On Stochastic Differential Equations with Reflecting Boundary Condition in Convex Domains
Volume 52 / 2004
Bulletin Polish Acad. Sci. Math. 52 (2004), 445-455
MSC: Primary 60H20.
DOI: 10.4064/ba52-4-11
Abstract
Let $D$ be an open convex set in $\mathbb R^d$ and let $F$ be a Lipschitz operator defined on the space of adapted càdlàg processes. We show that for any adapted process $H$ and any semimartingale $Z$ there exists a unique strong solution of the following stochastic differential equation (SDE) with reflection on the boundary of $D$: $$ X_t=H_t+\int_0^t\, \langle F(X)_{s-},dZ_s\rangle + K_t, \ \quad t \in \mathbb R^+. $$ Our proofs are based on new a priori estimates for solutions of the deterministic Skorokhod problem.
