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Let φt(x), t ≥ 0, x ∈ Rd+ be a flow generated by SDE in a halfspace R + with normal reflection at the hyperplane R×{0}. We prove that the flow φt is Sobolev differentiable with respect to a starting point x with probability 1 if coefficients of the corresponding SDE are Lipschitz continuous and diffusion is not degenerate. Stochastic equations for derivative ∇xφt(x) are obtained. We also discuss an existence of Fréchet derivative (not only Sobolev).

Differentiability of stochastic reflecting flow with respect to starting point