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Well-posedness and numerical schemes for one-dimensional McKean-Vlasov equations and interacting particle systems with discontinuous drift

Gunther Leobacher, Christoph Reisinger, Wolfgang Stockinger

TL;DR

This paper establishes well-posedness results for one-dimensional McKean-Vlasov stochastic differential equations (SDEs) and related particle systems with a measure-dependent drift coefficient that is discontinuous in the spatial component, and a diffusion coefficient which is a Lipschitz function of the state only.

Abstract

In this paper, we first establish well-posedness results for one-dimensional McKean-Vlasov stochastic differential equations (SDEs) and related particle systems with a measure-dependent drift coefficient that is discontinuous in the spatial component, and a diffusion coefficient which is a Lipschitz function of the state only. We only require a fairly mild condition on the diffusion coefficient, namely to be non-zero in a point of discontinuity of the drift, while we need to impose certain structural assumptions on the measure-dependence of the drift. Second, we study Euler-Maruyama type schemes for the particle system to approximate the solution of the one-dimensional McKean-Vlasov SDE. Here, we will prove strong convergence results in terms of the number of time-steps and number of particles. Due to the discontinuity of the drift, the convergence analysis is non-standard and the usual strong convergence order $1/2$ known for the Lipschitz case cannot be recovered for all schemes.

Well-posedness and numerical schemes for one-dimensional McKean-Vlasov equations and interacting particle systems with discontinuous drift

TL;DR

This paper establishes well-posedness results for one-dimensional McKean-Vlasov stochastic differential equations (SDEs) and related particle systems with a measure-dependent drift coefficient that is discontinuous in the spatial component, and a diffusion coefficient which is a Lipschitz function of the state only.

Abstract

In this paper, we first establish well-posedness results for one-dimensional McKean-Vlasov stochastic differential equations (SDEs) and related particle systems with a measure-dependent drift coefficient that is discontinuous in the spatial component, and a diffusion coefficient which is a Lipschitz function of the state only. We only require a fairly mild condition on the diffusion coefficient, namely to be non-zero in a point of discontinuity of the drift, while we need to impose certain structural assumptions on the measure-dependence of the drift. Second, we study Euler-Maruyama type schemes for the particle system to approximate the solution of the one-dimensional McKean-Vlasov SDE. Here, we will prove strong convergence results in terms of the number of time-steps and number of particles. Due to the discontinuity of the drift, the convergence analysis is non-standard and the usual strong convergence order known for the Lipschitz case cannot be recovered for all schemes.

Paper Structure

This paper contains 18 sections, 17 theorems, 130 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notions and notation
  4. Properties of the transformation G
  5. Existence and uniqueness results
  6. McKean--Vlasov SDEs and interacting particle systems with decomposable drift
  7. McKean--Vlasov SDE with non-decomposable drift
  8. Interacting particle system with non-decomposable drift
  9. Euler--Maruyama scheme with and without transformation
  10. Scheme 1: Euler--Maruyama after transformation (decomposable case)
  11. Scheme 2: Euler--Maruyama without transformation (decomposable case)
  12. Scheme 1 for the non-decomposable case
  13. Numerical illustration
  14. Neuronal interactions
  15. Systemic risk
  16. ...and 3 more sections

Key Result

Proposition 3.1

Let Assumption (H.Assum:A) be satisfied, let $\xi \in L_p^{0}(\mathbb{R})$ for a given $p \geq 2$ and assume $c < 1/|\alpha|$. Then, the McKean--Vlasov SDE defined in (eq:Model1) has a unique strong solution in $\mathcal{S}^{p}([0,T])$.

Figures (3)

  • Figure 1: Strong convergence of the Euler--Maruyama scheme applied to the particle system obtained by approximating the equation for the action potential of the neurons.
  • Figure 2: Sample trajectories of the particle system associated with (\ref{['eq:control']}) for $N=10$ and $a=1$ (left) and $a=10$ (right).
  • Figure 3: Strong convergence of the Euler--Maruyama scheme applied to the particle system obtained by approximating the equation (\ref{['eq:control']}) with $a=1,5,10$.

Theorems & Definitions (42)

  • Definition 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Remark 3.3
  • ...and 32 more