A strong approximation for double subfractional integrals

Abstract

A Riemann-Stieltjes integral strong approximation to double Stratonovich integrals with respect to odd and even fractional Brownian motions is considered. We prove the convergence in quadratic mean, uniformly on compact time intervals, of the ordinary double integral process obtained by linear interpolation of the odd and even fractional Brownian motions, to the double Stratonovich integral. The deterministic integrands are continuous or are given by bimeasures.

Multiple sub-fractional integrals and some approximations

“…The main result of this article (Theorem 4.1) extends the results from [25], for Wong-Zakai and mollifier approximations, to the stronger mean square convergence in the uniform norm. The technique utilizes the pathwise representation of multiple subfractional integrals as multiple Wiener-Itoˆintegral via an operator of fractional type (Proposition 3.1) and a maximal inequality for multiple Stratonovich integral process driven by the Brownian motion as obtained in [18].…”

“…Þ: Assume now that 1 j ½ðp=2Þ: We use Equation (25). The contribution 'of order 1=n' to (34) is given, taking into account the definition off m, k and (25) …”

On the convergence to the multiple subfractional Wiener–Itô integral

“…Tudor (2007a) considered that the double Wong-Zakai approximation converges in the L 2 sense and uniformly on every compact time interval, to the double Stratonovich subfractional integral, the integrands are continuous or given by bimeasures. Tudor (2008b) extended the results from Tudor (2007a), for Wong-Zakai and mollifier approximation, to the stronger mean square convergence in the uniform norm. On the other hand, Bardina, Es-Sebaiy, and Tudor (2010) researched approximation of the finite dimensional distributions of multiple fractional integrals.…”