{t) 6 S', then the n-dimensional reflected BSDE exists a triple of solution (Y, Z, K) satisfying (H2.5) - (H2.8).