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: \[ \cases{ \dfrac{\partial u}{\partial t}(t,x )+\mathcal{L}_{t}u(t,x )+f (t,x,u (t,x )) \in\partial\varphi (u (t,x)), & $\quad t\in[ 0,T) ,x\in\mathbb{R}^{d},$\cr u( T,x ) =h(x), & $\quad x\in\mathbb{R}^{d},$ } \] where $\partial\varphi$ is the subdifferential operator of the proper convex lower semicontinuous function $\varphi:\mathbb{R}^{k}\rightarrow (-\infty,+\infty]$ and $\mathcal{L}_{t}$ is a second differential operator given by $\mathcal{L}_{t}v_{i}(x)=\frac{1}{2}\operatorname{Tr}% [\sigma(t,x)\sigma^{\ast}(t,x)\mathrm{D}^{2}v_{i}(x) ]+ \langle b(t,x),\nabla v_{i}(x) \rangle$, $i\in\overline{1,k}$.

We prove the uniqueness of the viscosity solution and then, via a stochastic approach, prove the existence of a viscosity solution $u: [ 0,T ] \times\mathbb{R}^{d}\rightarrow\mathbb{R}^{k}$ of the above parabolic variational inequality.