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

TL;DR

This paper introduces a discrete approximation method for solving continuous-time, non-Markovian optimal stopping problems, providing convergence rates and enabling practical Monte Carlo implementations.

Contribution

It develops a novel discrete approximation scheme for non-Markovian optimal stopping problems with explicit convergence rates and practical Monte Carlo algorithms.

Findings

Explicit convergence rates for optimal values

Construction of $$-optimal stopping times

Monte Carlo schemes for non-Markovian problems

Abstract

In this paper, we present a discrete-type approximation scheme to solve continuous-time optimal stopping problems based on fully non-Markovian continuous processes adapted to the Brownian motion filtration. The approximations satisfy suitable variational inequalities which allow us to construct $\epsilon$-optimal stopping times and optimal values in full generality. Explicit rates of convergence are presented for optimal values based on reward functionals of path-dependent SDEs driven by fractional Brownian motion. In particular, the methodology allows us to design concrete Monte-Carlo schemes for non-Markovian optimal stopping time problems as demonstrated in the companion paper by Bezerra, Ohashi and Russo.