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

摘要

Multiple integrals with respect to several Holder continuous functions of exponent μ > 1/2 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 H > 1/2. 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 Ito integral with random kernels. The Hu-Meyer formulas between multiple Ito and multiple pathwise integrals with random kernels are also obtained.