Backward Doubly Stochastic Differential Equation With Non-Lipschitz Coefficients And Comparison Theorem

In this paper, (1) we prove the existence and uniqueness of backward doubly stochastic differential equations with non-Lipschitz coefficients, and the comparison theorem of this kind equations;(2) we prove the existence and uniqueness of infinite horizon backward doubly stochastic differential, continuous dependence theorem and convergence theorem for this class of equations.In 1990, Pardoux and Peng introduced the following backward stochastic differential equation:From then on, a lot of researchers have applied themselves to this field.In 1994, Pardoux and Peng also introduced the following backward stochastic differential equation :We call backward doubly stochastic differential equation. we note that the integral with respect to {B_t} is a "backward Ito integral" and the integral with respect to {W_t} is a standard forward Ito integral, (which will become clear below) the solution of this kind equation gave a probabilistic representation of such systems of quasilinear SPDES. i.e applied this to the Feynman-Kac formula of SPDES.Where T is a finite time interval, when T = ∞, i.e.infinite horizon backward doubly stochastic differential equation:Infinite horizon BDSDEs are also very interesting to produce a probabilistic representation of certain quasilinear stochastic partial differential equations. Where we have to restrictg to be independent of z. i.e.Yt = Z+ [' f(s,Y,,Za)ds + [ g(s,Ys)dBs - f°ZsdWs, t > 0. (0.2) Jt Jt JtThe aim of this paper is to study the existence and uniqueness of eq.(O.l) under kind of non-Lipschitz (Hl.l), (H1.3), (H2.1), (H2.2) and comparison theorem, and the existence and uniqueness of eq.(0.2)under(H3.1), (H3.2), (H3.3). this paper is composed of three section: Firstly we want to introduce the paper's background.In section 1: under conditions(Hl.l), (Hi.3), we apply the method of Picard-type iteration to get the existence and uniqueness of solution to (0.1), we also proved the comparison theorem of this kind equation.Theoreml.10 under the conditions (Hl.l), (HI.3), then there exists a unique solution{Yt,Zt) € S2{[0,T};Rk) x M2{0,T;Rkxd) to equation (0.1).Theorem 1.11 There are two backward doubly stochastic differential equation:Yt = Z+ [ f(s,Ys,Zs)ds+ [ g(s,Ys,Zs)dBs- f ZsdWs, te[0,T\, Jt Jt JtYt = ￡ + f f(s,Ys,Zs)ds+ [ g(s,Ys,Zs)dBs- f ZsdWs, te[0,T}. Jt Jt Jtsatisfy the conditions of theorem 1.10, let(YJ, Zt) and (Yt, Zt) be solutions respectively. If ￡ < ￡, f(t, Yt, Zt) < f{t, Yt, Zt), then Yt < Yt, a.s., V* € [0, T).In section 2: under the assumption (H2.1)(H2.2), we apply the method of from locally to globally to get the existence and uniqueness of solution to (0.1).Theorem2.4 under the condition of (H2.1) and(H2.2). Then there exists a unique solution(Yt,Zt) E S2([0,T);Rk) x M2(0,T;Rk*d)to equation (0.1).In section 3: under the condition of (H3.1), (H3.2)and (H3.3), we get the existence and uniqueness of solution to eq.(0.2), and continuous dependence theorem and convergence theorem for this class of equations.Theorem 3.2 Let f e L2(n,F00,P;Rk) be given, (H3.1),(H3.2) and (H3.3) hold for / and g. Then the BDSDE(0.2) has a unique solution (y., z.) e B2.Theorem'3.3 Suppose & € L2[Sl,1Fa0,P\I&), (i = 1,2), under'(H3.1)-(H3.3), let (y\2*) be the solutions of BDSDE (0.2) corresponding to the terminal data ( = fi,f = ￡2) respectively, then there exists a constant C > 0 such that\\{yl -y\zl -z2)\\2BTheorem 3.4 Suppose f, & € L2(Q, ??, P; Rk), (k = 1,2, ? ? ?), (H3.1)-(H3.3) hold for / and g. Let (yk,zk) be the solutions of the following BDSDEs:/(*. V*. -z*)rfs + / P(s> y*)d5f - / zkdWs, 0 0 as k -^ oc, then there exists a pair (y, z) € B2 such that\\(yk - y, zk - z)\\B ->? 0 as k -> oo. Furthermore , (y, z) is the solution of the followingBDSDE:/OO />OO />00/(s, ya> zs)ds + I g(s, ys)dBs - I zsd\Vs, 0 < t < oo.