# Optimal Stopping Theory: Methods and Techniques

## Spring 2024

Welcome to the 'Optimal Stopping Theory: Methods and Techniques' Reading Seminar, run by the students of Columbia University.

This Spring we will continue studying Optimal Stopping Theory by reading various papers. Our talks will be held in-person in Columbia University (Math 507) on Fridays from 4:30 p.m. to 5:30 p.m. EDT.

This seminar is the logical continuation of the seminars held in Fall 2021 - Optimal Stopping Seminar, Fall 2021. and Spring 2023 - OS and Control Seminar, Spring 2022.

If you would like to come or to be added on the mailing list, please email gg2793@columbia.edu.

## Next Seminar

Date and time Speaker Title and abstract
Friday, April 26, 4:30 p.m. EDT Graeme Baker Free boundary PDE and the American put exercise boundary

We show that the exercise boundary for the American put can be formulated as a free boundary PDE problem. This formulation leads to integral equation representations, which can be used for approximations. Using fixed point theorems, one can obtain existence, uniqueness, and regularity for some free boundary PDE. Chen and Chadam consider the regularity of the American put exercise boundary in a 2007 paper. That paper is quite technical, so we will only give a hint of the proof, by way of analogy with a free boundary problem for the heat equation from one of our own works (see on the Google Drive). Free boundary problems for the heat equation, known as Stefan problems, have an old history - for special initial conditions we'll show an explicit solution which was known in 1831!

## Past Seminars

Date and time Speaker Title and abstract
Friday, January 26, 4:30 p.m. EDT Ioannis Karatzas Goal Problems of Control and Stopping

If you can control the local characteristics of a diffusion process, within certain bounds mandated by your current position, how do you act if your aim is to
(i) reach a certain goal by a fixed deadline?
(ii) reach the goal in finite time?
(iii) minimize the expected time until you reach your goal?
(iv) maximize the expected reward you can collect by terminating the process at a stopping time of our choice?
We provide some answers to such stochastic optimization questions of the Dubbins-Savage variety, and pose open problems. (Based on joint work with William D. Sudderth.)
Friday, February 2, 4:30 p.m. EDT Georgy Gaitsgori Variational Inequalities in Optimal Stopping Problems

We will discuss how variational inequalities (VI) can help to solve optimal stopping (OS) problems. In short, this method can be described as follows: first, one conjectures that the value function of an OS problem must satisfy certain variational inequalities (which essentially arise from common sense and Ito's formula). Then one solves these inequalities and, moreover, completes the verification step, i.e., shows that if a function satisfies the VI, then it indeed provides the lower (upper) bound on achievable cost. We will demonstrate this method by discussing papers by Davis and Zervos '94, Karatzas and Wang '00, and others.
Friday, February 9, 4:30 p.m. EDT Richard Groenewald Free Boundary Problems in Optimal Stopping

We discuss commonly used techniques used to show that in the context of time-inhomogeneous optimal stopping problems, the value function is a solution to a free boundary problem. The general approach to show that the value function of a particular optimal stopping problem is continuous, assesses the existence and uniqueness questions pertaining to a related PDE on the interior of the continuation region and uses this PDE, along with an extension of Ito's formula (found in Peskir '05) to generate an integral equation for the free boundary. This is illustrated with the finite time horizon American option problem (see Methods of Mathematical Finance by Karatzas and Shreve) and in the setting of sequential testing of the drift of a Brownian path (see Ekstrom and Vaicenavacius '15).
Friday, February 16, 4:30 p.m. EDT Steven Campbell Bayesian Sequential Testing Problems

In this talk we will introduce and study several variants of the famous Bayesian sequential testing problem for a Brownian motion's drift. The problem puts us in the shoes of an agent who faces a Bayes risk and uses information from a signal process (the arithmetic Brownian motion) to determine an underlying and unobserved state of nature (the drift). Traditionally, the state of nature is binary, and the objective presents a hard-classification problem (in the sense that the observer makes a definite decision about the state of nature at their stopping time). We will consider both this classical setting and a relaxed formulation with soft classification loss functions (e.g., the cross-entropy loss) that are popular in machine learning. Our investigation will show that the value function is the unique non-trivial solution to a free boundary problem, and that the associated optimal stopping boundaries satisfy a pair of transcendental equations. A distinct feature of the soft classification setting is that stopping immediately is optimal if the signal-to-noise ratio is not strong enough.
Friday, February 23, 4:30 p.m. EDT Georgy Gaitsgori Superharmonicity, Scale functions, and Principle of Smooth Fit

