Feynman-Kac Representation of Fully Nonlinear PDEs and Applications

Feynman-Kac representation of fully nonlinear PDEs and applications

The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in...

Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE

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Solving Dirichlet problems numerically using the Feynman-Kac representation

In this paper we study numerical solutions of the Dirichlet problem in high dimensions using the Feynman-Kac representation. What is involved are Monte-Carlo simulations of stochastic differential equations and algorithms to accurately determine exit times and process values at the boundary. It is assumed that the radius of curvature of the boundary is much larger than the square root of the st...

Quantum Feynman-Kac perturbations

The Feynman-Kac formula

where ∆ is the Laplace operator. Here σ > 0 is a constant (the diffusion constant). It has dimensions of distance squared over time, so H0 has dimensions of inverse time. The operator exp(−tH0) for t > 0 is an self-adjoint integral operator, which gives the solution of the heat or diffusion equation. Here t is the time parameter. It is easy to solve for this operator by Fourier transforms. Sinc...