# Abstract

**12.10.2018** Teemu Pennanen (King's College London)

**Title: **Convex duality in nonlinear optimal transport

**Abstract: **This paper provides a unified treatment of a large class of related problems in probability theory and allows for generalizations of the classical problem formulations. General results on convex duality yield dual problems and optimality conditions for these problems. When the objective takes the form of a convex integral functional, we obtain more explicit optimality conditions and establish the existence of solutions for a relaxed formulation of the problem. This covers, in particular, the mass transportation problem and its nonlinear generalizations.

**19.10.2018** Tiziano De Angelis (University of Leeds)

**Title: **Dynkin games with incomplete and asymmetric information

**Abstract: **We study Nash equilibria for a two-player zero-sum optimal stopping game with incomplete and asymmetric information. In our set-up, the drift of the underlying diffusion process is unknown to one player (incomplete information feature), but known to the other one (asymmetric information feature). We formulate the problem and reduce it to a fully Markovian setup where the uninformed player optimises over stopping times and the informed one uses randomised stopping times (in order to hide their informational advantage).Then we provide a general verification result which allows us to find Nash equilibria by solving suitable quasi-variational inequalities with some non-standard constraints. Finally, we solve explicitly an example with linear payoffs with applications to short selling of stocks. (joint work with Erik Ekstrom (Uppsala) and Kristoffer Glover (Sidney))

**23.11.2018** Alexander Cox (University of Bath)

**Title:** Utility Maximisation with Model-Independent Trading Constraints

**Abstract:** In this work we consider the classical utility maximisation problem for a trader who is constrained by a model-independent portfolio constraint. Specifically, the trader aims to maximise her utility subject to the constraint that her portfolio value is bounded below when any derivative contracts are valued at their intrinsic value. Here, the intrinsic value is taken to be the model-independent super/sub-hedging price of the derivative. Using ideas of El Karoui and Meziou (2006), we are able to find explicit strategies for the trader. (Joint work with Daniel Hernandez-Hernandez).

**30.11.2018** Sigrid Källblad (Vienna University of Technology)

**Title: **Stochastic control of measure-valued martingales with applications to robust pricing and Skorokhod embedding problems

**Abstract:** We consider a stochastic control problem where the controlled processes are (probability) measure-valued martingales (MVMs). We will discuss a notion of (weakly) controlled MVMs and in particular how to obtain so-called terminating MVMs. We will then establish well-posedness of our problem, show that the dynamic programming principle holds and that the solution is a viscosity solution in a certain sense to an emerging HJB equation. A key motivation for the study of control problems featuring MVMs is that a number of interesting probabilistic problems can be formulated as such. In particular, this applies to the financially motivated problem of finding model-independent price bounds on options when call prices at the maturity of the option are known. It is also the case for the classical optimal Skorokhod embedding problem; we illustrate this by considering some key examples. The talk is based on joint work with A. Cox, M. Larsson and S. Svaluto.

**18.1.2019** Alex Mijatovic (University of Warwick)

Title: Stability of overshoots of zero mean random walks

Abstract: Take a one-dimensional random walk with zero mean increments, and consider the sizes of its overshoots over the zero level. It turns out that this sequence, which forms a Markov chain, always has a stationary distribution of a simple explicit form. We prove that this stationary distribution is unique using methods of infinite ergodic theory. In terms of stability, we were able to prove only the total variation convergence, which holds for lattice random walks and for the ones whose distribution, essentially, has density. We also obtained the rate of this convergence under additional mild assumptions. We will also discuss connections to related topics: local times of random walks, ergodic theory and renewal theory. This is joint work with Vlad Vysotsky.

**25.01.2019 **Michael Tehranchi (University of Cambridge)

Title: A Black--Scholes inequality: Applications and generalisations

Abstract: A curious inequality involving the Black--Scholes pricing function is explored. One way to understand the inequality is via a natural noncommutative semigroup structure of the space of space of call price functions. The binary operation is compatible with the convex order, and therefore a one-parameter sub-semigroup gives rise to an arbitrage-free market model. It is shown that each such one-parameter semigroup corresponds to a unique log-concave probability density, providing a family of tractable call price surface parametrisations in the spirit of the Gatheral--Jacquier SVI surface. An explicit example is given to illustrate the idea.

