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Unbiased Approximations for Stationary Distributions of McKean-Vlasov SDEs

Elsiddig Awadelkarim, Neil K. Chada, Ajay Jasra

TL;DR

This work addresses unbiased estimation of the stationary distribution for McKean-Vlasov SDEs by adapting a randomized multilevel Monte Carlo framework to the MV-SDE setting. It introduces a two-discretization scheme using Euler–Maruyama dynamics, constructs level-specific estimators through coupling, and proves unbiasedness of the overall estimator under geometric ergodicity and regularity assumptions. The theory is complemented by rigorous ergodicity results for both the continuous MV-SDE and its discretizations, and by extensive numerical experiments on Curie–Weiss dynamics, a parameter estimation MV-SDE, and a 3D neuroscience-inspired model. The findings demonstrate that the proposed unbiased estimator can effectively approximate invariant measures without discretization bias, with practical guidance on parameter choices and potential extensions to higher-order schemes and MVSPDEs.

Abstract

We consider the development of unbiased estimators, to approximate the stationary distribution of Mckean-Vlasov stochastic differential equations (MVSDEs). These are an important class of processes, which frequently appear in applications such as mathematical finance, biology and opinion dynamics. Typically the stationary distribution is unknown and indeed one cannot simulate such processes exactly. As a result one commonly requires a time-discretization scheme which results in a discretization bias and a bias from not being able to simulate the associated stationary distribution. To overcome this bias, we present a new unbiased estimator taking motivation from the literature on unbiased Monte Carlo. We prove the unbiasedness of our estimator, under assumptions. In order to prove this we require developing ergodicity results of various discrete time processes, through an appropriate discretization scheme, towards the invariant measure. Numerous numerical experiments are provided, on a range of MVSDEs, to demonstrate the effectiveness of our unbiased estimator. Such examples include the Currie-Weiss model, a 3D neuroscience model and a parameter estimation problem.

Unbiased Approximations for Stationary Distributions of McKean-Vlasov SDEs

TL;DR

This work addresses unbiased estimation of the stationary distribution for McKean-Vlasov SDEs by adapting a randomized multilevel Monte Carlo framework to the MV-SDE setting. It introduces a two-discretization scheme using Euler–Maruyama dynamics, constructs level-specific estimators through coupling, and proves unbiasedness of the overall estimator under geometric ergodicity and regularity assumptions. The theory is complemented by rigorous ergodicity results for both the continuous MV-SDE and its discretizations, and by extensive numerical experiments on Curie–Weiss dynamics, a parameter estimation MV-SDE, and a 3D neuroscience-inspired model. The findings demonstrate that the proposed unbiased estimator can effectively approximate invariant measures without discretization bias, with practical guidance on parameter choices and potential extensions to higher-order schemes and MVSPDEs.

Abstract

We consider the development of unbiased estimators, to approximate the stationary distribution of Mckean-Vlasov stochastic differential equations (MVSDEs). These are an important class of processes, which frequently appear in applications such as mathematical finance, biology and opinion dynamics. Typically the stationary distribution is unknown and indeed one cannot simulate such processes exactly. As a result one commonly requires a time-discretization scheme which results in a discretization bias and a bias from not being able to simulate the associated stationary distribution. To overcome this bias, we present a new unbiased estimator taking motivation from the literature on unbiased Monte Carlo. We prove the unbiasedness of our estimator, under assumptions. In order to prove this we require developing ergodicity results of various discrete time processes, through an appropriate discretization scheme, towards the invariant measure. Numerous numerical experiments are provided, on a range of MVSDEs, to demonstrate the effectiveness of our unbiased estimator. Such examples include the Currie-Weiss model, a 3D neuroscience model and a parameter estimation problem.

Paper Structure

This paper contains 25 sections, 9 theorems, 100 equations, 3 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Method
  4. Overview of the Method and the Notation
  5. Discretization for MVSDE
  6. Overall Strategy
  7. Approximation of $\Pi^{l_*}(\varphi)$
  8. Approximation of $[\Pi^l-\Pi^{l-1}](\varphi)$
  9. Final Methodology and Estimator
  10. Summary of Algorithms
  11. Theoretical Results
  12. Notation
  13. Assumptions
  14. Unbiasedness of the Estimator
  15. Discussion
  16. ...and 10 more sections

Key Result

Theorem 3.1

Assume assump:A1. Then for any $\varphi\in\mathcal{C}_b(\mathbb{R}^d,\mathbb{R})\cap\mathcal{C}^{\mathrm{Lip}}(\mathbb{R}^d,\mathbb{R})$ we have for $l_*$ large enough.

Figures (3)

  • Figure 1: Numerical simulations for the Curie-Weiss model \ref{['eq:cw']}. Top left: MSE approximation of $\pi(\varphi)$. Top right: Meeting time for Curie-Weiss model. Bottom left: comparison of exact and approximated invariant distribution. Bottom Right: Numerically verifying Assumption \ref{['assump:A3']} by plotting $\widehat{Q}_1$ and $\widehat{Q}_2$.
  • Figure 2: Numerical simulations for parameter estimation example \ref{['eq:kuntz_1']}. Top left: MSE approximation of $\phi$. Top right: Meeting time for parameter estimation example. Bottom: comparison of exact and approximated posterior distribution.
  • Figure 3: Numerical simulations for the 3D neuron model \ref{['eq:3d']}. Top left: approximated Invariant distribution for 1st component. Top right: approximated Invariant distribution for 2nd component. Bottom left: approximated Invariant distribution for 3rd component. Bottom right: Meeting time for parameter estimation example.

Theorems & Definitions (20)

  • Theorem 3.1
  • Proof 1
  • Remark 3.2
  • Remark 3.3
  • Definition A.1: $i$-Wasserstein distance
  • Definition A.2: $\mathcal{M}$ distance
  • Proposition A.3
  • Proof 2
  • Theorem A.4
  • Theorem A.5
  • ...and 10 more