On data-based optimal stopping under stationarity and ergodicity

TL;DR

This paper introduces a nonparametric, data-driven algorithm for optimal stopping in stationary and ergodic processes, achieving universal consistency and applicable to American options without relying on specific model assumptions.

Contribution

It presents a novel nonparametric algorithm for optimal stopping that guarantees convergence to the optimal expected gain under minimal process assumptions.

Findings

Algorithm is universally consistent for stationary ergodic processes.

Achieves convergence of expected gain to the optimal value as data size increases.

Demonstrated effectiveness through simulated American option exercises.

Abstract

The problem of optimal stopping with finite horizon in discrete time is considered in view of maximizing the expected gain. The algorithm proposed in this paper is completely nonparametric in the sense that it uses observed data from the past of the process up to time $-n+1$, $n\in\mathbb{N}$, not relying on any specific model assumption. Kernel regression estimation of conditional expectations and prediction theory of individual sequences are used as tools. It is shown that the algorithm is universally consistent: the achieved expected gain converges to the optimal value for $n\to\infty$ whenever the underlying process is stationary and ergodic. An application to exercising American options is given, and the algorithm is illustrated by simulated data.