Stochastic Differential Equations Driven by Fractional Brownian Motion and Standard Brownian Motion

Abstract

We prove an existence and uniqueness theorem for solutions of multidimensional, time dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst parameter H > 1/2 and a multidimensional standard Brownian motion. The proof relies on some a priori estimates, which are obtained using the methods of fractional integration and the classical Itô stochastic calculus. The existence result is based on the Yamada–Watanabe theorem.

Keywords:

• Fractional Brownian motion
• Stochastic differential equations

Mathematics Subject Classification:

• 60H05
• 60H10

J. G. was supported by FCT-(FEDER/POCI 2010) and by the MEC Grant No. MTM2006-03211. D. N. was supported by the NSF Grant No. DMS0604207.