带Poisson跳随机微分方程终值与边值问题的适应解

Adapted Solutions of Stochastic Differential Equations for Terminal and Boundary Value Problems with Poisson Jumps

Poisson跳的拟线性倒向随机微分方程x(t) +∫tf(s,x(s),,x(s))+y(s)]dMs =ξ,t∈[0,1],这里M = (W,Q)T,其中W为Wiener过程,Q为补偿Poisson过程.利用区间延拓和 Bihari 不等式证明了在某种弱于Lipschitz条件下方程存在唯一适应解,并给出了解的估计,从而将文章[1]的结论推广到带 Poission 跳的情形.另外,本文还讨论了以下形式的边值问题:dx(t) = f(t,x(t),y(t))dt + y(t)dMt,Ax(0) + Bx(1) =ξ*,t∈[0,1],并证明了在Lipschitz条件下适应解的存在唯一性.

In this paper,we consider a quasi linear backward stochastic differential equation with Poisson jumps:x(t)+∫ 1 tf(s,x(s),y(s)) d s+∫ 1 t[g(s,x(s))+y(s)] d M s=ξ,t∈[0,1],where M=(W,Q) T ,and W is a d dimensional standard Wiener process,Q is an (m-d) dimensional compensated Poisson process.By using the continuation method and Bihari inequality,the existence and uniqueness of the adapted solutions under a condition weaker than the Lipschitz one are proved,and the moment estimates of the solutions are also obtained.These results can be used to consider the solution of the boundary value problem: d x(t)=f(t,x(t),y(t)) d t+y(t) d M t, Ax(0)+Bx(1)=ξ *, t∈[0,1],where the Lipschitz condition is assumed.