A Probabilistic Scheme for Fully Non-linear Non-local Parabolic PDEs with singular Levy measures

Details A Probabilistic Scheme for Fully Non-linear Non-local Parabolic PDEs with singular Levy measures A Probabilistic Scheme for Fully Non-linear Non-local Parabolic PDEs with singular Levy measures

2010-10-29

arxiv.org/abs/1010.6225v3

Mathematics

Probability

Scientific paper

We introduce a Monte Carlo scheme for the approximation of the solutions of fully non-linear parabolic non-local PDEs. The method is the generalization of the method proposed by [Fahim-Touzi-Warin,2011] for fully non-linear parabolic PDEs. As an independent result, we also introduce a Monte Carlo Quadrature method to approximate the integral with respect to L\'evy measure which may appear inside the scheme. We consider the equations whose non-linearity is of the Hamilton-Jacobi-Belman type. We avoid the difficulties of infinite Levy measures by truncation of the Levy integral by some $\kappa>0$ near 0. The first result provides the convergence of the scheme for general parabolic non-linearities. The second result provides bounds on the rate of convergence for concave non-linearities from above and below. For both results, it is crucial to choose $\kappa$ appropriately dependent on $h$.

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