Viscosity solutions for systems of parabolic variational inequalities

Abstract

In this paper, we first define the notion of viscosity solution for the following system of partial differential equations involving a subdifferential operator:\[\{[c]{l}\dfrac{\partial u}{\partial t}(t,x)+\mathcal{L}_tu(t,x)+f(t,x,u(t,x))\in\partialϕ(u(t,x)),\quad t\in[0,T),x\in\mathbb{R}^d, u(T,x)=h(x),\quad x\in\mathbb{R}^d,\] where $\partial \varphi$ is the subdifferential operator of the proper convex lower semicontinuous function $\varphi :{\mathbb{R}}^k\to (-\infty ,+\infty ]$ and ${\mathcal{L}}_t$ is a second differential operator given by ${\mathcal{L}}_t{v}_i(x)=1/2\mathrm{Tr}[\sigma (t,x){\sigma }^{\ast }(t,x){\mathrm{D}}^2{v}_i(x)]+<b(t,x),\nabla v_i(x)>$, $i\in \overline{1,k}$. 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.

Published in at http://dx.doi.org/10.3150/09-BEJ204 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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