Self-Similar Factor Stabilization of Weak Third Order Explicit Integration of Stochastic Differential Equations with Additive Noise

Source abstract

A third-order weak stochastic Taylor expansion (WSTE) is derived for systems driven by additive Gaussian noise with an invertible noise-coupling matrix. In contrast to the classical Kloeden–Platen construction, the present expansion employs a single Gaussian vector whose dimensionality matches that of the state variable. Numerical tests in stiff nonlinear potentials reveal, however, that both the third-order WSTE and its second-order truncation have limited stability and cannot be used as practical integrators at moderate time steps. To overcome this limitation, a self-similar factor (SSF) resummation is applied to the WSTE coefficients. The resulting stabilized scheme preserves third-order weak accuracy while exhibiting dramatically improved robustness compared to standard Euler and Heun methods. The Kloeden-Platen version of the WSTE of order three is shown to be not SSF-stabilizable, because it involves two independent sets of Gaussian random variables. A possible way to achieve its SSF-stabilization is suggested. The results reported demonstrate that higher-order weak integrators can offer substantial practical benefits when combined with appropriate stabilization techniques, and that the WSTE framework provides a natural foundation for developing such methods.