Forward and symmetric Wick-Itô integrals with respect to fractional Brownian motion

Let BH = {BtH, t ⩾ 0} be a fractional Brownian motion with Hurst index H ∈ (0, 1). Inspired by pathwise integrals and Wick product, in this paper, we consider the forward and symmetric Wick-Itô integrals with respect to BH as follows: ∫0tus⋄d−BsH=limε↓01ε∫0tus⋄(Bs+εH−BsH)ds∫0tus⋄d°BsH=limε↓012ε∫0tus⋄(Bs+εH−B(s−ε)∨0H)ds\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eqalign{ \,\,\,\int_0^t {{u_s}} \diamondsuit {{\rm{d}}^ -}B_s^H = \mathop {\lim}\limits_{\varepsilon \downarrow 0} {1 \over \varepsilon}\int_0^t {{u_s}} \diamondsuit \left({B_{s + \varepsilon}^H - B_s^H} \right){\rm{d}}s \cr \int_0^t {{u_s}} \diamondsuit {{\rm{d}}^\circ}B_s^H = \mathop {\lim}\limits_{\varepsilon \downarrow 0} {1 \over {2\varepsilon}}\int_0^t {{u_s}} \diamondsuit \left({B_{s + \varepsilon}^H - B_{(s - \varepsilon) \vee 0}^H} \right){\rm{d}}s \cr} $$\end{document} in probability, where ⋄ denotes the Wick product. We show that the two integrals coincide with divergence-type integral of BH for all H ∈ (0, 1).