Chernoff-Mehler Approximation for L\'evy Processes with Drift

TL;DR

This paper develops a Chernoff product formula-based approximation scheme for Le9vy processes with drift, providing explicit convergence criteria and demonstrating its applicability to various numerical methods and classical limit theorems.

Contribution

It introduces a novel approximation framework for Le9vy processes with drift using Chernoff formulas, extending to deterministic components and broad classes of numerical schemes.

Findings

Provides necessary and sufficient conditions for convolution semigroup convergence.

Shows the framework encompasses Euler schemes, Runge-Kutta methods, and the Central Limit Theorem.

Establishes explicit criteria for convergence of approximations to Le9vy processes with drift.

Abstract

In this paper, we study an approximation scheme for L\'evy processes with drift in terms of a representation that is akin to the celebrated Mehler formula for L\'evy-Ornstein-Uhlenbeck processes. The approximation scheme is based on a variant of the Chernoff product formula on the space of bounded continuous functions. In a first step, we provide sufficient and necessary conditions for arbitrary families of probability measures, indexed by positive real numbers, to give rise to a convolution semigroup via a Chernoff approximation on the space of bounded continuous functions, equipped with the mixed topology. In this context, we provide explicit criteria both for the convergence of subsequences and the entire family, and discuss fine properties related to the domain of the associated generator of the L\'evy process and the infinitesimal behavior of the approximating family of measures. In a second step, we enrich the family of measures by a deterministic component and derive explicit conditions that ensure both the convergence of subsequences and the entire family to a L\'evy process with drift under a Chernoff approximation. In a series of examples, we show that our general conditions on the dynamics are satisfied, for example, by flows of Lipschitz ordinary differential equations, Euler schemes, and arbitrary Runge-Kutta methods, and that the Central Limit Theorem can be subsumed under our framework.