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Antithetic Multilevel Particle Filters

Ajay Jasra, Mohamed Maama, Hernando Ombao

TL;DR

This work develops the Antithetic Multilevel Particle Filter (AMLPF) for filtering partially observed diffusion processes discretized in time. By leveraging an antithetic truncated Milstein discretization and a novel maximal-coupling resampling strategy within a multilevel Monte Carlo framework, AMLPF delivers an $\mathbb{L}_2$-convergent estimator for the filtering distribution with cost scaling as $O(\epsilon^{-2}\log(\epsilon)^2)$. This improves upon Euler-based MLMF in multi-dimensional settings with non-constant diffusion coefficients, and is supported by theory and numerical experiments on GBM, Clark-Cameron, and nonlinear diffusion models. The approach also enables unbiased recursive estimation of the normalizing constant, with practical efficiency gains demonstrated in simulations.

Abstract

In this paper we consider the filtering of partially observed multi-dimensional diffusion processes that are observed regularly at discrete times. This is a challenging problem which requires the use of advanced numerical schemes based upon time-discretization of the diffusion process and then the application of particle filters. Perhaps the state-of-the-art method for moderate dimensional problems is the multilevel particle filter of \cite{mlpf}. This is a method that combines multilevel Monte Carlo and particle filters. The approach in that article is based intrinsically upon an Euler discretization method. We develop a new particle filter based upon the antithetic truncated Milstein scheme of \cite{ml_anti}. We show that for a class of diffusion problems, for $ε>0$ given, that the cost to produce a mean square error (MSE) in estimation of the filter, of $\mathcal{O}(ε^2)$ is $\mathcal{O}(ε^{-2}\log(ε)^2)$. In the case of multidimensional diffusions with non-constant diffusion coefficient, the method of \cite{mlpf} has a cost of $\mathcal{O}(ε^{-2.5})$ to achieve the same MSE. We support our theory with numerical results in several examples.

Antithetic Multilevel Particle Filters

TL;DR

This work develops the Antithetic Multilevel Particle Filter (AMLPF) for filtering partially observed diffusion processes discretized in time. By leveraging an antithetic truncated Milstein discretization and a novel maximal-coupling resampling strategy within a multilevel Monte Carlo framework, AMLPF delivers an -convergent estimator for the filtering distribution with cost scaling as . This improves upon Euler-based MLMF in multi-dimensional settings with non-constant diffusion coefficients, and is supported by theory and numerical experiments on GBM, Clark-Cameron, and nonlinear diffusion models. The approach also enables unbiased recursive estimation of the normalizing constant, with practical efficiency gains demonstrated in simulations.

Abstract

In this paper we consider the filtering of partially observed multi-dimensional diffusion processes that are observed regularly at discrete times. This is a challenging problem which requires the use of advanced numerical schemes based upon time-discretization of the diffusion process and then the application of particle filters. Perhaps the state-of-the-art method for moderate dimensional problems is the multilevel particle filter of \cite{mlpf}. This is a method that combines multilevel Monte Carlo and particle filters. The approach in that article is based intrinsically upon an Euler discretization method. We develop a new particle filter based upon the antithetic truncated Milstein scheme of \cite{ml_anti}. We show that for a class of diffusion problems, for given, that the cost to produce a mean square error (MSE) in estimation of the filter, of is . In the case of multidimensional diffusions with non-constant diffusion coefficient, the method of \cite{mlpf} has a cost of to achieve the same MSE. We support our theory with numerical results in several examples.
Paper Structure (21 sections, 17 theorems, 174 equations, 2 figures, 1 table, 5 algorithms)

This paper contains 21 sections, 17 theorems, 174 equations, 2 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Model and Discretization
  3. State-Space Model
  4. Time Discretization
  5. Antithetic Multilevel Particle Filters
  6. Multilevel Particle Filter
  7. Mathematical Results
  8. Numerical Results
  9. Models
  10. Model 1: Geometric Brownian motion (GBM) process
  11. Model 2: Clark-Cameron SDE
  12. Model 3: Multi-dimensional SDEs with a nonlinear diffusion term (NLMs)
  13. Simulation Settings
  14. Simulation Results
  15. Introduction and Some Notations
  16. ...and 6 more sections

Key Result

Theorem 4.1

Assume (Aass:diff1-ass:g). Then for any $(k,\varphi)\in\mathbb{N}\times \mathcal{B}_b(\mathsf{X})\cap\mathcal{C}_b^2(\mathsf{X},\mathbb{R})$ there exists a $C<+\infty$ such that for any $(\underline{L},\overline{L},l,N_{\underline{L}},\dots,N_{\overline{L}},\varepsilon)\in\mathbb{N}_0^2\times\{\unde where $\mathsf{D}_{\underline{L},\overline{L}}=\{(l,q)\in\{\underline{L},\dots,\overline{L}\}:l\neq

Figures (2)

  • Figure 1: Cost rates as a function of the mean squared error. The results are for the filter.
  • Figure 2: Cost rates as a function of the mean squared error of our algorithms with results for the normalizing constants.

Theorems & Definitions (38)

  • Theorem 4.1
  • proof
  • Lemma B.1
  • proof
  • Lemma B.2
  • proof
  • Proposition B.1
  • proof
  • Lemma C.1
  • proof
  • ...and 28 more