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Approximation of invariant probability measures for super-linear stochastic functional differential equations with infinite delay

Guozhen Li, Shan Huang, Xiaoyue Li, Xuerong Mao

Abstract

This paper studies explicit numerical approximations of the invariant probability measures (IPMs) for stochastic functional differential equations (SFDEs) with infinite delay under one-sided Lipschitz condition on the drift coefficient. To date, numerical approximations of IPMs for super-linear SFDEs have been focused to finite-delay cases and implicit schemes that require additional computational effort. To overcome these constraints, we propose an explicit truncated Euler-Maruyama (TEM) scheme employing both time and space truncation for SFDEs with infinite delay, which is explicit and requires only finite historical storage. Firstly, we establish the strong convergence of the numerical segment process and determine its convergence rate over any finite time horizon. Next, we show that the numerical segment process generated by the TEM scheme admits a unique numerical IPM. Leveraging these results, we then prove that the numerical IPM converges to the exact IPM in the Wasserstein distance, with an explicitly obtained convergence rate.

Approximation of invariant probability measures for super-linear stochastic functional differential equations with infinite delay

Abstract

This paper studies explicit numerical approximations of the invariant probability measures (IPMs) for stochastic functional differential equations (SFDEs) with infinite delay under one-sided Lipschitz condition on the drift coefficient. To date, numerical approximations of IPMs for super-linear SFDEs have been focused to finite-delay cases and implicit schemes that require additional computational effort. To overcome these constraints, we propose an explicit truncated Euler-Maruyama (TEM) scheme employing both time and space truncation for SFDEs with infinite delay, which is explicit and requires only finite historical storage. Firstly, we establish the strong convergence of the numerical segment process and determine its convergence rate over any finite time horizon. Next, we show that the numerical segment process generated by the TEM scheme admits a unique numerical IPM. Leveraging these results, we then prove that the numerical IPM converges to the exact IPM in the Wasserstein distance, with an explicitly obtained convergence rate.
Paper Structure (11 sections, 23 theorems, 268 equations, 3 figures)

This paper contains 11 sections, 23 theorems, 268 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Numerical segment process
  4. Truncated Euler-Maruyama scheme
  5. Strong convergence of numerical segment process
  6. Convergence rate of numerical segment process
  7. Numerical IPM
  8. Ergodicity of SFDEswID
  9. Convergence of numerical IPM
  10. Convergence rate of numerical IPM
  11. Numerical experiments

Key Result

Theorem 2.4

Let Assumptions a2.1-f, a2.1-g and a2.2 hold. Then the SFDEswID ISFDE has a unique global solution $x(t)$ on $t \in (-\infty, \infty)$. Furthermore, for any $p > 0$, and where and $\tau_h = \inf\{t \ge 0: |x(t)|\ge h \}$, for any $h > \|\xi\|_r$.

Figures (3)

  • Figure 1: (a) $\xi_1(u) = e^{0.2u}$. (b) $\xi_2(u) = - e^{0.2u}$. (c) $\xi_3(u) = u$.
  • Figure 2: (a) $cos(\|\cdot\|_r)$. (b) $\|\cdot\|_r \wedge 2$.
  • Figure 3: (a) $cos(\|\cdot\|_r)$. (b) $\|\cdot\|_r \wedge 2$.

Theorems & Definitions (43)

  • Theorem 2.4
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • Theorem 3.6
  • proof
  • Lemma 3.7
  • Lemma 3.8
  • proof
  • ...and 33 more