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Approximation of Invariant Measures for Stochastic Differential Equations with Piecewise Continuous Arguments via Backward Euler Method

Chuchu Chen, Jialin Hong, Yulan Lu

TL;DR

The paper analyzes stochastic differential equations with piecewise continuous arguments and proves that the integer-time process $X(k)$ forms a time-homogeneous Markov chain with a unique invariant measure. It then discretizes the system using the backward Euler method, showing the BE sequence $Y_k$ is also a Markov chain with a unique invariant measure and preserved exponential ergodicity. A time-independent weak-error framework yields a weak convergence order of 1 for the BE method and a first-order convergence between invariant measures, i.e., $|\int \phi \,d\pi - \int \phi \,d\pi^{\delta}| = O(\delta)$ for suitable test functions $\phi$, with the analysis supported by Malliavin calculus. Numerical experiments on 1D PCAs with additive and multiplicative noise confirm the theoretical results, illustrating both ergodicity and the anticipated weak convergence rate.

Abstract

For the stochastic differential equation (SDE) which has piecewise continuous arguments (PCAs), is driven by multiplicative noises and its drift coefficients are dissipative, we show that the solution at integer time is a Markov chain and admits a unique invariant measure. In order to inherit numerically the invariant measure of SDE with PCAs, we apply the backward Euler (BE) method to the equation, and prove that the numerical solution at integer time is not only Markovian but also reproduces a unique numerical invariant measure. We present the time-independent weak error analysis for the method under certain hypothesis. Further, we show that the numerical invariant measure converges to the original one with order 1. Numerical experiments verify the theoretical analysis.

Approximation of Invariant Measures for Stochastic Differential Equations with Piecewise Continuous Arguments via Backward Euler Method

TL;DR

The paper analyzes stochastic differential equations with piecewise continuous arguments and proves that the integer-time process forms a time-homogeneous Markov chain with a unique invariant measure. It then discretizes the system using the backward Euler method, showing the BE sequence is also a Markov chain with a unique invariant measure and preserved exponential ergodicity. A time-independent weak-error framework yields a weak convergence order of 1 for the BE method and a first-order convergence between invariant measures, i.e., for suitable test functions , with the analysis supported by Malliavin calculus. Numerical experiments on 1D PCAs with additive and multiplicative noise confirm the theoretical results, illustrating both ergodicity and the anticipated weak convergence rate.

Abstract

For the stochastic differential equation (SDE) which has piecewise continuous arguments (PCAs), is driven by multiplicative noises and its drift coefficients are dissipative, we show that the solution at integer time is a Markov chain and admits a unique invariant measure. In order to inherit numerically the invariant measure of SDE with PCAs, we apply the backward Euler (BE) method to the equation, and prove that the numerical solution at integer time is not only Markovian but also reproduces a unique numerical invariant measure. We present the time-independent weak error analysis for the method under certain hypothesis. Further, we show that the numerical invariant measure converges to the original one with order 1. Numerical experiments verify the theoretical analysis.

Paper Structure

This paper contains 7 sections, 15 theorems, 215 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notations and Invariant Measures of the Solution
  3. Invariant Measures of the Backward Euler Method
  4. Approximation of the Invariant Measures
  5. A Priori Estimates
  6. Errors of $\pi$ and $\pi^\delta$
  7. Numerical Simulations

Key Result

Theorem 2.1

Suppose that Assumptions assumption1 and assumption2 hold. Then $\{X(k)\}_{k\in\mathbb{N}}$ is a time-homogeneous Markov chain with the transition probability kernel $P(x,B)$.

Figures (5)

  • Figure 1: The expectations and variances of $X(t)$ and $X(k)$
  • Figure 2: Order of weak convergence of BE method
  • Figure 3: The mean values of $Y_k$ with different initial data
  • Figure 4: Order of weak convergence of BE method
  • Figure 5: The mean values of $Y_k$ with different initial data

Theorems & Definitions (28)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 18 more