Resource-scaling analysis for variational Feynman–Kac propagation

Establish general convergence rates and error bounds for the McLachlan-principle variational algorithm that propagates solutions of the Feynman–Kac/Kolmogorov forward equation ∂_t u = L u − V u, and determine how the number of time steps N_tau and the number of ansatz parameters N_theta must scale with model characteristics such as the local-volatility surface σ(x,t), payoff discontinuities, and stiffness, in order to achieve a target accuracy and clarify resource requirements.

Background

The paper contrasts its Schrödingerisation-based forward-PDE approach with a related variational method that applies McLachlan’s principle to the Feynman–Kac (Kolmogorov forward) equation ∂_t u = L u − V u. In that method, the ansatz state is advanced in time by iteratively solving for small parameter updates, avoiding an outer optimization loop and exhibiting practical short-horizon stability.

Despite empirical promise, the variational approach lacks theoretical guarantees: specifically, there are no established general convergence rates or error bounds. The authors highlight that an open issue is to rigorously relate the algorithm’s resource parameters—the number of time steps N_tau and the number of ansatz parameters N_theta—to concrete model features such as the local-volatility function σ(x,t), payoff discontinuities, or stiffness. Resolving this would clarify scaling, guide ansatz design, and inform practical parameter choices.