Stochastic,Averaging,Principle,for,Mixed,Stochastic,Differential,Equations
JING Yuanyuan,PENG Yarong and LI Zhi
School of Information and Mathematics,Yangtze University,Jingzhou 434023,China.
Abstract. In this paper, an averaging principle for the solutions to mixed stochastic differential equation involving standard Brownian motion,a fractional Brownian motion BH with the Hurst parameter H> 12 and a discontinuous drift was estimated.Under some proper assumptions, we proved that the solutions of the simplified systems can be approximated to that of the original systems in the sense of mean square by the method of the pathwise approach and the Itˆo stochastic calculus.
Key Words:Averaging principle;mixed stochastic differential equation;discontinuous drift;fractional Brownian motion.
In this paper,we consider the following mixed stochastic differential equation:
whereX0∈R,t∈[0,T],a,b,c:[0,T]×R →R, andBHis a fractional Brownian motion with the Hurst parameterH∈(12,1). The processesWandBHcan be dependent. The integral w.r.t.Wiener processWis a standard Itˆo integral,and the integral w.r.t.BHis the pathwise generalized Lebesgue-Stieltjes integral which is defined in[1,2,3].
The strongest motivation to study such equations comes, in particular, from financial modeling. It is well known that there are two kinds of sources that affect prices in financial markets. The random noise from the first source usually prevails in a shorter time periods and can be modeled by a Wiener processW. While the random noise of the second source is coming from economical background and usually has a long range dependence property,which can be modeled by ...
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