Viscosity solutions for systems of parabolic variational inequalities

Abstract

In this paper, we first define the notion of viscosity solution for the following system of partial differential equations involving a subdifferential operator: (equation) where ∂φ is the subdifferential operator of the proper convex lower semicontinuous function φ: ℝk → (-∞, +∞] and Lt is a second differential operator given by Ltvi(x) = 1/2 Tr[σ(t, x)σ*(t, x)D2vi(x)] + 〈b(t,x),∇vi(x)〉,i∈ 1,k.̄ We prove the uniqueness of the viscosity solution and then, via a stochastic approach, prove the existence of a viscosity solution u: [0, T] × ℝd → ℝk of the above parabolic variational inequality. © 2010 ISI/BS.