Convergence rate of multiple fractional Stratonovich type integral for Hurst parameter less than 1/2

Abstract

In this paper, we have investigated the problem of the convergence rate of the multiple integral { ∫ 0 T . ⋯ ∫ 0 T f ( t 1 , ⋯ , t n ) d B t 1 H , π } , where f ɛ C n+1([0, T] n) is a given function, π is a partition of the interval ([0, T]) and { B t i H , π } is a family of interpolation approximation of fractional Brownian motion B H t with Hurst parameter H<1/2. The limit process is the multiple Stratonovich integral of the function f. In view of known results, the convergence rate is different for different multiplicity n. Under some mild conditions, we obtain that the uniform convergence rate is Δ 2 H in the mean square sense, where Δ is the norm of the partition generating the approximations.