Multiple sub-fractional integrals and some approximations

Abstract

We study multiple fractional integrals with respect to the even fractional Brownian motion (also called sub-fractional Brownian motion). The multiple integrals are introduced by using a representation formula for the even fractional Brownian motion as a Wiener integral with respect to a Brownian motion defined on the same probability space and a transfer principle. Then, Riemann–Stieltjes integral approximations to multiple Stratonovich fractional integrals are also considered. For two standard approximations (Wong–Zakai and mollifier approximations) and continuous integrands, the mean square convergence in the uniform norm of these approximations to the multiple Stratonovich sub-fractional integral is shown.

Keywords:

• even fractional Brownian motion
• multiple Stratonovich integral
• Wong–Zakai approximation
• Mollifier approximation

AMS 2000 Subject Classification :

• 60H05

Acknowledgement

The author acknowledges support from the Romanian Ministry of Education and Research (CEEX Program, Project 2-CEx06-11-18/2006).