分数布朗运动驱动下z一致连续的BSDE解的存在性与唯一性

Existence and Uniqueness of the Solution to BSDE Driven by Fractional Brownian Motion with the Generator is Uniformly Continuous in z

本文研究一类由分数布朗运动驱动的一维倒向随机微分方程解的存在性与唯一性问题,在假设其生成元满足关于y Lipschitz连续,但关于z一致连续的条件下,通过应用分数布朗运动的Tanaka公式以及拟条件期望在一定条件下满足的单调性质,得到倒向随机微分方程的解的一个不等式估计,应用Gronwall不等式得到了一个关于这类方程的解的存在性与唯一性结果,推广了一些经典结果以及生成元满足一致Lipschitz条件下的由分数布朗运动驱动的倒向随机微分方程解的结果.

We study the existence and uniqueness problem of the solutions to a class of one dimensional backward stochastic differential equation driven by fractional Brownian motion. We assume the generator of theses equations satisfies the conditions,that is,the generator is Lipschitz continuous in y and uniformly continuous in z,by using the Tanaka＇s formula for fractional Brownian motion and the propositions of quasi-conditional expectation,particularly the monotone proposition which is satisfied only under some conditions,we obtain some inequality estimations for the solution to the backward stochastic differential equation, moreover,by using Gronwall＇s inequality,we obtain a result of existence and uniqueness for the solution to the backward stochastic differential equation.Our result generalizes some known result among the classical backward stochastic differential equation theory and the results which are obtained under the uniformly Lipschitz condition among the fractional Brownian motion ranges.