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Order-one explicit approximations of random periodic solutions of semi-linear SDEs with multiplicative noise

Yujia Guo, Xiaojie Wang, Yue Wu

TL;DR

The study addresses computing random periodic solutions for semi-linear SDEs with multiplicative noise under non-globally Lipschitz drift/diffusion. It introduces a projection-enhanced Milstein scheme (PMM) that remains explicit and proves pull-back convergence of the discretized process to the true RPS, achieving order-one mean-square accuracy on infinite time horizons without requiring high-moment bounds a priori. The authors further prove existence and uniqueness of the PMM's random periodic solution and establish $O(h)$ convergence of the numerical RPS to the exact RPS under a mild timestep constraint. Numerical experiments corroborate the theory, showing PMM outperforms PEM with clear near-linear convergence in practice, validating the method's efficiency for long-time simulations of RPS.

Abstract

This paper is devoted to order-one explicit approximations of random periodic solutions to multiplicative noise driven stochastic differential equations (SDEs) with non-globally Lipschitz coefficients. The existence of the random periodic solution is demonstrated as the limit of the pull-back of the discretized SDE. A novel approach is introduced to analyze mean-square error bounds of the proposed scheme that does not depend on a prior high-order moment bounds of the numerical approximations. Under mild assumptions, the proposed scheme is proved to achieve an expected order-one mean square convergence in the infinite time horizon. Numerical examples are finally provided to verify the theoretical results.

Order-one explicit approximations of random periodic solutions of semi-linear SDEs with multiplicative noise

TL;DR

The study addresses computing random periodic solutions for semi-linear SDEs with multiplicative noise under non-globally Lipschitz drift/diffusion. It introduces a projection-enhanced Milstein scheme (PMM) that remains explicit and proves pull-back convergence of the discretized process to the true RPS, achieving order-one mean-square accuracy on infinite time horizons without requiring high-moment bounds a priori. The authors further prove existence and uniqueness of the PMM's random periodic solution and establish convergence of the numerical RPS to the exact RPS under a mild timestep constraint. Numerical experiments corroborate the theory, showing PMM outperforms PEM with clear near-linear convergence in practice, validating the method's efficiency for long-time simulations of RPS.

Abstract

This paper is devoted to order-one explicit approximations of random periodic solutions to multiplicative noise driven stochastic differential equations (SDEs) with non-globally Lipschitz coefficients. The existence of the random periodic solution is demonstrated as the limit of the pull-back of the discretized SDE. A novel approach is introduced to analyze mean-square error bounds of the proposed scheme that does not depend on a prior high-order moment bounds of the numerical approximations. Under mild assumptions, the proposed scheme is proved to achieve an expected order-one mean square convergence in the infinite time horizon. Numerical examples are finally provided to verify the theoretical results.
Paper Structure (7 sections, 11 theorems, 123 equations, 3 figures)

This paper contains 7 sections, 11 theorems, 123 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Random Periodic Solutions of SDEs
  3. Numerical Approximation of Random Periodic solutions
  4. The local one-step approximation error
  5. The order-one convergence of PMM
  6. The Random Periodic Solution of the projected Milstein scheme
  7. Numerical experiments

Key Result

Lemma 2.4

Let Assumption ass_PMM hold and let $X_{t}^{-k\tau}$ be the exact solution of the SDE eq_PMM:Problem_SDE. If the initial value $X^{-k\tau}_{-k\tau}=\xi$, then for any $p \in [1,2p^*)$, there exists a positive constant $C$ such that

Figures (3)

  • Figure 1: Two paths generated by projected Milstein methods from differential initial conditions.
  • Figure 2: Simulations of the process ${\tilde{X}_{t}^{-20}(\omega,0.3),2 \leq t \leq 6}$ and ${\tilde{X}_{t}^{-20}(\theta_{-2}\omega,0.3),4 \leq t \leq 8}$.
  • Figure 3: The mean-square error plot of \ref{['eq_PMM:example_1']}.

Theorems & Definitions (18)

  • Definition 2.1
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • proof : Proof of Lemma \ref{['lem_PMM:the_esti_(X(t1)-PhiX(t2)']}
  • Lemma 3.3
  • Lemma 3.4
  • proof : Proof of Lemma \ref{['lem_PMM:the_esti_R_j+1']}
  • Theorem 3.5
  • ...and 8 more