On the continuity of the probabilistic representation of a semilinear Neumann-Dirichlet problem

Abstract

In this article we prove the continuity of the deterministic function $u:[0,T]\times \bar{\mathcal{D}}\to \mathbb{R}$, defined by $u(t,x):=Y_t^{t,x}$, where the process $(Y_s^{t,x}{)}_{s\in [t,T]}$ is given by the generalized multivalued backward stochastic differential equation: \begin{equation*} \left\{ \begin{array}{l} -dY_{s}^{t,x}+\partial φ(Y_{s}^{t,x})ds+\partialψ(Y_{s}^{t,x})dA_{s}^{t,x}\ni f(s,X_{s}^{t,x},Y_{s}^{t,x})ds \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+g(s,X_{s}^{t,x},Y_{s}^{t,x})dA_{s}^{t,x}-Z_{s}^{t,x}dW_{s}~,\;t\leq s < T, \\ {Y_{T}=h(X_{T}^{t,x}).} \end{array} \right. \end{equation*} The process $(X_s^{t,x},A_s^{t,x}{)}_{s\ge t}$ is the solution of a stochastic differential equation with reflecting boundary conditions.

Some proofs have been slighty changed

References in corpus (4)

• Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions
• Penalization method for a nonlinear Neumann PDE via weak solutions of reflected SDEs
• Cadlag Skorokhod problem driven by a maximal monotone operator
• Obstacle problem for SPDE with nonlinear Neumann boundary condition via reflected generalized backward doubly SDEs

Cited by in corpus (3)

• $L^p$-Variational Solutions of Multivalued Backward Stochastic Differential Equations
• Probabilistic representation of parabolic stochastic variational inequality with Dirichlet-Neumann boundary and variational generalized backward doubly stochastic differential equations
• $L^p$ - Variational Solution of Backward Stochastic Differential Equation driven by subdifferential operators on a deterministic interval time