Stochastic representations of derivatives of solutions of one-dimensional parabolic variational inequalities with Neumann boundary conditions

Abstract

International audienceIn this paper we explicit the derivative of the flows of one-dimensional reflected diffusion processes. We then get stochastic representations for derivatives of viscosity solutions of one-dimensional semilinear parabolic partial differential equations and parabolic variational inequalities with Neumann boundary conditions

Pathwise differentiability of reflected diffusions in convex polyhedral domains

“…In addition to these works, the following authors considered certain pathwise derivatives of normally reflected diffusions: Deuschel and Zambotti [11] characterized derivatives of stochastic flows for normally reflected diffusions with identity diffusion coefficient in the orthant; Burdzy [6] characterized derivatives of stochastic flows for normally reflected Brownian motions in smooth domains; Andres [2] generalized the results of [6] to allow state-dependent drifts; Pilipenko (see [29] and references therein) investigated derivatives of stochastic flows for normally reflected diffusions in the half space; and Bossy, Cissé and Talay [4] obtained an explicit representation for the derivatives of a one-dimensional reflected diffusion in a bounded interval. Lastly, Costantini, Gobet and Karoui [10] studied boundary sensitivities of normally reflected diffusions in a timedependent domain.…”

Multivalued monotone stochastic differential equations with jumps

“…Approximations of solutions of SDEs with reflecting boundary condition were studied in D. Lépingle [19], J.L. Menaldi [23], R. Pettersson [25], M. Bossy, E. Gobet, D. Talay [4], M. Bossy, M. Cissé, D. Talay [5], W. Laukajtys, L. S lomiński [17,18] and L. S lomiński [37,38]. It is worth noting that in all the papers devoted to reflecting SDEs in convex domains the projection used is the classical one, i.e.…”