Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Primal and dual optimal stopping with signatures

Christian Bayer, Luca Pelizzari, John Schoenmakers

TL;DR

This work addresses optimal stopping under memoryful, non-Markovian dynamics by leveraging rough-path signatures to lift path history into a Markovian-like feature space. It develops two complementary approaches: a primal Longstaff–Schwartz method using signature-based regression and a dual Rogers-type method with a sample-average approximation, both with proven convergence. The framework is instantiated in two non-Markovian finance settings—stopping fractional Brownian motion and pricing American options in rough Bergomi models—demonstrating reliable lower and upper bounds and highlighting the benefits of signature-based basis expansions. The results provide a principled, scalable pathwise approach for American-style pricing in memory-driven models and suggest avenues for further enhancement via richer basis functions or neural- signature methods.

Abstract

We propose two signature-based methods to solve the optimal stopping problem - that is, to price American options - in non-Markovian frameworks. Both methods rely on a global approximation result for $L^p-$functionals on rough path-spaces, using linear functionals of robust, rough path signatures. In the primal formulation, we present a non-Markovian generalization of the famous Longstaff-Schwartz algorithm, using linear functionals of the signature as regression basis. For the dual formulation, we parametrize the space of square-integrable martingales using linear functionals of the signature, and apply a sample average approximation. We prove convergence for both methods and present first numerical examples in non-Markovian and non-semimartingale regimes.

Primal and dual optimal stopping with signatures

TL;DR

This work addresses optimal stopping under memoryful, non-Markovian dynamics by leveraging rough-path signatures to lift path history into a Markovian-like feature space. It develops two complementary approaches: a primal Longstaff–Schwartz method using signature-based regression and a dual Rogers-type method with a sample-average approximation, both with proven convergence. The framework is instantiated in two non-Markovian finance settings—stopping fractional Brownian motion and pricing American options in rough Bergomi models—demonstrating reliable lower and upper bounds and highlighting the benefits of signature-based basis expansions. The results provide a principled, scalable pathwise approach for American-style pricing in memory-driven models and suggest avenues for further enhancement via richer basis functions or neural- signature methods.

Abstract

We propose two signature-based methods to solve the optimal stopping problem - that is, to price American options - in non-Markovian frameworks. Both methods rely on a global approximation result for functionals on rough path-spaces, using linear functionals of robust, rough path signatures. In the primal formulation, we present a non-Markovian generalization of the famous Longstaff-Schwartz algorithm, using linear functionals of the signature as regression basis. For the dual formulation, we parametrize the space of square-integrable martingales using linear functionals of the signature, and apply a sample average approximation. We prove convergence for both methods and present first numerical examples in non-Markovian and non-semimartingale regimes.
Paper Structure (18 sections, 13 theorems, 122 equations, 4 tables)

This paper contains 18 sections, 13 theorems, 122 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Global approximation with rough-path signatures
  4. The space of stopped rough paths
  5. A Stone--Weierstrass result for robust signatures
  6. Approximation with robust signatures
  7. Optimal stopping with signatures
  8. Framework and problem formulation
  9. Primal optimal stopping with signatures
  10. Longstaff-Schwartz with signatures
  11. Dual optimal stopping with signatures
  12. Sample average approximation (SAA)
  13. Numerical examples
  14. Optimal stopping of fractional Brownian motion
  15. American options in rough volatility models
  16. ...and 3 more sections

Key Result

Lemma 2.4

For any process $A\in \mathbb{H}^2$ and $\alpha \in (0,1)$, there exists a measurable function $f:(\Lambda^{\alpha}_T,\mathcal{B}(\Lambda^{\alpha}_T)) \longrightarrow (\mathbb{R},\mathcal{B}(\mathbb{R}))$, such that $A_t = f(\mathbf{X}|_{[0,t]})$ almost everywhere.

Theorems & Definitions (26)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Example 1
  • Theorem 2.7
  • Corollary 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 16 more