Discrete-type approximations for non-Markovian optimal stopping problems: Part II

TL;DR

This paper introduces a Longstaff-Schwartz-type algorithm for non-Markovian optimal stopping problems, extending applicability to complex path-dependent processes like fractional Brownian motions, with error estimates based on statistical learning theory.

Contribution

It develops a novel algorithm for non-Markovian optimal stopping, incorporating error bounds and analytical insights for path-dependent stochastic processes.

Findings

Algorithm applicable to non-Markovian processes

Provides error estimates based on VC dimension

Analyzes continuation values for path-dependent SDEs

Abstract

In this paper, we present a Longstaff-Schwartz-type algorithm for optimal stopping time problems based on the Brownian motion filtration. The algorithm is based on Le\~ao, Ohashi and Russo and, in contrast to previous works, our methodology applies to optimal stopping problems for fully non-Markovian and non-semimartingale state processes such as functionals of path-dependent stochastic differential equations and fractional Brownian motions. Based on statistical learning theory techniques, we provide overall error estimates in terms of concrete approximation architecture spaces with finite Vapnik-Chervonenkis dimension. Analytical properties of continuation values for path-dependent SDEs and concrete linear architecture approximating spaces are also discussed.