Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A path-dependent PDE solver based on signature kernels

Alexandre Pannier, Cristopher Salvi

TL;DR

A provably convergent kernel-based solver for path-dependent PDEs (PPDEs) that leverages signature kernels, a recently introduced class of kernels on path-space, and proves consistency of the proposed scheme, guaranteeing convergence to the PPDE solution as the number of collocation points increases.

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.

A path-dependent PDE solver based on signature kernels

TL;DR

A provably convergent kernel-based solver for path-dependent PDEs (PPDEs) that leverages signature kernels, a recently introduced class of kernels on path-space, and proves consistency of the proposed scheme, guaranteeing convergence to the PPDE solution as the number of collocation points increases.

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.
Paper Structure (37 sections, 12 theorems, 130 equations, 1 figure, 4 tables)

This paper contains 37 sections, 12 theorems, 130 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Path-dependent PDEs
  3. Signature kernels
  4. Kernels vs neural networks for PDEs
  5. Monte Carlo methods for pricing under rough volatility
  6. Contributions
  7. Related literature
  8. Outline
  9. Path-dependent PDEs
  10. Path spaces
  11. Functions on continuous paths
  12. The target PDEs
  13. Fractional Brownian motion
  14. VIX options under rough Bergomi model
  15. Options under rough Bergomi model
  16. ...and 22 more sections

Key Result

Lemma 3.2

Let $\gamma \in \mathrm{BV}(\mathbb{T},V)$. Then, the linear controlled differential equation admits $Y_s = A \cdot S(\gamma)_{[t,s]}$ as unique solution, where the product $\cdot$ on $\overline{T(V)}$ is defined for any $v=\left(v_{0},v_{1},...\right)$ and $w=\left( w_{0},w_{1},...\right)$ in $\overline{T(V)}$ as the element $v \cdot w=\left( z_{0},z_{1},...\right)$ in $\overline{T(V)}$ su and

Figures (1)

  • Figure 1: Analytic prices against Signature Kernel prices with $400$ collocation points.

Theorems & Definitions (32)

  • Remark 2.1
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Definition 3.4
  • Remark 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Definition 3.8
  • ...and 22 more