Generalized differentiability with respect to the initial data of a flow generated by a stochastic equation with reflection

Let $\varphi _t(x)$, $x\in \mathbb {R}^d_+$, be a solution of a stochastic differential equation in the half-space $\mathbb {R}^d_+$ with normal reflection in the boundary; the solution starts from a point $x$. We prove that the random mapping $\varphi _t(\boldsymbol \cdot ,\omega )$ is differentiable in the Sobolev sense for almost all $\omega$. We obtain a stochastic equation for the derivative $\nabla \varphi _t$.