FRE Special Seminar: Giorgio Ferrari & Emma Hubert

Giorgio Ferrari

Full Professor for Mathematical Finance at the Center for Mathematical Economics (IMW) at Bielefeld University

Title

Optimal Policy Characterization for a Class of Multi-Dimensional Ergodic Singular Stochastic Control Problems

Abstract

In ergodic singular stochastic control problems, a decision maker can instantaneously adjust the evolution of a state variable via a control of bounded variation, aiming to minimize a certain long-time average cost functional, where the cost of control is proportional to the effort. The constant optimal ergodic cost is typically part of the solution to a dynamic programming equation (DPE) involving an auxiliary function called the potential function. While the DPE is well understood through partial differential equation (PDE) theory in general settings, little is known about the characterization of the optimal control policy in multi-dimensional problems. In this talk, I will introduce a new methodology to address this characterization issue for a class of multi-dimensional ergodic singular stochastic control problems. In these problems, the coefficients of a linearly controlled one-dimensional stochastic differential equation and the cost functional to be optimized involve a multi-dimensional uncontrolled process Y. The optimal control is characterized in terms of a Skorokhod reflection at Y-dependent free boundaries, which arise from the study of an auxiliary Dynkin game. Applications to optimal inventory models will be presented.

Bio

Giorgio Ferrari is full professor for Mathematical Finance at the Center for Mathematical Economics (IMW) at Bielefeld University, where he is also Principal Investigator in the "Collaborative Research Center 1283" and Director of the "Bielefeld Graduate School in Theoretical Sciences". He obtained a Ph.D. in Mathematics for Economic-Financial Applications at the University of Rome "La Sapienza" in 2012. He then moved to IMW where he was first post-doctoral researcher (2012-2016), and then Junior-Professor (2016-2017) and Associate Professor (2017-2023). His research interests lie in the field of stochastic control theory and its applications to Economics and Finance. Particular attention is devoted to dynamic stochastic optimization problems and games involving singular controls and stopping rules, and to the analysis of the corresponding free-boundary problems.

Emma Hubert

Tenure-track Assistant Professor in the ORFE Department at Princeton University

Title

A New Approach to Principal-Agent Problems with Volatility Control

Abstract

The recent work by Cvitanic, Possamai, and Touzi (2018) [1] presents a general approach for continuous-time principal-agent problems, through dynamic programming and so-called second-order backward stochastic differential equations (2BSDEs). In this talk, we provide an alternative formulation of the principal-agent problem, which can be solved using more straightforward techniques, simply relying on the theory of BSDEs. This reformulation is strongly inspired by an important remark in [1], namely that if the principal observes the output process X in continuous-time, she can compute its quadratic variation pathwise. While in [1] this information is used in the contract, our reformulation consists in assuming that the principal could directly control this process, in a ‘first-best’ fashion. The resolution approach for this alternative problem actually follows the line of the so-called ‘Sannikov’s trick’ in the literature on continuous-time principal-agent problems, as originally introduced by Sannikov (2008) [2]. We then show that the solution to this ‘first-best’ formulation is identical to the solution of the original problem. More precisely, using the contract’s form introduced in [1] as penalisation contracts, we highlight that this ‘first-best’ scenario can be achieved even if the principal cannot directly control the quadratic variation. Nevertheless, we do not have to rely on the theory of 2BSDEs to prove that such contracts are optimal, as their optimality is ensured by showing that the ‘first-best’ scenario is achieved. We believe that this more straightforward approach to solve general continuous-time principal-agent problems with volatility control will facilitate the dissemination of these problems across many fields, and its extension to more intricate principal-agent problems. In particular, we will conclude this talk with some preliminary results on the extension to multi-agent frameworks. Joint work with Alessandro Chiusolo.

Bio

Emma Hubert is a tenure-track Assistant Professor in the ORFE Department at Princeton University. Her research, partially supported by the NSF grant DMS-2307736, focuses on stochastic control and games, particularly Principal-agent problems and mean-field games, with applications to economics, finance, and energy systems. She currently teaches the undergraduate course ORF 418: Optimal Learning in the Fall and the graduate course ORF 527: Stochastic Calculus in the Spring. She also serves as an Associate Editor for the IEEE Transactions on Network Science and Engineering (TNSE). Before joining Princeton in 2021, she was a Research Associate in the Department of Mathematics & CFM - Imperial Institute of Quantitative Finance at Imperial College London. She defended her PhD in Mathematics at Université Paris-Est in December 2020, for which her thesis was awarded the Prix de thèse SMAI-GAMNI, the Prix de thèse Paris-Est Sup, and the Prix Paul Caseau.