Stochastic differential equations with a convex constraint

Abstract

This work uses techniques from convex analysis to establish the existence and uniqueness of constrained strong solutions to stochastic ordinary differential equations in Hilbert space. The solution consists of a semimartingale u with a bounded variation component η, in which u must stay within the domain of a given convex function φ, and η satisfies the variational inequality a.s. for all suitable test functions v. Roughly speaking η is singular in some sense and represents the amount of pushing needed to maintain the constraint on u. The restrictions on φ are that it be lower semicontinuous and have a domain with non-empty interior. If is an open convex set and φ is the indicator of then u is reflected to stay in and η is non absolutely continuous and varies only when u is at the boundary of . We thus recover previous results on reflected diffusion processes in finite dimensions. If φ has an open domain and is Gateaux differentiable than η becomes the absolutely continuous process . In this situation is ...