A path-dependent PDE solver based on signature kernels

TL;DR

This paper introduces a novel kernel-based numerical method for solving path-dependent PDEs using signature kernels, providing convergence guarantees and demonstrating effectiveness in option pricing scenarios.

Contribution

It presents a new PPDE solver leveraging signature kernels with proven convergence and a closed-form solution in linear cases, offering an alternative to Monte Carlo methods.

Findings

Convergence of the scheme is proven under certain assumptions.

The method provides a closed-form solution in linear PPDE cases.

Numerical experiments show effectiveness in option pricing with rough volatility.

Abstract

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.