Generating Sparse Stochastic Processes Using Matched Splines

TL;DR

This paper introduces an off-the-grid algorithm leveraging B-spline representations to generate trajectories of sparse stochastic processes driven by Lévy white noises, providing accurate approximations of solutions to differential equations.

Contribution

It presents a novel off-the-grid method using B-splines to generate sparse stochastic processes driven by Lévy noises, based on recent theoretical limits.

Findings

Algorithm accurately approximates target processes

Numerical illustrations validate the approach

Generates trajectories close to solutions of differential equations

Abstract

We provide an algorithm to generate trajectories of sparse stochastic processes that are solutions of linear ordinary differential equations driven by L\'evy white noises. A recent paper showed that these processes are limits in law of generalized compound-Poisson processes. Based on this result, we derive an off-the-grid algorithm that generates arbitrarily close approximations of the target process. Our method relies on a B-spline representation of generalized compound-Poisson processes. We illustrate numerically the validity of our approach.