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 $\nabla\varphi_t$.