Optimal Stopping Reading Seminar

Fall 2021

Welcome to the Optimal Stopping Reading Seminar, run by the students of Columbia University.

This Fall we will continue studying Optimal Stopping Theory. We are going to read various books and papers, in particular Optimal Stopping and Free-Boundary Problems by Peskir and Shiryaev. Our talks will be held in hybrid form over Zoom and in Columbia University on Thursdays from 6p.m. to 7p.m. EDT.

This seminar is the continuation of the same seminar held in Summer - OS Seminar, Summer 2021.

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

Past Seminars

Date and time Speaker Title and abstract
Thursday, September 30, 6:00p.m. EDT Everyone Organizational Meeting

Thursday, October 7, 6:00p.m. EDT Georgy Gaitsgori Recalling the summer seminars

Since we have already started the book, but our last seminar was two months ago, I have decided to dedicate our first seminar to recalling the materials we covered before. Thus today we will recall the last three seminars, using the presentations by Hindy, Lane, and myself. We will discuss what is an optimal stopping problem and how it can be related to free-boundary problems. Then we will discuss a few examples of such free-boundary problems and the issue of the uniqueness of their solution.
Thursday, October 14, 6:00p.m. EDT Ioannis Karatzas A class of stochastic control problems, and their relation to optimal stopping

We present a class of so-called "singular" stochastic control problems, which go a long way back to Bather and Chernoff in the 1960's; then explain their rather unexpected connections to optimal stopping.
Thursday, October 21, 6:00p.m. EDT Sid Mane Masters of Time and Space

An introduction to probabilistic and analytic transformations of time and space as they arise in optimal stopping (following sections 4.10-11 of Peskir and Shiryaev).
Thursday, October 28, 6:00p.m. EDT Georgy Gaitsgori Extracting a convergent subsequence, Komlos Theorem and its applications

Sometimes in optimal stopping or optimization problems it is useful to extract a convergent subsequence of some given sequence. In this context, we will discuss three important theorems: Dunford-Pettis Theorem, "baby" Komlos Theorem and general Komlos Theorem (the latest ones were used in particular in the paper "Generalised Lyapunov Functions and Functionally Generated Trading Strategies" by Ruf and Xie, and in "Connections between bounded-variation control and Dynkin games" by Karatzas and Wang). We will show how these theorems can be applied to prove famous Doob-Meyer decomposition and probably sometimes in the future how they are applied in the aforementioned papers.
Thursday, November 04, 6:00p.m. EDT Shalin Parekh Examples of applying the time change method

Continuation of section IV.10 from Peskir-Shiryaev.
Thursday, November 11, 6:00p.m. EDT Georgy Gaitsgori On Hilbertian Komlos Lemma and Komlos Theorem

We continue to discuss Komlos Theorem. We will try to prove Hilbertian Komlos Lemma, Komlos Theorem and if time permits discuss the short proof of Doob-Meyer decomposition via these tools. We will also discuss some developments of the topic.
Thursday, November 18, 6:00p.m. EDT Ethan Woojin Che Optimal stopping and Bayesian sequential testing

We discuss applications of optimal stopping for Bayesian sequential testing of the drift of a Brownian motion, covered in Chapter 6 of Peskir-Shiryaev. We will discuss some motivation for this problem and derive explicit solutions in the case of a linear time cost of sampling and in the case where the experimenter has a discounted utility function. If time permits, we will also discuss the problem of detecting a change in the drift of a Browian motion as quickly as possible.
Thursday, November 25, 6:00p.m. EDT No seminar

Thursday, December 2, 6:00p.m. EDT Georgy Gaitsgori On Chow-Robbins Game, Part 1: history, current state and main ideas

This is the first part of a two-part talk about a famous Chow-Robbins game. Suppose you toss a fair coin, and you want to stop in such a way so that you maximize (#heads - #tails)/#tosses. This game was introduced by Chow and Robbins in 1965, and despite having a very transparent formulation, it is still not completely solved. However, the existence of a stopping time, some asymptotic properties, and generalizations of the game were established. Today we will discuss the history of this problem, the current "state of the art" and some generalizations. Then we will also discuss the main ideas behind proofs of some of these results. Next time we will discuss proofs and results in more detail.
Thursday, December 9, 6:00p.m. EDT Georgy Gaitsgori On Chow-Robbins Game, Part 2: Properties of the stopping region

Last time we discussed the history of the Chow-Robbins game and the main ideas behind the proofs of the existence of the optimal stopping time. This time we will discuss some properties of the stopping region. As we have already mentioned, there is no explicit formula for the stopping region (or optimal stopping time), but the asymptotic behavior of the stopping boundary and the explicit stopping region for small starting positions are known. We will discuss the main results and ideas of the proofs in this direction.