Optimal Stopping Theory: Methods and Techniques

Fall 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 on Fridays at 4:30 pm, the place will be announced later.

This seminar is the logical continuation of the seminar held in Spring 2024 - Optimal Stopping Theory: Methods and Techniques - Spring 2024.

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, December 6, 3:50 p.m. EDT Georgy Gaitsgori Testing Uncertain Rewards under a Stochastic Deadline

We consider a variation of a classical sequential estimation problem for the drift of a Wiener process in the presence of a random horizon. In our setup, the observations are modeled by an arithmetic Brownian motion with an unknown and unobservable drift, which can take one of the two predetermined values. The observer's goal is to estimate this drift and terminate the observations to get a reward that is equal to the drift's value. The role of time penalization is played by a stochastic deadline, meaning that the observer gets the reward only if the termination happens before some random time gamma. We assume that this time gamma has a known distribution and is independent of the driving Brownian motion, but might depend on the value of the drift. We discuss solution to this problem under appropriate assumption. Based on a joint work with S. Campbell, R. Groenewald, and I. Karatzas.

Past Seminars

Date and time Speaker Title and abstract
Friday, September 13, 4:30 p.m. EDT Georgy Gaitsgori Solving non-linear optimal stopping problems by the method of time-change

We will discuss the paper ``Solving non-linear optimal stopping problems by the method of time-change'' by Pederson and Peskir. We will show how time-change method can be applied in OS problems first in general, and then on several examples, including optimal stopping of Brownian motion in the presence of square-root time cost, and (veterans must remember) a continuous version of the so-called optimal stopping problem of dishonest statistician introduced by Chow and Robbins and solved by Shepp.
Friday, September 20, 4:30 p.m. EDT No seminar

Friday, September 27, 4:30 p.m. EDT Richard Groenewald Parametric Continuity of SDE Solutions with Applications to Optimal Stopping Problems

We will discuss some parts of the paper "Volatility Time and Properties of Option Prices" by Svante Janson and Johan Tysk. In particular, we will discuss the proof of their representation of a family of SDE solutions (with zero drift) using a single Brownian motion evaluated at a family of stopping times. Some continuity properties of the value function associated with American-type options contracts will be covered. Lastly, we will outline potential extensions to the case of non-zero drift.
Friday, October 4, 4:30 p.m. EDT Steven Campbell Parametric Continuity in Problems of Optimal Stopping

We use the dual perspective for optimal stopping, introduced by Davis and Karatzas, to derive pathwise conditions for commonly sought properties of optimal stopping problems. In particular, we show that the continuity of the value function, the continuity of the optimal stopping time, the continuity of the spatial derivative of the value function, and the existence of equilibria in a large class of stopping games can be derived from the path properties of the reward process and its Snell envelope. We provide several illustrative examples and applications of these results. This talk is based on ongoing joint work with G. Gaitsgori, R. Groenewald and I. Karatzas.
Friday, October 11, 4:30 p.m. EDT Georgy Gaitsgori Doob-Meyer decomposition and other fundamentals

Continuity of several stochastic processes is essential for problems of optimal stopping. Moreover, these continuities are closely related to some fundamental notions and results in probability. Therefore, this week we are going back to the origins to discuss Doob-Meyer decomposition and some other useful results. In particular, we will recall two proofs of Doob-Meyer decomposition via Dunford-Pettis' theorem and also Komlos' theorem.
Friday, October 18, 4:30 p.m. EDT Georgy Gaitsgori Doob-Meyer decomposition: Part 2

We continue our discussion from the last time. In particular, we give detailed proofs of the Doob-Meyer decomposition (DMD) via Hilbertian Komlos lemma, and give detailed proof of the result that the compensator in the DMD is contunuous if the process X is regular.
Friday, October 25, 4:30 p.m. EDT Graeme Baker A Resource Sharing Model with Local Time Interactions

We consider a resource sharing model for N financial firms which takes the form of a semimartingale reflected Brownian motion in the positive orthant. When the capital levels of any firm hits zero, the other firms contribute to a local time reflection term to keep the distressed firm's capital non-negative. Depending on a critical parameter alpha, which reflects either friction or subsidy, the interval of existence for this scheme is shown to be either infinite, or almost surely finite. In the mean-field limit, the behaviour of the Fokker--Planck equation also depends on alpha: either solutions exist for all time or the equation exhibits finite-time blowup. The passage from finite to mean-field model is investigated. We also analyze connections between our model and systemic risk models involving hitting times, the up-the-river problem of Aldous, various free boundary PDEs, and Atlas models from Stochastic Portfolio Theory.
Friday, November 1, 4:30 p.m. EDT Richard Groenewald More Results about Parametric Continuity of SDE Solutions

We will state and prove some continuity results for one dimensional SDEs that depend on a parameter, which may take values in an arbitrary topological space, under suitable assumptions on the drift and diffusion coefficients. This is part of ongoing joint work with Steven Campbell, Georgy Gaitsgori, and Ioannis Karatzas. It is our hope that some of the assumptions may be relaxed, and that we can extend to further cases. Some applications to optimal stopping problems will be briefly addressed.
Friday, November 8, 4:30 p.m. EDT No seminar

Friday, November 15, 4:30 p.m. EDT Manuel Arnese Quantitative Propagation of Chaos for conditional Mckean Vlasov equations

Motivated by the theory of Wasserstein gradient flows, we study the rate of propagation of chaos for a McKean-Vlasov equation with conditional expectation terms in the drift. We use a (regularized) Nadaraya-Watson estimator at a particle level to approximate the conditional expectations. The main tools are information theoretic inequalities and a strong result on propagation of chaos in total variation due to Jabir. The non-parametric nature of the problem requires us to obtain higher regularity for the density of the McKean Vlasov limit; we do so with a bootstrap argument and energy estimates.
Friday, November 22, 4:30 p.m. EDT No seminar (NESP conference)

Friday, November 29, 4:30 p.m. EDT No seminar (Thanksgiving)