Itô type stochastic differential equations driven by fractional Brownian motions of Hurst parameter $H>1/2$

This paper studies the existence and uniqueness of solution of Itô 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 Itô 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 Itô 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are solved by a careful analysis of Picard iteration, with a new technique to replace the Grönwall lemma which is no longer applicable. The solution of this system of equations is then applied to solve the original Itô 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.

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