**01.02.2019 **Neofytos Rodosthenous (Queen Mary University of London)

Title: Optimal timing for governmental control of the debt-to-GDP ratio

Abstract: We study the problem of a government wishing to control the country's debt-to-GDP ratio. The debt-to-GDP ratio evolves stochastically and the interest on debt is affected by an N-state continuous-time Markov chain, representing the country's credit ratings. The debt-to-GDP ratio can be reduced through fiscal interventions or increased by public investments. The government aims to choose a policy minimising the total expected cost of having debt and fiscal interventions counterbalanced by the gain from public investments. The problem is modelled by a bounded-variation stochastic control problem, that we explicitly solve through the analysis of an associated Dynkin game. This is joint work with Giorgio Ferrari.

**15.02.2019 **Matija Vidmar (University of Ljubljana)

Title: On a family of non-linear optimal martingale transport problems

Abstract: We investigate the structure of the solution to a class of "non-linear" one-step one-dimensional optimal martingale transport problems. The mentioned non-linearity is in the objective functional, and it comes from an application of a (non-linear) function to a conditional expectation, before the outer unconditional expectation (all

w.r.t. a martingale coupling) is finally taken. The class of problems is motivated by one of its particular cases that has to do with a robust (model-independent) pricing of a futures on the S&P 500 VIX volatility index (Guyon et al., Finance Stoch (2017) 21(3):593). This is joint work (in progress) with Alex Cox.

**22.02.2019 **Mihail Zervos (LSE)

Title: Risksharing with two-sided limited commitment: a duality approach in continuous time

Abstract: We characterise efficient risksharing under two-sided limited commitment in a continuous-time endowment economy. The dual formulation involves the Lagrange multipliers of the participation constraints and gives rise to a singular stochastic control problem. We prove that strong duality holds in a general setting. As an application, we consider a symmetric environment where endowment shares are driven by a mean-reverting diffusion process and aggregate uncertainty may be uncertain. The relevant co-state acts as a time-varying Pareto weight that determines the consumption allocation. We analyse the HJB equation associated with this problem and solve for the free-boundaries that delineate regions of the state space in which participation constraints become binding.

**01.03.2019 **Athena Picarelli (University of Verona)

Title: Optimal control under controlled loss constraints via reachability approach and compactification

Abstract: We study optimal control problems under controlled loss constraints at several fixed dates. It is well known that for such problems the characterization of the value function by a Hamilton-Jacobi-Bellman equation requires additional strong assumptions involving an interplay between the set of constraints and the dynamics of the controlled system. To treat the problem in absence of these assumptions we first translate it into a state-constrained stochastic target problem and then apply a level-set approach to describe the reachable set. The main advantage of our approach is that it allows us to easily handle the state constraints by an exact penalization. However, this target problem involves a new set of control variables that are unbounded. A ''compactification'' of the problem is then performed.

**05.04.2019 **Daniel Bartl (University of Vienna)

Title: Model uncertainty in mathematical finance via Wasserstein distances

Abstract: In this talk we model uncertainty through neighborhoods in Wasserstein distance within a one-period framework. After a short discussion on the choice of distance, we show (semi-)explicit formulas for some robust risk measures. We then conduct a sensitivity analysis (of e.g. utility maximization) and finally study a scaling limit in continuous time of Wasserstein neighborhoods. If time permits, we shortly elaborate why Wasserstein distances are not suited for a general multi-period analysis and introduce an adapted modification. Based on joint works with J.Backhoff, M.Beiglboeck, S.Drapeau, M.Eder, M.Kupper, J.Obloj, L.Tangpi, J.Wiesel.

