Integration by Parts Formula and Applications for SDEs Driven by Fractional Brownian Motions

Abstract

By using coupling by change of measures, the Driver-type integration by parts formula is established for a class of stochastic differential equations driven by fractional Brownian motions. As applications, (log) shift Harnack inequalities and estimates on the distribution density of the solutions are presented.

Harnack Inequality and Applications for Stochastic Retarded Differential Equations Driven by Fractional Brownian Motion

“…However, the Harnack inequality established in [26] is local in the sense that |x−y| is bounded by a constant. The Harnack inequality established in our Theorem 3.1 is global.…”

“…In [26], Fan established Harnack inequality for equation (2.4) of high dimensions and multiplicative noise in one-dimension without delay by using the method of derivative formula. However, the Harnack inequality established in [26] is local in the sense that |x−y| is bounded by a constant.…”

“…As applications, the global Harnack type inequalities are presented. However, in [28] the condition that b is Fréchet differentiable such that ∇b is bounded and Lipschitz continuous seems to be relatively strong. On the other hand, as far as I know, in the case that 1/2 < H < 1, using the approach of coupling for the segment process and Girsanov transformations, in virtue of irregularity of the operator K −1 H , it is very difficult to obtain the Harnack inequality.…”

“…driven by fractional Brownian motion with Hurst parameter 1/2< H<1 were established by Fan in [26]. Subsequently, using the approach of coupling and Girsanov transformations to fractional Brownian motion with Hurst parameter 1/2<H<1, the global Harnack inequalities for the following stochastic differential equations…”

“…Furthermore, using Malliavin calculus, Fan [28] established Bismut derivative formulae for the following stochastic differential equations…”