Monte Carlo for high-dimensional degenerated Semi Linear and Full Non Linear PDEs

Abstract

We extend a recently developed method to solve semi-linear PDEs to the case of a degenerated diffusion. Being a pure Monte Carlo method it does not suffer from the so called curse of dimensionality and it can be used to solve problems that were out of reach so far. We give some results of convergence and show numerically that it is effective. Besides we numerically show that the new scheme developed can be used to solve some full non linear PDEs. At last we provide an effective algorithm to implement the scheme.

23 pages, 13 figures

References in corpus (4)

• Solving high-dimensional partial differential equations using deep learning
• Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations
• Variations on branching methods for non linear PDEs
• Multi-level Picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities

Cited by in corpus (5)

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• An overview on deep learning-based approximation methods for partial differential equations
• Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks
• Deep backward schemes for high-dimensional nonlinear PDEs
• Strong $L^p$-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations