Welcome to the 'Optimal Stopping Theory: Methods and Techniques' Reading Seminar, run by the students and postdocs of Columbia University.
The seminar meets in-person in Columbia University on Fridays at 4:00 pm in Math room 520.
This seminar is the logical continuation of the seminar held in Fall 2024 - Optimal Stopping Theory: Methods and Techniques - Fall 2024.
If you would like to come or to be added on the mailing list, please email gg2793@columbia.edu.
| Date and time | Speaker | Title and abstract |
|---|---|---|
| Friday, March 28, 4:00 p.m. EDT | No seminar (Columbia Stats and OT Workshop) |
|
| Date and time | Speaker | Title and abstract |
|---|---|---|
| Friday, April 11, 4:00 p.m. EDT | Karl Kristian Engelund | TBA
TBA |
| TBA | Ioannis Karatzas (Columbia University) | TBA
TBA |
| Date and time | Speaker | Title and abstract |
|---|---|---|
| Friday, January 24, 4:30 p.m. EDT | Tiziano De Angelis (University of Torino) | A probabilistic view on free boundary problems
I will give an overview of some results concerning optimal stopping that can be obtained by purely probabilistic methods. In particular I will consider finite-time horizon problems and illustrate peculiar properties of the optimal stopping boundary, arising from low regularity of the gain function. If time allows, I will also illustrate some recent results on the continuous differentiability of the optimal stopping boundary and the link between optimal stopping and Stefan problem. |
| Friday, January 31, 4:00 p.m. EDT | Georgy Gaitsgori | Convexity of the optimal stopping boundary for the American put option
We briefly discuss some history of the theory of pricing American options (based on "The Pricing of the American Option" by Ravi Myneni) and then prove that the optimal stopping boundary for American put is a convex function (based on "Convexity of the optimal stopping boundary for the American put option" by Erik Ekström). |
| Friday, February 7, 4:00 p.m. EDT | Steven Campbell (Columbia University) | Optimal Execution among N Traders with Transient Price Impact
We study N-player optimal execution games in an Obizhaeva--Wang model of transient price impact. When the game is regularized by an instantaneous cost on the trading rate, a unique equilibrium exists and we derive its closed form. Whereas without regularization, there is no equilibrium. We prove that existence is restored if (and only if) a very particular, time-dependent cost on block trades is added to the model. In that case, the equilibrium is particularly tractable. We show that this equilibrium is the limit of the regularized equilibria as the instantaneous cost parameter ε tends to zero. Moreover, we explain the seemingly ad-hoc block cost as the limit of the equilibrium instantaneous costs. Notably, in contrast to the single-player problem, the optimal instantaneous costs do not vanish in the limit ε \to 0. We use this tractable equilibrium to study the cost of liquidating in the presence of predators and the cost of anarchy. Our results also give a new interpretation to the erratic behaviors previously observed in discrete-time trading games with transient price impact. |
| Friday, February 14, 4:00 p.m. EDT | No seminar. |
|
| Friday, February 21, 4:00 p.m. EDT | Renyuan Xu (NYU) | Exploratory Optimal Stopping: A Singular Control Formulation
We explore continuous-time and state-space optimal stopping problems from a reinforcement learning perspective. We begin by formulating the stopping problem using randomized stopping times, where the decision maker's control is represented by the probability of stopping within a given time--specifically, a bounded, non-decreasing, càdlàg control process. To encourage exploration and facilitate learning, we introduce a regularized version of the problem by penalizing it with the cumulative residual entropy of the randomized stopping time. The regularized problem takes the form of an (n+1)-dimensional degenerate singular stochastic control with finite-fuel. We address this through the dynamic programming principle, which enables us to identify the unique optimal exploratory strategy. For the specific case of a real option problem, we derive a semi-explicit solution to the regularized problem, allowing us to assess the impact of entropy regularization and analyze the vanishing entropy limit. Finally, we propose a reinforcement learning algorithm based on policy iteration. We show both policy improvement and policy convergence results for our proposed algorithms. |
| Friday, February 28, 4:00 p.m. EDT | Giorgio Ferrari (Bielefeld University) | On Mean-field Games with Singular Controls and Regime Switching
In this talk, I present results regarding the existence, uniqueness, and characterization of equilibria for stationary mean-field games with singular controls and regime switching. This class of problems has natural applications in Economics and Finance, such as investment problems in oligopolies. In such games, the representative agent employs a bounded-variation control to maximize an ergodic profit functional depending on the long-time distribution of the optimally controlled state process, which evolves as a one-dimensional Itô-diffusion process with Markov-modulated drift and volatility coefficients. The constructed mean-field equilibrium is then shown to provide an εN-Nash equilibrium for a suitable class of N-player ergodic games of singular controls with regime switching. Finally, some ongoing projects related to stationary mean-field games of singular controls will be introduced. |
| Friday, March 7, 4:00 p.m. EDT | Richard Groenewald | Some Problems of Optimal Stopping
We will discuss three problems in the field of optimal stopping. In the first, we study a class of optimal stopping problems which depend on a parameter taking values in an arbitrary topological space. Using the dual perspective for optimal stopping introduced by Davis and Karatzas (1994), we prove that the optimal stopping times of these problems are continuous with respect to their parameter, and discuss several applications to the smooth-fit principle and the existence of Nash equilibria in a large class of stopping games. In the second, we study a variant of the classical sequential estimation problem of the drift of an arithmetic Brownian motion. In our framework, the observer's objective is to estimate the drift and terminate the observations to maximize a reward equal to the drift's value, should they choose to accept it, subject to a random time horizon. In the third, we characterize the Nash equilibria of a two-player, nonzero-sum Dynkin game of stopping with incomplete information. The players observe independent Brownian motions which are not observable by the other player. The player who stops first receives a payoff that depends on the stopping position. This talk will serve as a rehearsal for the defense of my thesis. |
| Friday, March 14, 4:00 p.m. EDT | Rama Cont (University of Oxford) | Causal transport on path space
We study properties of causal couplings for probability measures on the space of continuous functions. We show that when these measures correspond to the law of the solution of a stochastic differential equation, causal transports between such measure inherit a differential structure which we characterize in detail. We first provide a characterization of bicausal couplings between weak solutions of stochastic differential equations. In particular, we show that bicausal Monge couplings of d-dimensional Wiener measures are induced by stochastic integrals of rotation-valued integrands. As an application, we give necessary and sufficient conditions for bicausal couplings to be induced by Monge maps and show that such bicausal Monge transports are dense in the set of bicausal couplings between laws of SDEs with regular coefficients. Joint work wth Fang Rui LIM (Oxford) |
| Friday, March 21, 4:00 p.m. EDT | No seminar (Spring break) |
|