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Numerical approximations to invariant measures of hybrid stochastic differential equations with superlinear coefficients via the backward Euler-Maruyama method

Wei Liu, Jie Xu

TL;DR

This work addresses invariant measures of hybrid SDEs with Markovian switching and superlinear coefficients by introducing a backward Euler-Maruyama scheme. It proves existence and uniqueness of a numerical invariant measure and shows exponential convergence of the numerical measure to the true invariant measure, while also establishing a rate of convergence in the Wasserstein metric with respect to the time step. Theoretical results are complemented by numerical simulations that confirm rapid convergence and robustness to initial conditions. The approach relaxes the global Lipschitz restriction on diffusion and suggests potential connections to numerically solving related PDEs.

Abstract

For stochastic differential equations (SDEs) with Markovian switching, whose drift and diffusion coefficients are allowed to contain superlinear terms, the backward Euler-Maruyama (BEM) method is proposed to approximate the invariant measure. The existence and uniqueness of the invariant measure of the numerical solution generated by the BEM method is proved. Then the convergence of the numerical invariant measure to its underlying counterpart is shown. Those results obtained in this work release the requirement of the global Lipschitz condition on the diffusion coefficient in [X. Li et al. SIAM J. Numer. Anal. 56(3)(2018), pp. 1435-1455] and can also be regarded as a non-trivial extension of [W. Liu et al. Appl. Numer. Math. 184(2023), pp. 137-150] to the case of hybrid SDEs.

Numerical approximations to invariant measures of hybrid stochastic differential equations with superlinear coefficients via the backward Euler-Maruyama method

TL;DR

This work addresses invariant measures of hybrid SDEs with Markovian switching and superlinear coefficients by introducing a backward Euler-Maruyama scheme. It proves existence and uniqueness of a numerical invariant measure and shows exponential convergence of the numerical measure to the true invariant measure, while also establishing a rate of convergence in the Wasserstein metric with respect to the time step. Theoretical results are complemented by numerical simulations that confirm rapid convergence and robustness to initial conditions. The approach relaxes the global Lipschitz restriction on diffusion and suggests potential connections to numerically solving related PDEs.

Abstract

For stochastic differential equations (SDEs) with Markovian switching, whose drift and diffusion coefficients are allowed to contain superlinear terms, the backward Euler-Maruyama (BEM) method is proposed to approximate the invariant measure. The existence and uniqueness of the invariant measure of the numerical solution generated by the BEM method is proved. Then the convergence of the numerical invariant measure to its underlying counterpart is shown. Those results obtained in this work release the requirement of the global Lipschitz condition on the diffusion coefficient in [X. Li et al. SIAM J. Numer. Anal. 56(3)(2018), pp. 1435-1455] and can also be regarded as a non-trivial extension of [W. Liu et al. Appl. Numer. Math. 184(2023), pp. 137-150] to the case of hybrid SDEs.

Paper Structure

This paper contains 7 sections, 7 theorems, 79 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Main results
  4. Existence and uniqueness of numerical invariant measure
  5. Convergence of numerical invariant measure
  6. Numerical simulations
  7. Conclusion and future works

Key Result

Lemma 2.1

Let Assumption KhCon hold. If $\Delta \in (0, 1/(n_M+2))$, then the BEM method eq:2 is well defined.

Figures (3)

  • Figure 1: Left: one sample path of $r(t)$. Right: proportions of visits to each state for one path of $r(t)$
  • Figure 2: Left: empirical density functions at different time points. Right: empirical density functions at t=40 for SDEs with different initial values.
  • Figure 3: K-S tests for samples at consecutive time points.

Theorems & Definitions (15)

  • Remark 1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • ...and 5 more