We will start by proving a classical result of Dynkin and Yushkevich that the value function of the OS problem of stopping a Brownian motion, defined on a bounded interval with absorption at the endpoints, is the smallest concave majorant of the gain function. We will then discuss the results of Dayanik and Karatzas, who generalize the above statement to general diffusion processes with the help of scale functions. We will finish by discussing how the results of Dayanik and Karatzas are related to the principle of smooth fit. Moreover, we will discuss a result by Peskir about the existence of a regular diffusion and a differentiable gain function, such that the value function in the corresponding OS problem fails to satisfy the smooth fir condition.
Friday, March 1, 4:30 p.m. EDT Richard Groenewald Applications of an Optimal Stopping Problem to the Analysis of Corporate Debt Pricing

We will discuss the paper "Term Structures of Credit Spreads with Incomplete Accounting Information" by Duffie and Lando (2001) which analyzes corporate debt pricing in the setting where investors only observe nooisy estimates of a given firm's assets at specific times. The authors use the model of Leland and Toft (1996), where the firm defaults (liquidates all assets) at a stopping time which maximizes the expected return for shareholders, which is of course an optimal stopping problem. We will show that the value function of this problem satisfies an HJB equation and obtain it along with the optimal liquidation time explicitly. We will also briefly discuss the valuation of debt instruments issued by this firm as a consequence of this behaviour, under the assumption that bond market participants are restricted to a filtration which is strictly smaller than that of the firm.
Friday, March 8, 4:30 p.m. EDT No seminar

Friday, March 15, 4:30 p.m. EDT No seminar (Spring break).

Friday, March 22, 4:30 p.m. EDT Georgy Gaitsgori Some explicitly solvable problems of optimal stopping

We will discuss some explicitly solvable problems of optimal stopping, in particular, some problems considered by G. Peskir and his coathors. We will discuss papers "Optimal Stopping Inequalities for the Integral of Brownian Paths" by G. Peskir, "On Wald-Type Optimal Stopping for Brownian Motion" by S.E. Graversen and G. Peskir, and others.
Friday, March 29, 4:30 p.m. EDT Ioannis Karatzas Deterministic and Integral Representation Approaches to Optimal Stopping

The problem of optimal stopping is of fundamental importance in sequential analysis, and has found many applications in many other fields as well. We review two relatively new methodologies for it: one uses a deterministic approach, and identifies the Lagrange multiplier that enforces the non-anticipativity constraint; the second proceeds via an integral representation. Both approaches have found interesting applications, make contact with the classical Snell envelope, and shed light on it.
Friday, April 5, 4:30 p.m. EDT Richard Groenewald The American Put Optimal Stopping Problem with Stochastic Interest Rates

We will discuss the paper ``The American put with finite-time maturity and stochastic interest rate'' by Cheng Cai, Tiziano De Angelis and Jan Palczewski (2022). In particular, we will go through the technical details of the proof that the optimal stopping time associated with a given starting point for the problem is continuous as a function of this starting point. This fact may be used to show that the value function is continuously differentiable on the interior of its domain. If time permits, we will also discuss regularity properties of the boundary associated with this optimal stopping problem.
Friday, April 12, 4:30 p.m. EDT No seminar

Friday, April 19, 4:30 p.m. EDT Steven Campbell A Mean Field Game of Sequential Testing

In this talk we will introduce a mean field game for a family of optimal stopping problems related to the classic sequential testing of the drift of a Brownian motion. In our formulation, the players' stopping decisions can influence the information structure of the population. We will show that the game is well-posed, characterize the solution, and establish the existence of an equilibrium under certain assumptions. Illustrations of the solution behavior for several examples of interest will also be provided. To the best of our knowledge this work presents the first treatment of a mean field game with filtering, stopping and an unobserved common noise in the literature.
This talk is based on joint work with Yuchong Zhang: https://arxiv.org/pdf/2403.18297.pdf