Linear Skorohod stochastic differential equations (Q1178980)

Let \(\sigma\) and \(b\) be bounded processes on the Wiener space \((\Omega,{\mathcal F},P)\), \(\Omega=C([0,1])\), which are possibly anticipating the Brownian motion \(W_ t(\omega)=\omega(t)\), and let \(\eta\) be a bounded random variable. We deduce the existence and uniqueness of a solution \(X\) for the linear equation with Skorokhod integral \[ X_ t=\eta+\int^ t_ 0\sigma_ sX_ sdW_ s+\int^ t_ 0b_ sX_ sds,\quad t\in[0,1],\leqno (1) \] under rather weak assumptions on \(\sigma\) and no additional requirement on \(b\) and \(\eta\). The descriptions on \(X\) requires to study the family \(\{T_ t, t\in[0,1]\}\) of transformations \(T_ t\) of \(\Omega\) into itself associated to (1) by the equation \[ T_ t=\omega+\int^{t\wedge.}_ 0\sigma_ s(T_ s\omega)ds, \quad \omega\in\Omega,\quad t\in[0,1]. \]