# Semi-tractability of optimal stopping problems via a weighted stochastic mesh algorithm

## TL;DR

This paper introduces a Weighted Stochastic Mesh algorithm for approximating optimal stopping problems, demonstrating semi-tractability in discrete cases and providing tight complexity bounds in continuous cases, supported by numerical examples.

## Contribution

The paper presents a novel Weighted Stochastic Mesh algorithm that achieves semi-tractability for discrete optimal stopping problems and offers the tightest known complexity bounds for continuous problems.

## Key findings

- Discrete case complexity bounded by ε^{-4} log^{d+2}(1/ε)
- Continuous case bounds are the tightest known
- Numerical example validates theoretical results

## Abstract

In this article we propose a Weighted Stochastic Mesh (WSM) Algorithm for approximating the value of a discrete and continuous time optimal stopping problem. We prove that in the discrete case the WSM algorithm leads to semi-tractability of the corresponding optimal problems in the sense that its complexity is bounded in order by $\varepsilon^{-4}\log^{d+2}(1/\varepsilon)$ with $d$ being the dimension of the underlying Markov chain. Furthermore we study the WSM approach in the context of continuous time optimal stopping problems and derive the corresponding complexity bounds. Although we can not prove semi-tractability in this case, our bounds turn out to be the tightest ones among the bounds known for the existing algorithms in the literature. We illustrate our theoretical findings by a numerical example.