A path-dependent PDE solver based on signature kernels
This provides a new computational tool for solving path-dependent PDEs, which is important for quantitative finance applications like option pricing, though it appears incremental as an extension of kernel methods to this domain.
The authors developed a kernel-based solver for path-dependent PDEs using signature kernels, proving convergence under strict assumptions and demonstrating its effectiveness as an alternative to Monte Carlo methods in option pricing under rough volatility.
We develop a kernel-based solver for path-dependent PDEs (PPDEs) along with a convergence theory. Our numerical scheme leverages signature kernels, a recently introduced class of kernels on path-space. Specifically, we solve an optimal recovery problem by approximating the solution of a PPDE with an element of minimal norm in the signature reproducing kernel Hilbert space constrained to satisfy the PPDE at a finite collection of collocation paths. In the linear case, we show that the optimisation has a unique closed-form solution expressed in terms of signature kernel evaluations at the collocation paths. Under strict assumptions, we prove consistency of the proposed scheme, guaranteeing convergence to the PPDE solution as the number of collocation points increases. Finally, several numerical examples are presented, in particular in the context of option pricing under rough volatility. Our numerical scheme constitutes a valid alternative to the ubiquitous Monte Carlo methods.