Strong convergence rate of the explicit adaptive time-stepping methods for stochastic diffusion systems with locally Lipschitz coefficients

TL;DR

This work addresses numerical approximation of $d$-dimensional SDEs with drift and diffusion that are locally Lipschitz and may grow polynomially. It introduces an explicit adaptive time-stepping scheme (ATS) paired with a backstop method (e.g., TEM) to control growth and maintain stability, proving strong convergence to the true solution on finite intervals with order $1/2$ under suitable conditions. The authors establish $p$-th moment bounds for the numerical solutions and derive a finite-interval convergence rate, further showing that a stronger set of assumptions yields the optimal $1/2$ rate with a precise $L^q$-bound for $q$ in an admissible range. They also demonstrate that backstop options like TaEM preserve the convergence properties, and provide numerical experiments on stiff, non-stiff, and high-dimensional systems that confirm efficiency gains and the theoretical results. The approach offers a practical, explicit scheme for challenging SDEs with non-globally Lipschitz coefficients and superlinear diffusion growth, with potential extensions to long-time behavior and ergodicity.

Abstract

This paper proposes an adaptive time-stepping mothods for stochastic diffusion systems whose drift and diffusion coefficients are locally Lipschitz continuous and may exhibit polynomial growth. By controlling the growth of both the drift and diffusion coefficients, we give the choice of the state-dependent adaptive timestep and establish strong convergence of the proposed scheme with the optimal order $1/2$. The performance of the adaptive time-stepping scheme is compared with several widely used explicit and implicit schemes, including tamed EM, truncated EM, and backward EM schemes. Numerical experiments on stiff, non-stiff and high-dimensional stochastic diffusion systems verify the improved computational efficiency of the proposed scheme and validate the theoretical results.

Figures (8)

• Figure 1: A sample path of \ref{['ex0']}.
• Figure 2: (Left) One sample path generated by the ATS-TEM \ref{['scheme']} for SDE \ref{['ex0']}; (Right) The corresponding adaptive timesteps generated along the same path, together with the average timestep ${\delta _{mean}} = 1.2 \times {10^{ - 3}}$.
• Figure 3: (Left) The RMSEs for 500 sample paths between the exact solution of SDE \ref{['ex0']} and ATS-TEM, FS-TaEM, respectively, as functions of runtime for $\check{\delta} \in \{ {2^{ - 10}},{2^{ - 9}},{2^{ - 8}},{2^{ - 7}},{2^{ - 6}}\}$.
• Figure 4: The RMSE for 1000 sample paths between the exact solution \ref{['ex1_1']} of SDE \ref{['ex1']} and the numerical solutions generated by ATS-TEM, FS-TEM, FS-TaEM and FS-BEM, respectively, as functions of runtime for ${\check{\delta}} \in \left\{ {{2^{ - 11}},{2^{ - 10}},{2^{ - 9}},{2^{ - 8}},{2^{ - 7}}} \right\}$.
• Figure 5: The RMSE for 1000 sample paths between the exact solution of SDE \ref{['ex1']} and the ATS-TEM numerical solution with $\check{\delta} \in \left\{ {{2^{ - 11}},{2^{ - 10}},{2^{ - 9}},{2^{ - 8}},{2^{ - 7}}} \right\}$. The least squares line for ATS-TEM is $y = 0.5001x + 0.0667$.

Theorems & Definitions (22)

• Remark 2.2
• Lemma 2.3
• Lemma 2.4
• Remark 2.5
• Remark 2.6
• Remark 2.7
• Theorem 3.1
• Remark 3.2
• Lemma 3.3
• Theorem 3.4