Paper: arXiv 2403.11738

Abstract

We develop a provably convergent kernel-based solver for path-dependent PDEs (PPDEs). 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 (RKHS) 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. 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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces a novel kernel-based solver for path-dependent PDEs, involving advanced functional analysis, signature kernels, and reproducible closed-form solutions, which is highly mathematically dense. However, the empirical evidence is limited to theoretical consistency proofs and a few numerical examples without comprehensive backtesting or real-world data validation.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>Solve Path-Dependent PDEs<br>without Monte Carlo"] --> B["Methodology:<br>Formulate Optimal Recovery<br>in Signature RKHS"]
  B --> C["Input:<br>Collocation Paths &<br>Boundary Conditions"]
  C --> D["Compute:<br>Signature Kernel<br>Evaluations"]
  D --> E["Solve Linear System:<br>Kernel Matrix Inversion"]
  E --> F["Outcome:<br>Closed-Form Solution<br>for PPDE"]
  F --> G["Validation:<br>Consistency Proof &<br>Option Pricing Examples"]

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4 of 84 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.