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:$$\{[c]l{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}}(t,x)+{\mathcal{L}}_{t}u(t,x)+f(t,x,u(t,x))\in \mathrm{\partial }\varphi (u(t,x)),t\in [0,T),x\in {\mathbb{R}}^d,u(T,x)=h(x),x\in {\mathbb{R}}^d,$$ where $\mathrm{\partial }\varphi$ is the subdifferential operator of the proper convex lower semicontinuous function $\varphi :{\mathbb{R}}^k\to (-\mathrm{\infty },+\mathrm{\infty }]$ and ${\mathcal{L}}_t$ is a second differential operator given by $$\mathcal{L}$_tv_i(x)={1/2}\operatorname {Tr}[\sigma(t,x)\sigma^*(t,x)$\mathrm{D}$^2v_i(x)]+< b(t,x),\nabla v_i(x)>$,$i\in\bar{1,k}$. We prove the uniqueness of the viscosity solution and then, via a stochastic approach, prove the existence of a viscosity solution $u:[0,T]\times {\mathbb{R}}^d\to {\mathbb{R}}^k$ of the above parabolic variational inequality.