Probability computation for high-dimensional semilinear SDEs driven by isotropic $\alpha-$stable processes via mild Kolmogorov equations

TL;DR

This paper develops a novel probabilistic numerical method for high-dimensional semilinear SDEs driven by isotropic alpha-stable processes, enabling efficient probability computations via mild Kolmogorov equations and Monte Carlo simulations.

Contribution

It introduces an iterative approximation technique linking Markov semigroups and Kolmogorov equations, improving computational efficiency for high-dimensional stochastic systems.

Findings

Effective approximation of probabilities in 100-dimensional SDEs

Single batch Monte Carlo method reduces computational cost

Satisfactory results for nonlinear vector fields in high dimensions

Abstract

Semilinear, $N-$dimensional stochastic differential equations (SDEs) driven by additive L\'evy noise are investigated. Specifically, given $\alpha\in\left(\frac{1}{2},1\right)$, the interest is on SDEs driven by $2\alpha-$stable, rotation-invariant processes obtained by subordination of a Brownian motion. An original connection between the time-dependent Markov transition semigroup associated with their solutions and Kolmogorov backward equations in mild integral form is established via regularization-by-noise techniques. Such a link is the starting point for an iterative method which allows to approximate probabilities related to the SDEs with a single batch of Monte Carlo simulations as several parameters change, bringing a compelling computational advantage over the standard Monte Carlo approach. This method also pertains to the numerical computation of solutions to high-dimensional integro-differential Kolmogorov backward equations. The scheme, and in particular the first order approximation it provides, is then applied for two nonlinear vector fields and shown to offer satisfactory results in dimension $N=100$.