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On the infinite time horizon approximation for Lévy-driven McKean-Vlasov SDEs with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients

Ngoc Khue Tran, Trung-Thuy Kieu, Duc-Trong Luong, Hoang-Long Ngo

Abstract

This paper studies the numerical approximation for McKean-Vlasov stochastic differential equations driven by Lévy processes. We propose a tamed-adaptive Euler-Maruyama scheme and consider its strong convergence in both finite and infinite time horizons when applying for some classes of Lévy-driven McKean-Vlasov stochastic differential equations with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients.

On the infinite time horizon approximation for Lévy-driven McKean-Vlasov SDEs with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients

Abstract

This paper studies the numerical approximation for McKean-Vlasov stochastic differential equations driven by Lévy processes. We propose a tamed-adaptive Euler-Maruyama scheme and consider its strong convergence in both finite and infinite time horizons when applying for some classes of Lévy-driven McKean-Vlasov stochastic differential equations with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients.
Paper Structure (8 sections, 17 theorems, 189 equations, 1 figure)

This paper contains 8 sections, 17 theorems, 189 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model assumptions and moment estimates
  3. Propagation of chaos
  4. Tamed-adaptive Euler-Maruyama scheme
  5. Definition
  6. Moments of the tamed-adaptive Euler-Maruyama scheme
  7. Convergence
  8. Numerical experiment

Key Result

Proposition 2.3

(NBKGR20) Assume Conditions A1, A3 and that Condition A2 holds for $\kappa_1=\kappa_2=1, L_1= L_2>0$. Then, there exists a unique càdlàg process $X=(X_t)_{t \ge 0}$ taking values in $\mathbb{R}^d$ satisfying the McKean-Vlasov SDE with jumps eqn1 such that where $T>0$ is a fixed constant and $K:=K(\vert x_0\vert^2,d,L,L_1,T)$ is a positive constant.

Figures (1)

  • Figure 1: Error $\log_2 \textrm{MSE}(\textbf{l}, 10)$ plotted against $\textbf{l}=1,...,6$.

Theorems & Definitions (38)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 28 more