Weak Solutions for Stochastic Differential Equations Driven by Fractional Brownian Motion

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\mathbbR 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.