In this paper, we first define the notion of viscosity solution for the following system of partial differential equations involving a subdifferential operator:
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\[\catcode`\*=4\cases{\dfrac{\partial u}{\partial t}(t,x )+\mathcal{L}_{t}u(t,x )+f (t,x,u (t,x ))\in\partial\varphi (u (t,x)),*\quad $t\in[ 0,T) ,x\in\mathbb{R}^{d}$,\cr u( T,x ) =h(x),*\quad $x\in\mathbb{R}^{d}$,}\]
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where ∂φ is the subdifferential operator of the proper convex lower semicontinuous function φ : ℝk→(−∞, +∞] and $\mathcal{L}_{t}$ is a second differential operator given by $\mathcal{L}_{t}v_{i}(x)=\frac{1}{2}\operatorname{Tr}[\sigma(t,x)\sigma^{\ast}(t,x)\mathrm{D}^{2}v_{i}(x)]+\langle b(t,x),\nabla v_{i}(x)\rangle$ , $i\in\overline{1,k}$ .
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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.