In this paper we prove an approximation result for the viscosity solution of a system of semi-linear partial differential equations with continuous coefficients and nonlinear Neumann boundary condition. The approximation we use is based on a penalization method and our approach is probabilistic. We prove the weak uniqueness of the solution for the reflected stochastic differential equation and we approximate it (in law) by a sequence of solutions of stochastic differential equations with penalized terms. Using then a suitable generalized backward stochastic differential equation and the uniqueness of the reflected stochastic differential equation, we prove the existence of a continuous function, given by a probabilistic representation, which is a viscosity solution of the considered partial differential equation. In addition, this solution is approximated by solutions of penalized partial differential equations.
In this paper, we first define the notion of viscosity solution for the
following system of partial differential equations involving a subdifferential
operator:\[\{[c]{l}\dfrac{\partial u}{\partial
t}(t,x)+\mathcal{L}_tu(t,x)+f(t,x,u(t,x))\in\partial\phi (u(t,x)),\quad
t\in[0,T),x\in\mathbb{R}^d, u(T,x)=h(x),\quad x\in\mathbb{R}^d,\] where
$\partial\phi$ is the subdifferential operator of the proper convex lower
semicontinuous function $\phi:\mathbb{R}^k\to (-\infty,+\infty]$ and
$\mathcal{L}_t$ is a second differential operator given by
$\mathcal{L}_tv_i(x)={1/2}\operatorname
{Tr}[\sigma(t,x)\sigma^*(t,x)\mathrm{D}^2v_i(x)]+< b(t,x),\nabla v_i(x)>$,
$i\in\bar{1,k}$. 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.Comment: 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
We prove the existence of a viscosity solution of the following path dependent nonlinear Kolmogorov equation:The result is obtained by a stochastic approach. In particular we prove a new type of nonlinear Feynman-Kac representation formula associated to a backward stochastic differential equation with time-delayed generator which is of non-Markovian type. Applications to the large investor problem and risk measures via g-expectations are also provided.
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