A stochastic approach to a new type of parabolic variational inequalities

We study the following quasilinear partial differential equation with two subdifferential operators: { partial derivative u/partial derivative s (s, x) + (Lu)(s, x, u(s, x), (del u(s, x))* sigma(s, x, u(s, x))) +f(s, x, u(s, x), (del u(s, x))* sigma (s, x, u(s, x))) is an element of partial derivative phi(u(s, x)) + <partial derivative phi(x), del u(s, x)>, (s, x) is an element of[0, T] X Dom psi, u(T, x) = g(x), x is an element of Dom phi, where for u is an element of C-1,C-2 ([0, T] X Dom phi) and (s, x, y, z) is an element of [0, T] X Dom phi X Dom phi X R-1xd (Lu) (s, x, y, z) : = 1/2 Sigma(n)(i, j=1) (sigma sigma*)(i,j)(s, x, y) partial derivative(2)u/partial derivative x(i)partial derivative x(j) (s, x) + Sigma(n)(i=1) b(i)(s, x, y, z) partial derivative u/partial derivative x(i) (s, x). The operator partial derivative phi (resp. partial derivative phi) is the subdifferential of the convex lower semicontinuous function psi : R-n -> (-infinity, +infinity) (resp. phi : R -> (-infinity, +infinity). We define the viscosity solution for such kind of partial differential equation and prove the uniqueness of the viscosity solution when s 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.