A stochastic approach to a new type of parabolic variational inequalities

Abstract

We study the following quasilinear partial differential equation with two subdifferential operators: $${\frac{\partial u}{\partial s}(s,x)} + (\mathcal{L}u)(s,x,u(s,x),(\nabla u(s,x))^\ast\sigma(s,x,u(s,x))) + f(s,x,u(s,x),(\nabla u(s,x))^\ast\sigma(s,x,u(s,x))) \in \partial\varphi(u(s,x)) + <\partial\psi(x),\nabla u(s,x)>, (s,x) \in[0,T]\times Dom\psi, u(T,x) =g(x),\quad x\in Dom\psi.$$ where for $u\in C^{1,2}\big([0,T]\times Dom\psi\big)$ and $(s,x,y,z)\in [0,T]\times Dom\psi\times Dom\varphi\times\mathbb{R}^{1\times d}$, $$(\mathcal{L}u)(s,x,y,z) := 1/2\sum_{i,j=1}^n (\sigma\sigma^\ast)_{i,j}(s,x,y)\frac{\partial^2u}{\partial x_{i}\partial x_{j}}(s,x) +\sum_{i=1}^n b_i(s,x,y,z)\frac{\partial u}{\partial x_i}(s,x). $$ The operator $\partial\psi$ (resp. $\partial\varphi$) is the subdifferential of the convex lower semicontinuous function $\psi:\mathbb{R}^{n}\to (-\infty,+\infty]$ (resp. $\varphi:\mathbb{R}\to(-\infty,+\infty]$). We define the viscosity solution for such kind of partial differential equations and prove the uniqueness of the viscosity solutions when $\sigma$ does not depend on $y$. To prove the existence of a viscosity solution, a stochastic representation formula of Feymann-Kac type will be developed. For this end, we investigate a fully coupled forward-backward stochastic variational inequality.