Optimal stopping times for stochastic processes with path-dependent payoff functions.

Project Abstract

In this work, we investigate the problem of optimal stopping times for stochastic processes with path-dependent payoff functions, focusing on a broad class of Markovian and non-Markovian dynamics where the reward depends not only on the current state but also on the history of the process. We establish a unified analytical framework that combines stochastic calculus, functional analysis, and dynamic programming in Banach spaces to derive necessary and sufficient conditions for the existence and characterization of optimal stopping strategies. Our approach extends classical Snell envelope constructions to path-dependent payoffs by leveraging functional Itô calculus and pathwise representations of conditional expectations, enabling the treatment of both continuous and jump-type stochastic processes. We begin by formulating the stopping problem in terms of a gain functional that is non-anticipative and depends on the entire path segment up to the stopping time. We delineate the regularity properties required of the payoff functional, including measurability, adaptiveness, and continuity with respect to the Skorokhod topology, and we examine how path-dependence alters the structure of the value process. The core contributions include (i) deriving a generalized Snell envelope for path-dependent payoffs and proving its right-continuity with left limits under suitable compactness and boundedness assumptions; (ii) establishing a verification theorem that connects the optimal stopping region to the least excessive majorant of the gain process in an appropriate function space; (iii) identifying conditions under which optimal stopping times are Markovian, semi-Markovian, or highly non-Markovian, and providing constructive schemes to approximate optimal policies via discretization, regression-based Monte Carlo methods, and, where applicable, lattice and tree-based schemes. We conduct extensive theoretical analysis to compare the impact of path-dependence versus state-only payoffs on the smooth-fit principle, continuity of the value function, and the geometry of the stopping boundary. Special attention is given to payoff functionals incorporating running maxima/minima, realized variance, and functionals of the empirical distribution of the path, as well as scenarios with regime-switching dynamics. Numerical experiments illustrate the efficiency and accuracy of proposed algorithms in extracting near-optimal stopping rules, with demonstrations on models including geometric Brownian motion with memory, Ornstein–Uhlenbeck processes influenced by historical averages, and jump-diffusion models with path-dependent rewards. We also explore applications in finance, such as American-style options with path-dependent features (Asian, robust, and lookback options) and real options under uncertainty with history-based performance criteria, as well as in sequential decision-making problems in engineering and economics where the objective is to maximize a cumulative, history-aware payoff. The results contribute to a deeper understanding of how historical information shapes decision rules in stochastic timing problems and provide practical computational tools for implementing optimal stopping strategies in complex, path-dependent environments.

Project Overview

What This Project Is About

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A simple, approachable look at how and when to stop a process that evolves randomly over time. The project studies rules for choosing the best time to stop so as to maximize a payoff that depends not just on the current state but also on the path the process has taken.

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The Problem It Addresses

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Many real-world decisions depend on the history of a process, not only its current value. Traditional stopping rules assume instant, memoryless outcomes. This project explores how past behavior affects the best stopping choice and why incorporating the history leads to better, more robust decisions in finance, biology, and engineering.

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Objectives of the Project

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1. Explain what stopping times are in simple terms.
2. Introduce path-dependent payoffs and why they matter.
3. Describe the main ideas behind optimal stopping with paths.
4. Illustrate basic examples with intuition, not heavy math.
5. Discuss potential real-world applications.

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What You Will Do Step by Step

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1) Learn the core ideas of stochastic processes and stopping rules with plain language.

2) Look at simple examples where payoff depends on the path taken.

3) Build small, intuitive models that demonstrate the concepts without advanced formulas.

4) Compare stopping strategies using simple simulations or thought experiments.

5) Summarize how path-dependent payoffs change the decision rule.

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Expected Outcome

A clear, beginner-friendly understanding of how path history influences optimal stopping, with accessible examples and potential applications highlighted for further study.