Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multilevel Particle Filters for Partially Observed McKean-Vlasov Stochastic Differential Equations

Elsiddig Awadelkarim, Ajay Jasra

TL;DR

This work addresses filtering for partially observed McKean-Vlasov SDEs by introducing a particle filter (PF) and a multilevel PF (MLPF) that estimate conditional state-functionals under Euler-Maruyama discretization. The authors establish mean-squared error bounds and derive cost scalings, showing PF costs scale as $O( ext{ε}^{-5})$ while MLPF attains $O( ext{ε}^{-4})$ in favorable cases or $O( ext{ε}^{-4} ext{log}( ext{ε})^2)$ in general, with potential for $O( ext{ε}^{-4})$ under stronger assumptions. The approach uses a discretized-law propagation, coupled sampling and resampling, and telescopic MLMC decompositions to reduce overall computational burden. Numerical experiments on Kuramoto-type MK-V SDEs corroborate the theoretical gains and illustrate substantial efficiency improvements of ML over PF in filtering tasks.

Abstract

In this paper we consider the filtering problem associated to partially observed McKean-Vlasov stochastic differential equations (SDEs). The model consists of data that are observed at regular and discrete times and the objective is to compute the conditional expectation of (functionals) of the solutions of the SDE at the current time. This problem, even the ordinary SDE case is challenging and requires numerical approximations. Based upon the ideas in [3, 12] we develop a new particle filter (PF) and multilevel particle filter (MLPF) to approximate the afore-mentioned expectations. We prove under assumptions that, for $ε>0$, to obtain a mean square error of $\mathcal{O}(ε^2)$ the PF has a cost per-observation time of $\mathcal{O}(ε^{-5})$ and the MLPF costs $\mathcal{O}(ε^{-4})$ (best case) or $\mathcal{O}(ε^{-4}\log(ε)^2)$ (worst case). Our theoretical results are supported by numerical experiments.

Multilevel Particle Filters for Partially Observed McKean-Vlasov Stochastic Differential Equations

TL;DR

This work addresses filtering for partially observed McKean-Vlasov SDEs by introducing a particle filter (PF) and a multilevel PF (MLPF) that estimate conditional state-functionals under Euler-Maruyama discretization. The authors establish mean-squared error bounds and derive cost scalings, showing PF costs scale as while MLPF attains in favorable cases or in general, with potential for under stronger assumptions. The approach uses a discretized-law propagation, coupled sampling and resampling, and telescopic MLMC decompositions to reduce overall computational burden. Numerical experiments on Kuramoto-type MK-V SDEs corroborate the theoretical gains and illustrate substantial efficiency improvements of ML over PF in filtering tasks.

Abstract

In this paper we consider the filtering problem associated to partially observed McKean-Vlasov stochastic differential equations (SDEs). The model consists of data that are observed at regular and discrete times and the objective is to compute the conditional expectation of (functionals) of the solutions of the SDE at the current time. This problem, even the ordinary SDE case is challenging and requires numerical approximations. Based upon the ideas in [3, 12] we develop a new particle filter (PF) and multilevel particle filter (MLPF) to approximate the afore-mentioned expectations. We prove under assumptions that, for , to obtain a mean square error of the PF has a cost per-observation time of and the MLPF costs (best case) or (worst case). Our theoretical results are supported by numerical experiments.
Paper Structure (21 sections, 16 theorems, 62 equations, 1 figure, 5 algorithms)

This paper contains 21 sections, 16 theorems, 62 equations, 1 figure, 5 algorithms.

Table of Contents

  1. Introduction
  2. Approach
  3. State-Space Model
  4. Approximating the Law
  5. Particle Filter
  6. Multilevel Particle Filter
  7. Theoretical Analysis
  8. Notation
  9. Assumptions
  10. Main Theorems
  11. Comments on Computational Cost
  12. Numerical Results
  13. Models
  14. Simulation Results
  15. Proofs
  16. ...and 6 more sections

Key Result

Theorem 3.1

Assume assump:A1 and assump:W1. Then for any $\varphi\in\mathcal{C}^{\infty}_b(\mathbb{R}^d)\cap\mathcal{B}_b(\mathbb{R}^d)$ and $t\in\{1,\dots,T\}$ there exists a constant $C<+\infty$ such that for every $(L,M,N)\in\mathbb{N}_0\times \mathbb{N}\times \mathbb{N}$:

Figures (1)

  • Figure 1: Left: Kuramoto Model: Single level (SL) for $2\leq L\leq 6$, Multi-level (ML) for $3\leq L\leq 6$. Right: Modified Kuramoto Model: Single level (SL) for $2\leq L\leq 6$, Multi-level (ML) for $3\leq L\leq 6$.

Theorems & Definitions (28)

  • Theorem 3.1
  • Theorem 3.2
  • Lemma A.1
  • Lemma A.2
  • Lemma A.3
  • proof
  • Lemma A.4
  • proof
  • Lemma A.5
  • Lemma A.6
  • ...and 18 more