On nonlinear Feynman–Kac formulas for viscosity solutions of semilinear parabolic partial differential equations
C Beck, M Hutzenthaler, A Jentzen - Stochastics and Dynamics, 2021 - World Scientific
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
paper, we extend the classical Feynman–Kac formula to certain semilinear Kolmogorov …
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
paper, we extend the classical Feynman–Kac formula to certain semilinear Kolmogorov …
On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations with gradient-dependent nonlinearities
M Hutzenthaler, K Pohl - arXiv preprint arXiv:2310.18197, 2023 - arxiv.org
The classical Feynman-Kac identity represents solutions of linear partial differential
equations in terms of stochastic differential euqations. This representation has been
generalized to nonlinear partial differential equations on the one hand via backward
stochastic differential equations and on the other hand via stochastic fixed-point equations.
In this article we generalize the representation via stochastic fixed-point equations to allow
the nonlinearity in the semilinear partial differential equation to depend also on the gradient …
equations in terms of stochastic differential euqations. This representation has been
generalized to nonlinear partial differential equations on the one hand via backward
stochastic differential equations and on the other hand via stochastic fixed-point equations.
In this article we generalize the representation via stochastic fixed-point equations to allow
the nonlinearity in the semilinear partial differential equation to depend also on the gradient …