Monte Carlo method for parabolic equations involving fractional Laplacian

TL;DR

This paper develops Monte Carlo schemes for solving parabolic equations with fractional Laplacian, using jump-adapted methods and jump removal techniques to handle different regularity levels, with demonstrated efficiency in high dimensions.

Contribution

It introduces new Monte Carlo schemes with jump-adapted and jump removal techniques for fractional Laplacian equations, achieving higher convergence orders based on solution regularity.

Findings

Numerical schemes confirm theoretical convergence rates.

Methods are effective in high-dimensional cases.

Proposed schemes improve computational efficiency.

Abstract

We apply the Monte Carlo method to solving the Dirichlet problem of linear parabolic equations with fractional Laplacian. This method exploit- s the idea of weak approximation of related stochastic differential equations driven by the symmetric stable L\'evy process with jumps. We utilize the jump- adapted scheme to approximate L\'evy process which gives exact exit time to the boundary. When the solution has low regularity, we establish a numeri- cal scheme by removing the small jumps of the L\'evy process and then show the convergence order. When the solution has higher regularity, we build up a higher-order numerical scheme by replacing small jumps with a simple process and then display the higher convergence order. Finally, numerical experiments including ten- and one hundred-dimensional cases are presented, which confirm the theoretical estimates and show the numerical efficiency of the proposed schemes for high dimensional parabolic equations.