**24.05.2019 **Peter Tankov (ENSAE ParisTech)

Title: Mean field games of optimal stopping and industry dynamics in the electricity market

Abstract: In this talk, we shall first discuss the mean field games of optimal stopping of the "war of attrition" type, introduced in [1], and present the relaxed solution approach to these games, developed in [2]. We shall then describe an application of these games to the modeling of long-term dynamics of the electricity industry, where the renewable producers look for the optimal moment to enter the market, the conventional producers look for the optimal moment to exit, and the interaction between the two types of producers takes place through the market price determined by an exogeneous demand curve and an endogeneous merit order supply curve. Based on joint works with R. Aid, G. Bouveret and R. Dumitrescu.

[1] Bertucci, C. (2018). Optimal stopping in mean field games, an obstacle problem approach. J. Math. Pures Appl., 120, 165-194.

[2] Bouveret, G., Dumitrescu, R., & Tankov, P. (2018). Mean-eld games of optimal stopping: a relaxed solution approach. arXiv:1812.06196.

**07.06.2019 **Jukka Lempa(University of Turku)

Title: *A Class of Solvable Multiple Entry Problems with Forced Exits*

Abstract: We study an optimal investment problem with multiple entries and forced exits. A closed form solution of the optimisation problem is presented for general underlying diffusion dynamics and a general running payoff function in the case when forced exits occur on the jump times of a Poisson process. Furthermore, we allow the investment opportunity to be subject to the risk of a catastrophe that can occur at the jumps of the Poisson process. More precisely, we attach IID Bernoulli trials to the jump times and if the trial fails, no further re-entries are allowed. Interestingly, we find in the general case that the optimal investment threshold is independent of the success probability is the Bernoulli trials. The results are illustrated with explicit examples.

**14.06.2019 Roxana Dumitrescu **(KCL)

Title: A dynamic dual representation of the buyer's price of American options in a nonlinear incomplete market

Abstract: In this paper we study the problem of nonlinear pricing of an American option with a right-continuous left-limited (RCLL) payoff process in an incomplete market with default, from the buyer's point of view. We show that the buyer's price process can be represented as the value of a stochastic control/optimal stopping game problem with nonlinear expectations, which corresponds to the maximal subsolution of a constrained reflected Backward Stochastic Differential Equation (BSDE). We then deduce a nonlinear optional decomposition of the buyer's price process. To the best of our knowledge, no dynamic dual representation (resp. no optional decomposition) of the buyer's price process can be found in the literature, even in the case of a linear incomplete market and brownian filtration. Finally, we prove the "infimum" and the "supremum" in the definition of the stochastic game problem can be interchanged. Our method relies on new tools, as simultaneous nonlinear Doob-Meyer decompositions of processes which have a Y-submartingale property for each admissible control.

**14.06.2019 Frank Riedel** (Bielefeld University)

Title: Viability and Arbitrage under Knightian Uncertainty

Abstract: We reconsider the microeconomic foundations of financial economics under Knightian Uncertainty. We do not assume that agents (implicitly) agree on a common probabilistic description of the world. We rather base our analysis on a common ordering of contracts, a much weaker requirement. The economic viability of asset prices and the absence of arbitrage are equivalent; both are closely related to the existence of nonlinear pricing measures. We show how the different versions of the Efficient Market Hypothesis are related to the assumptions we are willing to impose on the market’s ordering of contracts. Our approach also unifies recent versions of the Fundamental Theorem of Asset Pricing under a common framework.

**14.06.2019 Christoph Czichowsky** (LSE)

Title: Rough volatility and portfolio optimisation under transaction costs

Abstract: Rough volatility models have become quite popular recently, as they capture both the fractional scaling of the time series of the historic volatility (Gatheral et al. 2018) and the behavior of the implied volatility surface (Fukasawa 2011, Bayer et al. 2016) remarkably well. In contrast to classical stochastic volatility models, the volatility process is neither a Markov process nor a semimartingale. Therefore, these models fall outside the scope of standard stochastic analysis and provide new mathematical challenges. In this talk, we investigate the impact of rough volatility processes on portfolio optimisation under transaction costs. The talk is based on joint work with Johannes Muhle-Karbe and Denis Schelling.