## One World Optimal Stopping and Related Topics

## Online seminars

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.

**Organizers:**

**Organizers:**

Tiziano De Angelis (University of Turin)

Roxana Dumitrescu (King's College)

Yerkin Kitapbayev (North Carolina State University)

Mikhail Zhitlukhin (Steklov Mathematical Institute)

**Schedule for Spring 202****2**

**Schedule for Spring 202**

**2**

### January 18, 2022, 5 pm London time

**Ioannis Karatzas****, **Columbia University

**Ioannis Karatzas**

**,**Columbia University

Title: *Sequential estimation with control and discretionary stopping*

Abstract: We show that “full-bang” control is optimal in a problem that combines features of (i) sequential least-squares estimation with Bayesian updating, for a random quantity observed in white noise; (ii) bounded control of the rate at which observations are received, with a superquadratic cost per unit time; and (iii) “fast” discretionary stopping. We describe also the optimal filtering and stopping rules in this context. [Joint work with Erik Ekstrom and Juozas Vaicenavicius.]

### February 1, 2022, 5 pm London time

**Bruno Bouchard**, Paris Dauphine University

**Bruno Bouchard**, Paris Dauphine University

Title: *Itô-Dupire formula for $C^1$-functionals and approximate viscosity solutions of PPDE *

Abstract: We review some recent result on the Itô’s formula for path-dependent functionals that are either only $C^1$, or, concave in space and non-increasing in time. This leads to the study of the regularity of candidate solutions to path-dependent parabolic PDEs for which we introduce a notion of approximate viscosity solutions. Applications to perfect hedging in markets with price impact and to super-hedging under model uncertainty will be discussed.

### February 15, 2022, 5 pm London time

**Dylan Possamai, **ETH Zurich

**Dylan Possamai,**ETH Zurich

Title: *Is there a Golden Parachute in Sannikov’s principal-agent problem?*

Abstract: This paper provides a complete review of the continuous-time optimal contracting problem introduced by Sannikov (2008) in the extended context allowing for possible different discount factors of both parties. The agent’s problem is to seek for optimal effort, given the compensation scheme proposed by the principal over a random horizon. Then, given the optimal agent’s response, the principal determines the best compensation scheme in terms of running payment, retirement, and lump-sum payment at retirement. A Golden parachute is a situation where the agent ceases any efforts at some positive stopping time, and receives a payment afterwards, possibly consisting of a lump sum and/or a continuous stream of payments. We show that a Golden Parachute only exists in certain specific circumstances. This is in contrast with the results claimed by Sannikov (2008) where the only requirement is a positive agent’s marginal cost of effort at zero. Namely, we show that there is no Golden Parachute if this parameter is too large. Similarly, in the context of a concave marginal utility, there is no Golden Parachute if the agent’s utility function has a too negative curvature at zero. In the general case, we provide a rigorous analysis of this problem and we prove that an agent with positive reservation utility is either never retired by the principal, or retired above some given threshold (as in Sannikov (2008)’s solution). In particular, different discount factors induce naturally a face-lifted utility function, which allows to reduce the whole analysis to the equal-discount factors setting. Finally, we also confirm that an agent with small reservation utility does have an informational rent, meaning that the principal optimally offers him a contract with strictly higher utility value.

### March 1, 2022, 11 am London time

**Anna Aksamit****, **University of Sydney

**Anna Aksamit**

**,**University of Sydney

Title:* **Information modelling: new type of filtration enlargement and applications*

Abstract: In this talk I will review the classical results about enlargement of filtration and present some new development with applications. I will focus on conditions under which martingales in the reference filtration remain semimartingales in the large filtration, in which case, the canonical decomposition is of particular interest. I will then present enlargement of a reference filtration through the observation of a random time and a mark. Random time considered is such that its graph is included in the countable union of graphs of stopping times. Mark revealed at this random time is assumed to satisfied generalised Jacod’s condition. Our relaxation of Jacod’s condition accounts for the dynamic structure of the problem.

### March 29, 2022, 5 pm London time

**Patrick Cheridito, **ETH Zurich

**Patrick Cheridito,**ETH Zurich

Title: *Optimal stopping with neural networks*

Abstract: Two different approaches to solving optimal stopping problems with neural networks are presented. Both are broadly applicable in situations where the underlying randomness can efficiently be simulated and yield good results in high dimensions. As an applications the pricing and hedging of American-style options is discussed.

### April 26, 2022, 5 pm London time

**Jan Palczewski, **University of Leeds

**Jan Palczewski,**University of Leeds

Title: *Non-Markovian Dynkin games with partial and asymmetric information*

Abstract: In the talk I will show that a zero-sum Dynkin game in continuous time with partial and asymmetric information admits a saddle point (and, consequently, a value) in randomised stopping times when stopping payoffs of players are general càdlàg adapted processes. We do not assume a Markovian nature of the game nor a particular structure of the information available to the players. As our arguments are topological, I will start from a discussion of related classical results by Baxter, Chacon (1977) and Meyer (1978) for optimal stopping problems. I will also discuss examples demonstrating the necessity of randomisation of players' strategies.

### May 10, 2022, 2 pm London time

**Vicky Henderson, **University of Warwick

**Vicky Henderson,**University of Warwick

Title: *Regret in Trading Decisions: A Model and Empirical Study*

Abstract: In this talk we will present a simple model of dynamic regret. We will use this to motivate our empirical study of trading decisions using a large discount brokerage dataset. Our focus is to test how regret induced by not selling a stock at its maximum price shapes the propensity to sell.

We undertake a number of descriptive analyses and more formal analysis via proportional hazard modelling.

### May 24, 2022, 5 pm London time

**Svetlana Boyarchenko****, **The University of Texas at Austin

**Svetlana Boyarchenko**

**,**The University of Texas at Austin

Title: *Optimal stopping for Stieltjes-Levy processes*

### June 8, 2022, 5 pm London time

**Ernesto Mordecki,**** **Universidad de la República, Uruguay

**Ernesto Mordecki,**

**Universidad de la República, Uruguay**

Title: *An algorithm to solve optimal stopping problems for** **one-dimensional diffusions*

Abstract: Considering a real-valued diffusion, a real-valued reward function and a positive discount rate r>0, we provide an algorithm to solve the optimal stopping problem consisting in finding the optimal expected discounted reward and the optimal stopping time at which it is attained. Our approach is based on Dynkin's characterization of the value function. The combination of Riesz's representation of r-excessive functions and the inversion formula gives the density of the representing measure, being only necessary to determine its support. This last task is accomplished through an algorithm. The proposed method always arrives to the solution, thus no verification is needed, giving, in particular, the shape of the stopping region. Generalizations to diffusions with atoms in the speed measure and to non-smooth payoffs are analyzed. Examples with non-monotonous payoffs for Brownian motion, and for a diffusion with discontinuous drift is examined.