A stochastic approach to a new type of parabolic variational inequalities

Abstract

We study the following quasilinear partial differential equation with two subdifferential operators: $$\frac{\partial u}{\partial s}(s,x)+(\mathcal{L}u)(s,x,u(s,x),(\nabla u(s,x){)}^{\ast }\sigma (s,x,u(s,x)))+f(s,x,u(s,x),(\nabla u(s,x){)}^{\ast }\sigma (s,x,u(s,x)))\in \partial \phi (u(s,x))+<\partial \psi (x),\nabla u(s,x)>,(s,x)\in [0,T]\times Dom\psi ,u(T,x)=g(x),x\in Dom\psi .$$ where for $u\in C^{1,2}([0,T]\times Dom\psi )$ and $(s,x,y,z)\in [0,T]\times Dom\psi \times Dom\phi \times {\mathbb{R}}^{1\times d}$, $$(\mathcal{L}u)(s,x,y,z):=1/2\sum _{i,j=1}^n(\sigma {\sigma }^{\ast }{)}_{i,j}(s,x,y)\frac{{\partial }^{2}u}{\partial x_i\partial x_j}(s,x)+\sum _{i=1}^n{b}_i(s,x,y,z)\frac{\partial u}{\partial x_i}(s,x).$$ The operator $\partial \psi$ (resp. $\partial \phi$) is the subdifferential of the convex lower semicontinuous function $\psi :{\mathbb{R}}^n\to (-\infty ,+\infty ]$ (resp. $\phi :\mathbb{R}\to (-\infty ,+\infty ]$). 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.

38 pages

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