Multilevel Particle Filters for L\'evy-driven stochastic differential equations

TL;DR

This paper introduces a multilevel particle filter method for efficiently computing expectations in Le9vy-driven SDEs, improving computational cost and convergence rates over standard approaches.

Contribution

The paper develops a novel multilevel particle filter algorithm tailored for Le9vy-driven SDEs, achieving optimal convergence and reduced computational complexity.

Findings

Achieves optimal convergence rates for Le9vy-driven SDEs.

Reduces computational cost from O(ε^{-3}) to O(ε^{-2}) for MSE O(ε^2).

Validated through numerical simulations and theoretical analysis.

Abstract

We develop algorithms for computing expectations of the laws of models associated to stochastic differential equations (SDEs) driven by pure L\'evy processes. We consider filtering such processes and well as pricing of path dependent options. We propose a multilevel particle filter (MLPF) to address the computational issues involved in solving these continuum problems. We show via numerical simulations and theoretical results that under suitable assumptions of the discretization of the underlying driving L\'evy proccess, our proposed method achieves optimal convergence rates. The cost to obtain MSE $O(\epsilon^2)$ scales like $O(\epsilon^{-2})$ for our method, as compared with the standard particle filter $O(\epsilon^{-3})$.