Bismut–Elworthyʹs formula and random walk representation for SDEs with reflection

We study the existence of first derivatives with respect to the initial condition of the solution of a finite system of SDEs with reflection. We prove that such derivatives evolve according to a linear differential equation when the process is away from the boundary and that they are projected to the tangent space when the process hits the boundary. This evolution, rather complicated due to the structure of the set at times when the process is at the boundary, admits a simple representation in terms of an auxiliary random walk. A probabilistic representation formula of Bismut–Elworthyʹs type is given for the gradient of the transition semigroup of the reflected process.