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A projected Euler Method for Random Periodic Solutions of Semi-linear SDEs with non-globally Lipschitz coefficients

Yujia Guo, Xiaojie Wang, Yue Wu

Abstract

The present work introduces and investigates an explicit time discretization scheme, called the projected Euler method,to numerically approximate random periodic solutions of semi-linear SDEs under non-globally Lipschitz conditions. The existence of the random periodic solution is demonstrated as the limit of the pull-back of the discretized SDE. Without relying on a priori high-order moment bounds of the numerical approximations, the mean square convergence rate of the approximation scheme is proved to be order $0.5$ for SDEs with multiplicative noise and order $1$ for SDEs with additive noise. Numerical examples are also provided to validate our theoretical findings.

A projected Euler Method for Random Periodic Solutions of Semi-linear SDEs with non-globally Lipschitz coefficients

Abstract

The present work introduces and investigates an explicit time discretization scheme, called the projected Euler method,to numerically approximate random periodic solutions of semi-linear SDEs under non-globally Lipschitz conditions. The existence of the random periodic solution is demonstrated as the limit of the pull-back of the discretized SDE. Without relying on a priori high-order moment bounds of the numerical approximations, the mean square convergence rate of the approximation scheme is proved to be order for SDEs with multiplicative noise and order for SDEs with additive noise. Numerical examples are also provided to validate our theoretical findings.

Paper Structure

This paper contains 10 sections, 15 theorems, 121 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Random Periodic Solutions of SDEs
  3. Numerical Approximation of Random Periodic Solution
  4. Mean square convergence order of Projected Euler Method
  5. Convergence rates for SDEs with multiplicative noise
  6. Convergence rates for SDEs with additive noise
  7. Numerical experiments
  8. Example 1
  9. Example 2
  10. Proof of Lemma \ref{['lem::coupled_monononcity_condition']}

Key Result

Lemma 2.3

Let Assumption ass_PEM be fulfilled, for any $p_2 \in [1,p_1)$, there exists a small positive constant $\epsilon$ such that where $\alpha_2 =\alpha_1+\epsilon<\lambda_1$, $c_0= \tfrac{\|f(t,0)\|^2}{2\epsilon} +\tfrac{(2p_1-1)^2}{4(p_1-p_2)}\|g(t,0)\|^2 +\tfrac{2p_1-1}{2}\|g(t,0)\|^2$.

Figures (5)

  • Figure 1: Two paths generated by projected Euler methods from differential initial conditions.
  • Figure 2: Simulations of the process ${\tilde{X}_{t}^{-10}(\omega,0.3),2 \leq t \leq 6}$ and ${\tilde{X}_{t}^{-10}(\theta_{-2}\omega,0.3),4 \leq t \leq 8}$.
  • Figure 3: The mean-square error plot of the projected Euler method \ref{['eq:the_projected_euler_method']} for simulating the solution of \ref{['eq_PEM:example_1']}.
  • Figure 4: Simulations of the processes $\tilde{X}^{-5}_{t}(\omega,0.5),10\leq t\leq 13$ and $\tilde{X}^{-5}_{t}(\theta_{-1}\omega,0.5),11\leq t\leq 14$.
  • Figure 5: The mean-square error plot of the projected Euler method \ref{['eq:the_projected_euler_method']} for simulating the solution of \ref{['eq_PEM:example_2']}.

Theorems & Definitions (26)

  • Definition 2.1
  • Lemma 2.3
  • Lemma 2.5
  • Lemma 2.6
  • proof : Proof of Lemma \ref{['lem_PEM:the_pth_of_exact_solution']}.
  • Lemma 2.7
  • proof : Proof of Lemma \ref{['lem_PEM:the_esti_(X(t1)-X(t2)']}
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 16 more