Stochastic differential equation with piecewise continuous arguments: Markov property, invariant measure and numerical approximation

Chuchu Chen; Jialin Hong; Yulan Lu

doi:10.3934/dcdsb.2022098

Article Contents

2023, Volume 28, Issue 1: 765-807. Doi: 10.3934/dcdsb.2022098

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Stochastic differential equation with piecewise continuous arguments: Markov property, invariant measure and numerical approximation

1.
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100049, China
2.
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

^* Corresponding author: Yulan Lu
^* Corresponding author: Yulan Lu

Received: January 31, 2020

Revised: October 31, 2021

Early access: June 2022

Published: January 2023

This work is funded by National Natural Science Foundation of China (No. 11971470, No. 11871068, No. 12031020, No. 12022118, No. 12026428), by Youth Innovation Promotion Association CAS, and by National Postdoctoral Program for Innovative Talents (No. BX20180347)

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Abstract

For the stochastic differential equation with piecewise continuous arguments, multiplicative noises and dissipative drift coefficients, we show that the solution at integer time is a Markov chain and admits a unique invariant measure. In order to numerically preserve the invariant measure, we apply the backward Euler method to the equation, and prove that the numerical solution at integer time is also a Markov chain and possesses a unique numerical invariant measure. By establishing several a priori estimations, we present the time-independent weak error analysis for the method via Malliavin calculus, which implies that the numerical invariant measure converges to the original one with weak order 1. Numerical experiments verify the theoretical analysis.

Keywords:

Invariant measure,
Markov chain,
weak convergence,
backward Euler method,
stochastic differential equations with piecewise continuous arguments.

Mathematics Subject Classification: Primary: 60H35, 37M25; Secondary: 65C30.

Citation:

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Figure 1. The expectations and variances of $ X(t) $ and $ X(k) $

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Figure 2. The expectations (left column) and variances (right column) of $ X(t) $ with different $ \theta_2 $

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Figure 3. Order of weak convergence of BE method

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Figure 4. The evolution of $ \mathbb{E}\phi(Y_k) $ started from different initial data

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Figure 5. Order of weak convergence of the BE method

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Figure 6. The evolution of $ \mathbb{E}\phi(Y_k) $ started from different initial data

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Figure 7. Order of weak convergence of the BE method

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Figure 8. The evolution of $ \mathbb{E}\phi(Y_k) $ started from different initial data

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