Multiple integrals and expansion of solutions of differential equations driven by rough paths and by fractional Brownian motions

Abstract

Multiple integrals with respect to several Hölder continuous functions of exponent are studied by using fractional calculus. They are applied to obtain the Volterra expansion (with remainder) for the solution of a differential system driven by rough paths. The results are applied to stochastic differential equations driven by fractional Brownian motions of Hurst parameter . For the solution of a stochastic differential equation driven by fractional Brownian motion, we obtain its chaos expansion, as well as the finite chaos expansion with remainder. To this end, we study the multiple Itô integral with random kernels. The Hu–Meyer formulas between multiple Itô and multiple pathwise integrals with random kernels are also obtained.

Keywords:

• fractional Brownian motions
• fractional calculus
• differential equations driven by rough pathes
• Hu–Meyer formula
• Itô and Stratonovich multiple integral
• Malliavin calculus

2000 Mathematics Subject Classification::

• 26A33
• 26A42
• 34A34
• 34G20
• 41A58
• 60G22
• 60H05
• 60H07
• 60H10
• 60H30
• 93C15
• 93E03

Acknowledgements

I would like to thank an anonymous referee for his/her valuable and useful comments that helped to improve this paper significantly.

Notes

¹This work was supported, in part, by the National Science Foundation under Grant No. DMS0504783 and by a grant from the Simons Foundation #209206.