Optimal stopping with signatures

TL;DR

This paper introduces a novel approach to optimal stopping problems using signature methods and neural networks, enabling solutions under minimal assumptions and applicable to complex processes like fractional Brownian motion.

Contribution

It develops a signature-based framework for optimal stopping, recasting the problem as a deterministic optimization over expected signatures, and demonstrates an efficient neural network approach for numerical solutions.

Findings

Applicable to non-semimartingale processes like fractional Brownian motion

Recasts optimal stopping as a deterministic optimization problem

Uses neural networks to efficiently approximate signature functionals

Abstract

We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process $X$. We consider classic and randomized stopping times represented by linear and non-linear functionals of the rough path signature $\mathbb{X}^{<\infty}$ associated to $X$, and prove that maximizing over these classes of signature stopping times, in fact, solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a (deterministic) optimization problem depending only on the (truncated) expected signature $\mathbb{E}\left[ \mathbb{X}^{\le N}_{0,T} \right]$. By applying a deep neural network approach to approximate the non-linear signature functionals, we can efficiently solve the optimal stopping problem numerically. The only assumption on the process $X$ is that it is a continuous (geometric) random rough path. Hence, the theory encompasses processes such as fractional Brownian motion, which fail to be either semi-martingales or Markov processes, and can be used, in particular, for American-type option pricing in fractional models, e.g. on financial or electricity markets.