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__Monotone approximation schemes for linear parabolic PDEs by weak SDE approximation methods__

K. Debrabant and E.R. Jakobsen

**Abstract**

In this talk we consider equations of the form
\begin{gather*}
\label{E}
u_t=\frac12\mathrm{tr} [\sigma(t,x)\sigma^\top(t,x) D^2u(t,x)] +
b(t,x) D
u(t,x) +
c(t,x) u +
f(t,x)\text{ in} (0,T]\times\mathbb{R}^N,
u(0,x)=g(x)\text{ in}\mathbb{R}^N,
\end{gather*}
containing e.g. the Black-Scholes PDE.
We show how weak approximation schemes for the solution of stochastic differential equations can be used to obtain new higher order monotone PDE approximation methods. Some stability and convergence results are given, including convergence for coefficients which are only bounded and Lipschitz continuous, and the methods are applied to some examples.