Derivatives of Solutions of Semilinear Parabolic PDEs and Variational Inequalities with Neumann Boundary Conditions

Abstract

This chapter is a survey of the results obtained by Bossy et al. [Ann. Inst. H. Poincare Probab. Stat. 47(2), 395–424 (2011)]. We explicit the derivative of the flows of one-dimensional reflected diffusion processes. This allows us to get stochastic representations for space derivatives of viscosity solutions of one-dimensional semilinear parabolic partial differential equations and parabolic variational inequalities with Neumann boundary conditions. These results are applied to estimate American options hedging errors resulting from artificial Neumann boundary conditions which are necessary to localize numerical resolutions in bounded domains.