Viscosity solutions for systems of parabolic variational inequalities

In this paper, we first define the notion of viscosity solution for the following system of partial differential equations involving a subdifferential operator: $\cases \frac{\partial u}{\partial t}(t,x)+{\cal L}_{t}u(t,x)+f(t,x,u(t,x))\in \partial \varphi (u(t,x)), & t\in [0,T),x\in {\Bbb R}^{d},\\ u(T,x)=h(x), & x\in {\Bbb R}^{d}, \endcases $ where $\partial \varphi $ is the subdifferential operator of the proper convex lower semicontinuous function $\varphi \colon {\Bbb R}^{k}\rightarrow (-\infty,+\infty]$ and ${\cal L}_{t}$ is a second differential operator given by ${\cal L}_{t}\upsilon _{i}(x)=\frac{1}{2}\text{Tr}[\sigma (t,x)\sigma ^{\ast}(t,x)\text{D}^{2}\upsilon _{i}(x)]+\langle b(t,x),\nabla \upsilon _{i}(x)\rangle ,i\in \overline{l,k}$ . We prove the uniqueness of the viscosity solution and then, via a stochastic approach, prove the existence of a viscosity solution $u\colon [0,T]\times {\Bbb R}^{d}\rightarrow {\Bbb R}^{k}$ of the above parabolic variational inequality.