Optimal stopping with signatures

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 nonlinear 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 $\mathbb{E}[{\mathbb{X}}_{0,\mathit{T}}^{\le \mathit{N}}]$. By applying a deep neural network approach to approximate the nonlinear 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 semimartingales or Markov processes, and can be used, in particular, for American-type option pricing in fractional models, for example, on financial or electricity markets.