On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations

Abstract

The classical Feynman-Kac identity builds a bridge between stochastic analysis and partial differential equations (PDEs) by providing stochastic representations for classical solutions of linear Kolmogorov PDEs. This opens the door for the derivation of sampling based Monte Carlo approximation methods, which can be meshfree and thereby stand a chance to approximate solutions of PDEs without suffering from the curse of dimensionality. In this article we extend the classical Feynman-Kac formula to certain semilinear Kolmogorov PDEs. More specifically, we identify suitable solutions of stochastic fixed point equations (SFPEs), which arise when the classical Feynman-Kac identity is formally applied to semilinear Kolmorogov PDEs, as viscosity solutions of the corresponding PDEs. This justifies, in particular, employing full-history recursive multilevel Picard (MLP) approximation algorithms, which have recently been shown to overcome the curse of dimensionality in the numerical approximation of solutions of SFPEs, in the numerical approximation of semilinear Kolmogorov PDEs.

arXiv comment54 pages

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Cited by in corpus (10)

• Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning
• 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
• Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations
• Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing
• Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities
• Deep learning based numerical approximation algorithms for stochastic partial differential equations
• On existence and uniqueness properties for solutions of stochastic fixed point equations
• Non-standard Stochastic Control with Nonlinear Feynman-Kac Costs
• Strong $L^p$-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations