This is an international online seminar where recent work on optimal stopping theory and related topics is presented. The seminar is a part of One World Seminars that were inspired by One World Probability project. The previous talks can be found here.
The seminar series is supported by the De Castro Statistics Initiative at Collegio Carlo Alberto.
Organizers:
Tiziano De Angelis (University of Turin)
Roxana Dumitrescu (King's College)
Yerkin Kitapbayev (Khalifa University)
Mikhail Zhitlukhin (Steklov Mathematical Institute)
November 29, 2022, 5 pm London time
Giorgio Ferrari (Bielefeld University)
Title: A Stationary Mean-Field Equilibrium Model of Irreversible Investment in a Two-State Economy
Abstract: In this talk, I present results on a stationary mean-field model with singular controls for a Markov modulated Itô-diffusion, in which the representative agent interacts with a long-time conditional weighted average of the population through a discounted performance criterion. Natural applications are in the context of irreversible production expansion in dynamic oligopolies, where the dynamics of the production capacity is affected by the market's business cycles and the price of the produced good depends on the aggregate stationary production of the whole economy. We prove existence and uniqueness of the mean-field stationary equilibrium and we characterize it through a system of nonlinear equations. Along the way, explicit results for the joint stationary distribution of the controlled production capacity and the Markov chain at equilibrium are also derived. A numerical analysis allows to understand the dependency of the mean-field equilibrium with respect to the model's parameters. [This is based on a joint (ongoing) work with René Aid and Matteo Basei.]
December 13, 2022, 5 pm London time
Said Hamadene (Universite’ du Maine, LMM)
Title: On the existence of Nash Equilibria in a Multi-player Nonzero-sum Dynkin Game in Discrete Time
Abstract: In this talk we will discuss the problem of existence of Nash equilibria or $\eps$-Nash equilibria in the infinite horizon discrete time $N$-player nonzero-sum Dynkin game ($N\geq 2$) with stopping times as strategies (or pure strategies). The payoffs depend on the coalition of players that make the decision to stop the game. Under appropriate conditions we prove existence of a Nash equilibrium or $\eps$-Nash equilibrium point for the game by presenting a constructive algorithm. The existence result is extended to the case of a nonzero-sum game with finite horizon.
January 10, 2023, 5 pm London time
Andreea Minca (Cornell University)
Title: Leveraging-Deleveraging Games
Abstract: We introduce a new mechanism for leverage dynamics, based on a multiperiod game of lenders with differentiated beliefs about the firm’s fundamental returns. The game features strategic substitutability for low existing leverage and strategic complementarity for high existing leverage. The resulting leverage process exhibits a mean-reverting regime around a long-run level, as long as it stays below an instability level. Above the instability level, leverage becomes explosive. We validate our model empirically using aggregate returns of financial firms over the 10-year period 2001–2010. Our model is consistent with the leveraging/deleveraging of this period and with the 2008 collapse in short-term debt. [Joint with J. Wiissel.]
February 14, 2022, 5 pm London time
A. Max Reppen (Boston University)
Title: Neural Optimal Stopping Boundary
Abstract: A method based on deep artificial neural networks and empirical risk minimization is developed to calculate the boundary separating the stopping and continuation regions in optimal stopping. The algorithm parameterizes the stopping boundary as the graph of a function and introduces relaxed stopping rules based on fuzzy boundaries to facilitate efficient optimization. Several financial instruments, some in high dimensions, are analyzed through this method, demonstrating its effectiveness. The existence of the stopping boundary is also proved under natural structural assumptions.
March 21, 2023, 5 pm London time
Katia Colaneri (University of Rome - Tor Vergata)
Title: On the impact of tax uncertainty on investment into carbon abatement technologies
Abstract: In this paper we study the problem of a profit maximizing electricity producer who decides on investments in technologies for abatement of CO_2 emissions based on the taxation policy which is in force. We compare two scenarios: in the first scenario, the taxation policy is deterministic; in the second scenario we allow for exogenous deviations from the deterministic setting, which arrive at exponential times, and may either increase or decrease the taxes. We show that in certain scenarios the uncertainty on the future taxation makes the company less willing to make investment and hence a clear and a priori fixed strategy would instead maximise the actions for emission reduction. [This presentation is based on a joint work with Ruediger Frey and Verena Koeck.]
April 11, 2022, 5 pm London time
Diogo Gomes (King Abdullah University of Science and Technology (KAUST))
Title: Machine Learning architectures for price formation models with common noise
Abstract: We propose a machine learning method to solve a mean-field game price formation model with common noise. This involves determining the price of a commodity traded among rational agents subject to a market clearing condition imposed by random supply, which presents additional challenges compared to the deterministic counterpart. Our approach uses a dual recurrent neural network architecture encoding noise dependence and a particle approximation of the mean-field model with a single loss function optimized by adversarial training. We provide a posteriori estimates for convergence and illustrate our method through numerical experiments
May 16, 2023, 5 pm London time
Gechun Liang (University of Warwick)
Zoom link: https://us02web.zoom.us/meeting/register/tZclduGvrDkuGtAQJ6Fo5jsvzucUeGF-MYme
Title: Recursive Optimal Stopping with Poisson Stopping Constraints
Abstract: We focus on the representation of the solution for a recursive optimal stopping problem with Poisson stopping constraints using the penalized backward stochastic differential equation (PBSDE). The stopping in this problem is only allowed at Poisson random intervention times, which introduce jumps into the recursive utility and affect the model coefficients, making them dependent on not only a Brownian motion but also the associated Poisson process. To solve the problem, we propose a decomposition method that allows us to separate the problem into sub-problems between each two adjacent Poisson arrival times. The comparison theorem of BSDEs with jumps is essential in representing the value function of the optimal stopping problem when the initial time lies between two adjacent Poisson arrival times. We then apply this representation result to three types of recursive optimal control problems: (1) optimal intensity control problem, (2) optimal mixed control problem with distance penalization and Poisson stopping constraints, and (3) optimal switching problem. We provide the corresponding penalized equation representations for their solutions. [Based on joint work with Zhen Wu and Zhenda Xu from Shandong University.]