Deterministic and stochastic differential inclusions with multiple surfaces of discontinuity

Abstract

We consider a class of deterministic and stochastic dynamical systems with discontinuous drift f and solutions that are constrained to live in a given closed domain G in \({\mathbb{R}}^{n}\) according to a constraint vector field D(·) specified on the boundary \(\partial G\) of the domain. Specifically, we consider equations of the form \(\phi = \theta + \eta + u , \quad \dot{\theta}(t) \in F(\phi(t)), \quad \mbox{a.e. } t\) for u in an appropriate class of functions, where η is the “constraining term” in the Skorokhod problem specified by (G, D) and F is the set-valued upper semicontinuous envelope of f. The case \(G ={\mathbb{R}}^{n}\) (when there is no constraining mechanism) and u is absolutely continuous corresponds to the well known setting of differential inclusions (DI). We provide a general sufficient condition for uniqueness of solutions and Lipschitz continuity of the solution map, in the form of existence of a Lyapunov set. Here we assume (i) G is convex and admits the representation \(G=\cup_i\overline{C_i}\) , where \(\{C_i,i\in {\mathbb{I}}\}\) is a finite collection of disjoint, open, convex, polyhedral cones in \({\mathbb{R}}^{n}\) , each having its vertex at the origin; (ii) f = b + f c is a vector field defined on G such that b assumes a constant value on each of the given cones and f c is Lipschitz continuous on G; and (iii) D is an upper semicontinuous, cone-valued vector field that is constant on each face of ∂G. We also provide existence results under much weaker conditions (where no Lyapunov set condition is imposed). For stochastic differential equations (SDE) (possibly, reflected) that have drift coefficient f and a Lipschitz continuous (possibly degenerate) diffusion coefficient, we establish strong existence and pathwise uniqueness under appropriate conditions. Our approach yields new existence and uniqueness results for both DI and SDE even in the case \(G = {\mathbb{R}}^{n}.\) The work has applications in the study of scaling limits of stochastic networks.