Pathwise skew-symmetric discretisation for SDEs with superlinear drift

2026-07-20

AI Summary

This work addresses the instability of conventional numerical methods for stochastic differential equations with superlinear drift under non-globally Lipschitz conditions. To overcome this issue, the authors propose a path-dependent skew-symmetric discretization scheme that models the noise increment via a skew-normal distribution coupled with the Brownian motion increment, thereby substantially reducing the probability of erroneous excursions in regions of strong inward drift. Theoretical analysis establishes strong convergence of order 1/2 in the L² sense, rendering the method suitable for multilevel Monte Carlo (MLMC) implementations with a computational complexity of 𝒪(ε⁻²(log(1/ε))²). Numerical experiments confirm its superior performance over the tamed Euler scheme in terms of stability, convergence, and computational efficiency.

Abstract

The skew-symmetric discretisation has recently been proposed as a new robust simulation method for weakly approximating stochastic differential equations (SDEs) with non-globally Lipschitz drift. This work develops a pathwise version of the scheme by representing the noise increment as a skew-normal distribution and coupling it with the driving Brownian increments, thereby enabling its use in the multilevel Monte Carlo (MLMC) framework. Under suitable conditions, we establish strong convergence of order 1/2 in $L^2$. Subsequently, the associated MLMC estimator is shown to have computational complexity $\mathcal{O} \bigl(\varepsilon^{-2} (\log (1/\varepsilon))^2 \bigr)$ to achieve a mean-squared error $\varepsilon^2$. We then analytically compare the proposed scheme with the tamed Euler scheme, another benchmark for robust discretisation. Under a strong inward-drift regime with the current state being far from the stable region, we show that the probability of moving in the wrong direction tends to vanish in the skew-symmetric scheme, whereas the tamed Euler scheme makes such moves with a non-trivial probability. Furthermore, in the MLMC setting, employing a one-dimensional stochastic Ginzburg-Landau model, we specify the range of step sizes for which the asymptotic variance of the coupled level difference obtained via the pathwise skew-symmetric scheme is lower than that obtained via the tamed Euler scheme. Numerical experiments on several model examples support the theoretical rate of strong convergence and demonstrate the stability and effectiveness of the resulting MLMC in the superlinear drift setting.

Problem

• stochastic differential equations
• superlinear drift
• strong convergence
• multilevel Monte Carlo
• numerical stability

Innovation

• pathwise skew-symmetric discretisation
• strong convergence
• multilevel Monte Carlo
• superlinear drift
• skew-normal coupling