Convergence analysis and remedies for higher-mode smooth Brownian bridges

Establish the convergence behavior of SDC-SBB methods using additional terms in the Karhunen–Loève smooth Brownian bridge expansion, and determine effective remedies for the resulting high-frequency quadrature and substepping difficulties as the time step decreases.

Background

The paper analyzes the SDC-SBB method primarily for a linear smooth Brownian bridge approximation. Including more Karhunen–Loève modes can improve the approximation of the Brownian bridge, but introduces rapidly oscillating terms that complicate spectral quadrature and impose restrictions on the number of collocation nodes used with explicit Euler substepping.

The authors note that a complete analysis of these convergence effects and possible remedies—including lower-frequency variance-reduction terms and higher-order substeppers—has not been provided. This leaves unresolved how to retain higher temporal convergence orders while controlling the computational and discretization costs associated with additional bridge modes.

References

Additional terms in the expansion not only require specialized quadrature for the computation of $I_S$, but pose restrictions on the number of collocation nodes due to substepping by an Euler method in the computation of the correction sweep, see eq.~eq:correctionsweep. As a rule of thumb, for an explicit Euler method, the stepsize, i.e., here the distance between two quadrature nodes, should be less than $1/4$-th of the oscillation period. For a term involving $\cos(j\pi t/\Delta T)$ with $t\in [0,\Delta T]$ this would result in $M>2j$ collocation nodes. Options to avoid the need for fine discretization include introducing terms to the expansion that reduce the variance error without high-frequency oscillations, or using higher-order substeppers. A thorough analysis of these convergence issues and potential remedies is out of scope for this paper, but will be a topic for future research.

— Toward Spectral Deferred Correction Methods for Stochastic Differential Equations  (2610.23693 - Binkowski et al., 16 Oct 2026) in Section 4.2, “Smooth Brownian bridge approximation error”