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Particle approximation for a conditional McKean--Vlasov stochastic differential equation

Kai Du, Yunzhang Li, Yuyang Ye

Abstract

In this paper, we construct a type of interacting particle systems to approximate a class of stochastic different equations whose coefficients depend on the conditional probability distributions of the processes given partial observations. After proving the well-posedness and regularity of the particle systems, we establish a quantitative convergence result for the empirical measures of the particle systems in the Wasserstein space, as the number of particles increases. Moreover, we discuss an Euler--Maruyama scheme of the particle system and validate its strong convergence. A numerical experiment is conducted to illustrate our results.

Particle approximation for a conditional McKean--Vlasov stochastic differential equation

Abstract

In this paper, we construct a type of interacting particle systems to approximate a class of stochastic different equations whose coefficients depend on the conditional probability distributions of the processes given partial observations. After proving the well-posedness and regularity of the particle systems, we establish a quantitative convergence result for the empirical measures of the particle systems in the Wasserstein space, as the number of particles increases. Moreover, we discuss an Euler--Maruyama scheme of the particle system and validate its strong convergence. A numerical experiment is conducted to illustrate our results.
Paper Structure (12 sections, 13 theorems, 166 equations, 1 figure)

This paper contains 12 sections, 13 theorems, 166 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Conditional propagation of chaos
  4. Sequential importance sampling: full discretization
  5. Auxiliary lemmas
  6. Estimates for the weight of empirical measure
  7. Wasserstein convergence of weighted empirical measures
  8. Proof of main results
  9. Proof of Theorem \ref{['Theorem exist-unique']}
  10. Proof of particle approximation
  11. Proof of Theorem \ref{['Euler scheme']}
  12. Numerical Experiment

Key Result

Theorem 2.1

The particle system (particle) admits a unique strong solution $\{(X^{i,N},L^{i,N})\}_{i=1}^N$.

Figures (1)

  • Figure 1: Euler Scheme with Particle Approximation

Theorems & Definitions (25)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Corollary 2.1
  • Remark 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 15 more