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Convergence rate for random walk approximations of mean field BSDEs

Boualem Djehiche, Hannah Geiss, Stefan Geiss, Céline Labart, Jani Nykänen

TL;DR

This paper analyzes the convergence rate of scaled random-walk approximations to mean-field backward stochastic differential equations (BSDEs) in Wasserstein-type distances, using a freezing technique instead of particle methods. It extends Donsker-type convergence results to the mean-field setting, where a time-singularity in the generator arises from Hölder-terminal data, and introduces a modified Hölder continuity |r|_{(ε,β)} to recover polynomial rates up to logarithmic factors. The authors establish coupling-based pathwise and Wasserstein bounds, with rates that include a log term and depend on ε, α, and β, and they provide a practical, non-particle algorithm along with numerical validation showing the expected rate behavior. The work advances numerical methods for nonlocal mean-field BSDEs and yields tools that are broadly applicable to the probabilistic analysis of mean-field problems and their associated nonlocal PDEs.

Abstract

We study the rate of convergence w.r.t.~a Wasserstein type distance for random walk approximations of mean field BSDEs. Our method does not use the particle method but instead a freezing technique. We extend results by Briand, Ch. Geiss, S. Geiss, and Labart [Bernoulli, 27(2) 2021] about the rate of convergence of a Donsker-type theorem for BSDEs from the classical setting to the mean field setting. In this connection the mean field setting leads to new phenomena and requires new techniques that should be of independent interest: The Hölder continuous terminal condition causes a singularity in time of the generator when seen as a generator in the non-mean field setting. To handle this singularity we introduce a concept of modified Hölder continuity by which we are able to achieve, up to a logarithmic term, the same polynomial approximation rates as in the classical non-mean field setting (in fact, already when approximating the Brownian motion itself a logarithmic term is necessary). Moreover, the exploited freezing technique of the mean field terms yields to the problem to handle the quantitative behavior of several different generators. Using BMO-techniques we obtain the rate of convergence for the integrated gradient process in the scale of Lorentz (type) spaces of exponential type.

Convergence rate for random walk approximations of mean field BSDEs

TL;DR

This paper analyzes the convergence rate of scaled random-walk approximations to mean-field backward stochastic differential equations (BSDEs) in Wasserstein-type distances, using a freezing technique instead of particle methods. It extends Donsker-type convergence results to the mean-field setting, where a time-singularity in the generator arises from Hölder-terminal data, and introduces a modified Hölder continuity |r|_{(ε,β)} to recover polynomial rates up to logarithmic factors. The authors establish coupling-based pathwise and Wasserstein bounds, with rates that include a log term and depend on ε, α, and β, and they provide a practical, non-particle algorithm along with numerical validation showing the expected rate behavior. The work advances numerical methods for nonlocal mean-field BSDEs and yields tools that are broadly applicable to the probabilistic analysis of mean-field problems and their associated nonlocal PDEs.

Abstract

We study the rate of convergence w.r.t.~a Wasserstein type distance for random walk approximations of mean field BSDEs. Our method does not use the particle method but instead a freezing technique. We extend results by Briand, Ch. Geiss, S. Geiss, and Labart [Bernoulli, 27(2) 2021] about the rate of convergence of a Donsker-type theorem for BSDEs from the classical setting to the mean field setting. In this connection the mean field setting leads to new phenomena and requires new techniques that should be of independent interest: The Hölder continuous terminal condition causes a singularity in time of the generator when seen as a generator in the non-mean field setting. To handle this singularity we introduce a concept of modified Hölder continuity by which we are able to achieve, up to a logarithmic term, the same polynomial approximation rates as in the classical non-mean field setting (in fact, already when approximating the Brownian motion itself a logarithmic term is necessary). Moreover, the exploited freezing technique of the mean field terms yields to the problem to handle the quantitative behavior of several different generators. Using BMO-techniques we obtain the rate of convergence for the integrated gradient process in the scale of Lorentz (type) spaces of exponential type.
Paper Structure (29 sections, 22 theorems, 276 equations, 2 figures)

This paper contains 29 sections, 22 theorems, 276 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. The random walk BSDE
  3. Previous results on random walk approximations
  4. Main results
  5. The three steps from continuous to discrete
  6. New techniques and obstacles caused by the mean field setting
  7. Organization of the article
  8. Open problems
  9. Some notation and assumptions
  10. The function spaces Mphi
  11. Random walk approximation of the Brownian motion
  12. Regularity properties of u and nabla-u under A1
  13. Local Hölder continuity of the generator with frozen mean field terms
  14. The discrete scheme
  15. From continuous to discrete in three steps
  16. ...and 14 more sections

Key Result

Theorem 1.1

Suppose H:A1, $\beta>\frac{1}{2}$, $n\ge n_0$, and that $\|\cdot\|$ is a monotone semi-norm on $(\Omega, \mathcal{F}, \mathbb{P})$ with $\|\cdot\| \le \kappa \|\cdot \|_{L^{\exp(1)}}$ for some $\kappa>0$. Then, for $t \in [t_o,T[$ and $x,y\in {\mathbb{R}}$, one has and, for each coupling $(\mathcal{B}, \mathcal{B}^n)\in \Gamma(B,B^n)$, where $c_eqn:2:statement:main_result_local_version, c_eqn:1:

Figures (2)

  • Figure 1: Evolution of $\log(E_Y)$ w.r.t. $\log(n)$
  • Figure 2: Evolution of $\log(E_Y)$ w.r.t. $\log(n)$

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Example 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 4.1
  • proof
  • Lemma 5.1: BGGL21
  • ...and 36 more