关键词: fractional Brownian motion, trace, Stratonovich multiple integral, conver-gence rate

Abstract:

In this paper, we have investigated the problem of the convergence rate of the multiple integral

{∫^T₀···∫^T₀ f(t₁, ··· , t_n)dB^H, π_t1 ···dB^H, π_tn^},
where f ∈ ^Cn+1([0, T]ⁿ) is a given function, π is a partition of the interval [0, T] and {B^H, π_ti } 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^2H in the mean square sense, where Δ is the norm of the partition generating the approximations.

Key words: fractional Brownian motion, trace, Stratonovich multiple integral, conver-gence rate