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

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