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Strong convergence of multiscale truncated Euler-Maruyama method for super-linear slow-fast stochastic differential equations

Yuanping Cui, Xiaoyue Li, Xuerong Mao

TL;DR

This work tackles numerical approximation of slow components in slow-fast stochastic differential equations with super-linear drift, by formulating an explicit multiscale Euler–Maruyama scheme (MTEM) that combines a truncation of the averaged drift with a micro-solver-based estimator. Using the averaging principle, the authors prove strong convergence of the MTEM solution to the exact slow dynamics in the $p$-th moment and derive a strong error bound that scales with the macro step $\\Delta_1$ and micro-parameters $\\Delta_2$, $M$. The analysis relies on ergodicity and invariant measures for the frozen fast dynamics, together with rigorous control of the estimator $B_M$ and the truncated drift $T_{\\Delta_1}$, yielding optimal convergence rates under fairly mild conditions. Numerical experiments corroborate the theory, showing suppression of divergence due to truncation and confirming the predicted convergence rates.

Abstract

This manuscript is dedicated to the numerical approximation of super-linear slow-fast stochastic differential equations (SFSDEs). Borrowing the heterogeneous multiscale idea, we propose an explicit multiscale Euler-Maruyama scheme suitable for SFSDEs with locally Lipschitz coefficients using an appropriate truncation technique. By the averaging principle, we establish the strong convergence of the numerical solutions to the exact solutions in the pth moment. Additionally, under lenient conditions on the coefficients, we also furnish a strong error estimate. In conclusion, we give two illustrative examples and accompanying numerical simulations to affirm the theoretical outcomes.

Strong convergence of multiscale truncated Euler-Maruyama method for super-linear slow-fast stochastic differential equations

TL;DR

This work tackles numerical approximation of slow components in slow-fast stochastic differential equations with super-linear drift, by formulating an explicit multiscale Euler–Maruyama scheme (MTEM) that combines a truncation of the averaged drift with a micro-solver-based estimator. Using the averaging principle, the authors prove strong convergence of the MTEM solution to the exact slow dynamics in the -th moment and derive a strong error bound that scales with the macro step and micro-parameters , . The analysis relies on ergodicity and invariant measures for the frozen fast dynamics, together with rigorous control of the estimator and the truncated drift , yielding optimal convergence rates under fairly mild conditions. Numerical experiments corroborate the theory, showing suppression of divergence due to truncation and confirming the predicted convergence rates.

Abstract

This manuscript is dedicated to the numerical approximation of super-linear slow-fast stochastic differential equations (SFSDEs). Borrowing the heterogeneous multiscale idea, we propose an explicit multiscale Euler-Maruyama scheme suitable for SFSDEs with locally Lipschitz coefficients using an appropriate truncation technique. By the averaging principle, we establish the strong convergence of the numerical solutions to the exact solutions in the pth moment. Additionally, under lenient conditions on the coefficients, we also furnish a strong error estimate. In conclusion, we give two illustrative examples and accompanying numerical simulations to affirm the theoretical outcomes.
Paper Structure (8 sections, 33 theorems, 268 equations, 4 figures)

This paper contains 8 sections, 33 theorems, 268 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. The construction of explicit multiscale scheme
  4. Some preliminary estimates
  5. Strong convergence in $p$th moment
  6. Strong error bounds
  7. Numerical examples
  8. Concluding remarks

Key Result

Lemma 2.1

If $({\bf F3})$ hold, then for any $x\in \mathbb{R}^{n_1}$, $y_0\in \mathbb{R}^{n_2}$, there exist positive constants $\tilde{\alpha}_k$ and $C_k$ such that

Figures (4)

  • Figure 1: The sample paths of the PI numerical solution $Z(t)$ on $t\in [0,3]$ with $\Delta_1=2^{-6}$, $\Delta_2=2^{-6}$ and $M=2^{18}$.
  • Figure 2: The sample paths of the MTEM numerical solution $X(t)$ on $t\in [0,3]$ with $\Delta_1=2^{-6}$, $\Delta_2=2^{-6}$ and $M=2^{18}$.
  • Figure 3: The SMSE for $q=2,3,4,5,6,7$ with $500$ sample points. The red dashed line is the reference with slope -1.
  • Figure 4: Four pairs of sample paths of $\bar{x}(t)$ and $X(t)$ for $t\in [0, 5]$ with $(\Delta_1,\Delta_2,M)=(2^{-8},2^{-6},2^{12})$.

Theorems & Definitions (66)

  • Remark 2.1
  • Lemma 2.1: MR4047972
  • Lemma 2.2: MR4047972
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5: MR4047972
  • Lemma 2.6
  • proof
  • ...and 56 more