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

摘要

Let 4t(x),x∈R~d_+ , be a solution of a stochastic differential equation in the half-space R~d_+ with normal reflection in the boundary; the solution starts from a point 4_t(·,ω). We prove that the random mapping 4_t(·,ω) is differentiable in the Sobolev sense for almost all ω. We obtain a stochastic equation for the derivative ▽_(4t).