分数布朗运动驱动的一类随机微分方程的弱解问题

Weak Solutions for Stochastic Differential Equations Driven by Fractional Brownian Motion

本文,我们研究如下分数布朗运动驱动的一类随机微分方程的弱解问题[X_t=x+B_t^H+\int_0^tb(s,X_s)\md s,]其中B^H=\{B_t^H,\,0\leq t\leq T\}是Hurst指数为H\in(0,1/2)\cup(1/2,1)的分数布朗运动,b是Borel可测函数且满足线性增长条件|b(t,x)|\leq(1+|x|)f(t),其中x\in\mathbb{R}且0<t<T,f是非负Borel函数.值得注意的是f是无界的,比如函数f(t)=(T-t)^{-\beta}或f(t)=t^{-\alpha},对于一些0<\alpha,\beta<1无界.这个问题对于分数布朗运动驱动的随机微分方程来说是有意义的.

Let B^H=\{B_t^H,\,0\leq t\leq T\}$be a fractional Brownian motion with Hurst index H\in(0,1/2)\cup(1/2,1)and let b be a Borel measurable function such that|b(t,x)|\leq(1+|x|)f(t)$for x\in\mathbb{R}$and$0<t<T$,where$f$is a non-negative Borel function.In this note,we consider the existence of a weak solution for the stochastic differential equation of the form\[X_t=x+B_t^H+\int_0^tb(s,X_s)\md s.\]It is important to note that$f$can be unbounded such as f(t)=(T-t)^{-\beta}and f(t)=t^{-\alpha}for some 0<\alpha,\beta<1.This question is not trivial for stochastic differential equations driven by fractional Brownian motion.