It\^o type stochastic differential equations driven by fractional Brownian motions of Hurst parameter $H>1/2$

TL;DR

This paper establishes existence and uniqueness results for Itô type stochastic differential equations driven by fractional Brownian motions with Hurst parameter greater than 1/2, using novel techniques to handle anticipative coefficients and Malliavin derivatives.

Contribution

It introduces a new approach to solve Itô type SDEs with anticipative coefficients driven by fractional Brownian motion, including a novel technique replacing Grönwall's lemma.

Findings

Existence and uniqueness of solutions up to a positive random time.

Global solutions in linear and quasilinear cases.

A new method to analyze equations involving Malliavin derivatives.

Abstract

This paper studies the existence and uniqueness of solution of It\^o type stochastic differential equation $dx(t)=b(t, x(t), \om)dt+\si(t,x(t), \om) d B(t)$, where $B(t)$ is a fractional Brownian motion of Hurst parameter $H>1/2$ and $dB(t)$ is the It\^o differential defined by using Wick product or divergence operator. The coefficients $b$ and $\si$ are random and can be anticipative. Using the relationship between the It\^o type and pathwise integrals we first write the equation as a stochastic differential equation involving pathwise integral plus a Malliavin derivative term. To handle this Malliavin derivative term the equation is then further reduced to a system of (two) equations without Malliavin derivative. The reduced system of equations are solved by a careful analysis of Picard iteration, with a new technique to replace the Gr\"onwall lemma which is no longer applicable. The solution of this system of equations is then applied to solve the original It\^o type stochastic differential equation up to a positive random time. In the special linear and quasilinear cases the global solutions are proved to exist uniquely.