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arXiv · 1612.00078

Full-Projection explicit FBSDE scheme for parabolic PDEs with superlinear nonlinearities

Arnaud Lionnet·Gonçalos dos Reis·Lukasz Szpruch
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Abstract

Developing efficient and stable approximations for high dimensional PDEs is of key importance for numerous applications. The language of Forward-Backward Stochastic Differential Equations (FBSDE), with its nonlinear Feynman-Kac formula, allows for purely probabilistic representations of the solution and its gradient for parabolic nonlinear PDEs. In this work we build on the recent results of [Lionnet, dos Reis and Szpruch 2015] by introducing and studying a Full-Projection explicit time-discretization scheme for the approximation of FBSDEs with non-globally Lipschitz drivers of polynomial growth. We establish convergence rates and we show that, unlike classical explicit schemes, it preserves stability properties present in the continuous-time dynamics, in particular, the scheme is able to preserve the possible coercivity/contraction property of the PDE's coefficients. The scheme is then coupled with a quantization-type approximation of the conditional expectations on a space-time grid in order to provide a complete approximation scheme for these FBSDEs/nonlinear PDEs and a full analysis is also carried out. We illustrate our findings with numerical examples.

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Arnaud Lionnet, Gonçalos dos Reis, Lukasz Szpruch. 2016-11-30. Full-Projection explicit FBSDE scheme for parabolic PDEs with superlinear nonlinearities. https://arxiv.org/abs/1612.00078

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