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

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.

💡 Research Summary

The paper addresses the numerical approximation of stochastic diffusion systems whose drift and diffusion coefficients are only locally Lipschitz continuous and may grow polynomially. Classical explicit Euler–Maruyama (EM) schemes fail to converge strongly for such SDEs because the coefficients can become unbounded, especially in stiff or high‑dimensional problems. Existing remedies—tamed EM, truncated EM, backward EM—modify the coefficients or solve implicit equations, but they either introduce bias, require very small fixed steps, or are computationally expensive.

The authors propose a fully explicit adaptive time‑stepping method (A‑TS) that adjusts the step size based on the current state of the numerical solution. They introduce an increasing control function ϕ(r) that dominates the growth of both drift f and diffusion g, and define a state‑dependent step function
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