Several Properties Of The Wick-type Integration And The Solution For Stochastic Differential Equations With Respect To Fractional Brownian Motion

On the basis of the pioneering work of Duncan and Hu,this paper is concerned with several properties of the Wick-type integration and the solution for stochastic differential equations with respect to fractional Brownian motion.It(?)'s forumula is necessity for the problems related stochastic integration.An m-dimensional It(?) formula with expanded component is given.Using this formula several inequalities for the integration with respect to fractional Brownian motions(FBMs) for H>1/2 are gained.These extend the Burkholder-Davis-Gundy inequalities for fractional Brownian motions.Stochastically stability for the m-dimensional linear stochastic differential equations with respect to fractional Brownian motion(FBM) with Hurst parameter H>1/2 has also been studied.An improved derivative operator to Lyapunov functions is constructed, using the operator the sufficient conditions for the stochastically stability of linear stochastic differential equations driven by FBM are established.At last,the comparison theorem of two 1-dimensional SDEs with respect to fBm is obtained by using the properties of viscosity solutions of PDE.