Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Antithetic Multilevel Methods for Elliptic and Hypo-Elliptic Diffusions with Applications

Yuga Iguchi, Ajay Jasra, Mohamed Maama, Alexandros Beskos

TL;DR

This work addresses efficient estimation of expectations for diffusion processes that may be elliptic or hypo-elliptic, focusing on forward and filtering problems. It introduces a locally non-degenerate, weak-second-order scheme that avoids Lévy-area simulations and augments MLMC with antithetic couplings (AMLMC), achieving weak order 2 and strong order 1. The authors prove strong convergence for the scheme and develop an AMLMC estimator with optimal $O(\varepsilon^{-2})$ cost for the forward problem, and extend the approach to an AMLPF for diffusion filtering with costs near $O(\varepsilon^{-2}\log(\varepsilon)^2)$. Numerical experiments on hypo-elliptic and elliptic models demonstrate substantial efficiency gains over existing methods, particularly when hypo-ellipticity is present, and underscore the practical impact for high-accuracy diffusion simulations and sequential inference.

Abstract

We present a new antithetic multilevel Monte Carlo (MLMC) method for the estimation of expectations with respect to laws of diffusion processes that can be elliptic or hypo-elliptic. In particular, we consider the case where one has to resort to time discretization of the diffusion and numerical simulation of such schemes. Inspired by recent works, we introduce a new MLMC estimator of expectations, which does not require any Lévy area simulation and has a strong error of order 2 and a weak error of order 2. We then show how this approach can be used in the context of the filtering problem associated to partially observed diffusions with discrete time observations. We illustrate that in numerical simulations our new approaches provide efficiency gains for several problems, particularly when the diffusion process is hypo-elliptic, relative to some existing methods.

Antithetic Multilevel Methods for Elliptic and Hypo-Elliptic Diffusions with Applications

TL;DR

This work addresses efficient estimation of expectations for diffusion processes that may be elliptic or hypo-elliptic, focusing on forward and filtering problems. It introduces a locally non-degenerate, weak-second-order scheme that avoids Lévy-area simulations and augments MLMC with antithetic couplings (AMLMC), achieving weak order 2 and strong order 1. The authors prove strong convergence for the scheme and develop an AMLMC estimator with optimal cost for the forward problem, and extend the approach to an AMLPF for diffusion filtering with costs near . Numerical experiments on hypo-elliptic and elliptic models demonstrate substantial efficiency gains over existing methods, particularly when hypo-ellipticity is present, and underscore the practical impact for high-accuracy diffusion simulations and sequential inference.

Abstract

We present a new antithetic multilevel Monte Carlo (MLMC) method for the estimation of expectations with respect to laws of diffusion processes that can be elliptic or hypo-elliptic. In particular, we consider the case where one has to resort to time discretization of the diffusion and numerical simulation of such schemes. Inspired by recent works, we introduce a new MLMC estimator of expectations, which does not require any Lévy area simulation and has a strong error of order 2 and a weak error of order 2. We then show how this approach can be used in the context of the filtering problem associated to partially observed diffusions with discrete time observations. We illustrate that in numerical simulations our new approaches provide efficiency gains for several problems, particularly when the diffusion process is hypo-elliptic, relative to some existing methods.
Paper Structure (25 sections, 14 theorems, 89 equations, 2 figures, 2 tables, 6 algorithms)

This paper contains 25 sections, 14 theorems, 89 equations, 2 figures, 2 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Numerical schemes
  3. Basic assumptions and error
  4. Discretizations
  5. Milstein scheme
  6. Truncated Milstein scheme
  7. Second order weak scheme
  8. Strong convergence of the weak second order scheme and summary
  9. Antithetic MLMC with weak second order scheme
  10. AMLMC for forward problem
  11. Application to filtering
  12. State-space model
  13. Multilevel particle filters
  14. New multilevel particle filter
  15. Numerical results
  16. ...and 10 more sections

Key Result

Proposition 2.1

For any $p \ge 1$, there exists a constant $C > 0$ such that

Figures (2)

  • Figure 1: Cost versus MSE for the forward problem.
  • Figure 2: Cost versus MSE for the filtering problem.

Theorems & Definitions (28)

  • Proposition 2.1
  • Lemma 2.1
  • proof
  • Theorem 2.1
  • proof
  • Corollary 2.1
  • Remark 2.1
  • Remark 2.2
  • proof
  • Lemma B.1
  • ...and 18 more