Anticipative Stochastic Differential Equations with Non-smooth Diffusion Coefficient

In this paper we prove the existence and uniqueness of the solutions to the one-dimensional linear stochastic differential equation with Skorohod integral \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ X_{t} {\left( w \right)} = \eta {\left( w \right)} + {\int_0^t {a_{s} X_{s} {\left( w \right)}d} }{\text{\bf W}}_{s} + {\int_0^t {b_{s} X_{s} {\left( w \right)}ds,t \in {\left[ {0,1} \right]}} }, $$\end{document} where (Ws) is the canonical Wiener process defined on the standard Wiener space (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\fancyscript W}$$\end{document} ,ℋ , μ), a is non-smooth and adapted, but η and b may be anticipating to the filtration generated by (Ws). The intention of the paper is to eliminate the regularity of the diffusion coefficient a in the Malliavin sense, in the existing literature. The idea is to approach the non-smooth diffusion coefficient a by smooth ones.