Implicit SDC remedies for stiff smooth-Brownian-bridge problems

Determine whether implicit or implicit-explicit spectral deferred correction methods can remedy the order reduction caused by unresolved fast oscillations when solving stiff SBB-ODEs for stochastic differential equations.

Background

In the Ornstein–Uhlenbeck experiment, the SDC-SBB method exhibits an order reduction relative to the expected order when the reference solution uses the true nonsmooth Brownian motion. The paper attributes this reduction to the inability of a stiff solver to resolve the fast oscillations introduced by the higher-mode smooth Brownian bridge approximation.

The authors identify implicit SDC as a possible remedy but do not analyze or test whether implicit or IMEX variants recover the expected convergence order. Establishing their effectiveness is therefore an unresolved methodological question for stiff SBB-ODEs.

References

However, such an order reduction is not unexpected for a stiff equation, where the fast oscillations cannot be resolved. Potential remedies include using implicit SDC, and will be investigated in future work.

— Toward Spectral Deferred Correction Methods for Stochastic Differential Equations  (2610.23693 - Binkowski et al., 16 Oct 2026) in Section 5, “Numerical Examples,” final paragraph of the Ornstein–Uhlenbeck example