# Abstracts

**29.06.2018:** Cosimo-Andrea Munari (Zürich)

**Title: **Existence, uniqueness and stability of optimal portfolios of eligible assets

**Abstract: **In a capital adequacy framework, risk measures are used to determine the minimal amount of capital that a financial institution has to raise and invest in a portfolio of pre-specified eligible assets in order to pass a given capital adequacy test. From a capital efficiency perspective, it is important to be able to do so at the lowest possible cost and to identify the corresponding portfolios. We study the existence and uniqueness of such optimal portfolios as well as their stability behaviour under a perturbation of the underlying capital position. This behavior is naturally linked to the continuity properties of the set-valued map that associates to each capital position the corresponding set of optimal portfolios. Upper semicontinuity can be established under fairly natural assumptions. Lower semicontinuity is more elusive. While it is always satisfied in a polyhedral setting, it generally fails otherwise, even when the reference risk measure is convex. However, lower semicontinuity can often be established for portfolios that are close to being optimal. Besides capital adequacy, our results have a variety of natural applications to pricing, hedging, and capital allocation problems.

**25.05.2018:** Florian Stebegg (Columbia)

**Title: **Strong Duality and Relaxations in Constrained Transport

**Abstract: **Martingale Optimal Transport and Skorokhod Embedding Problems are naturally studied alongside their dual problems, also referred to as Robust Hedging Problems. Classically, the primal transport problem was relaxed to guarantee the existence of a primal optimizer (assuming continuity). We will revisit the analogon of this relaxation for Skorokhod Embedding Problems. We will furthermore explore the relaxations required for attainment in the dual problem, in both discrete and continuous time. In particular we will discuss how the choice of the dual domain can affect Strong Duality and how the existence of a dual optimizer allows for an easy derivation of Monotonicity Principles.

**11.05.2018:** Thaleia Zariphopoulou (Austin)

**Title: **Mean-field and n-agent games for optimal investment under relative performance criteria

**Abstract: **In this talk, I will discuss a family of portfolio management problems, under relative performance criteria, for fund managers having CARA or CRRA utilities and trading in a common investment horizon in log-normal markets. Explicit constant equilibrium strategies will be constructed, for both the finite population games and the corresponding mean-field games, and it will be shown that are unique in the class of constant equilibria. In the CARA case, competition drives agents to invest more in the risky asset than they would otherwise while in the CRRA case competitive agents may over- or under-invest, depending on the levels of their risk tolerance. This is joint work with D. Lacker.

**04.05.2018, 11am:** Hao Xing (LSE)

**Title: **Optimal contracting with unobservable managerial hedging

**Abstract:** We develop a continuous-time model where a risk-neutral principal contracts with a CARA manager protected by limited liability to elicit hidden effort. The manager can trade a market portfolio in an unobservable private account to hedge market risk in his compensation. New to the literature, our model incorporates simultaneously private saving and hedging, manager's risk aversion, and inefficient project liquidation. The inefficient project liquidation endogenously induces effective risk aversion of the risk-neutral principal. Due to a trade-off between incentive provision and risk sharing, the optimal contract is loaded dynamically on market return and displays a dynamic mixture of relative and absolute performance evaluations. This provides a new explanation to the lack of relative performance evaluation in practice. Moreover, the market component in the contract dynamically regulates the timing of liquidation and compensation to the manager. In particular, the contract sensitivity of market return is positive near the liquidation boundary, entailing increased probability of project liquidation after negative market shocks. This is consistent with the heightened managerial turnover in bad market conditions.

**04.05.2018, 2pm:** Sebastian Herrmann (Michigan)

**Title: **Inventory Management for High-Frequency Trading with Imperfect Competition

**Abstract:** We study Nash equilibria for inventory-averse high-frequency traders (HFTs), who trade to exploit information about future price changes. For discrete trading rounds, the HFTs' optimal trading strategies and their equilibrium price impact are described by a system of nonlinear equations; explicit solutions obtain around the continuous-time limit. Unlike in the risk-neutral case, the optimal inventories become mean-reverting and vanish as the number of trading rounds becomes large. In contrast, the HFTs' risk-adjusted profits and the equilibrium price impact converge to their risk-neutral counterparts. Compared to a social planner solution for cooperative HFTs, Nash competition leads to excess trading, so that marginal transaction taxes in fact benefit the HFTs. Joint work in progress with Johannes Muhle-Karbe, Dapeng Shang, and Chen Yang

**16.03.2018:** Dörte Kreher (HU Berlin)

**Title: **First and second order approximations for a Markovian limit order book model

**Abstract:** We present first and second order approximations for an inﬁnite dimensional, multiscale, Markovian limit order book model, in which the dynamics of the incoming order ﬂow is allowed to depend on the current market price as well as on a volume indicator. The ﬂuctuations of the price and volume process relative to their ﬁrst order approximation given in ODE-PDE form are studied under two diﬀerent rescaling regimes. In the ﬁrst case we suppose that price changes are really rare, yielding a constant ﬁrst order approximation for the price process and a measure-valued SDE in the second order approximation for the volume process. In the second case we use a slower rescaling rate, which allows for a non-degenerate ﬁrst order approximation and gives a PDE with random coeﬃcients in the second order approximation for the volume process. The talk is based on joint work with Ulrich Horst.

**09.03.2018:** Sören Christensen (Hamburg)

**Title: **Non-Smooth Verification for Impulse Control Problems

**Abstract:** Stochastic impulse control problems are continuous-time optimization problems in which a stochastic system is controlled through finitely many impulses causing a discontinuous displacement of the state process. The objective is to choose the impulses optimally so as to maximize a reward functional of the state process. This type of optimization problem arises in many branches of applied probability and economics such as optimal portfolio management under transaction costs, optimal forest harvesting, inventory control, and real options analysis. In this talk, I will give an introduction to optimal impulse control and discuss classical solution techniques. I will then introduce a new method to construct maximizers for impulse control problems under general assumptions based on superharmonic functions and a stochastic analogue of Perron's method. Finally, I will show how the general results can be applied to a problem of optimal investment in the presence of constant and proportional transaction costs. This talk is based on joint work with Christoph Belak and Frank T. Seifried (University of Trier).

**09.02.2018:** Samuel Drapeau (Shanghai Jiao Tong)

**Title: **Computational Aspects of Robust Optimized Certainty Equivalent

**Abstract:** An extension of the expected shortfall as well as the value at risk to model uncertainty has been proposed by P. Shige. In this talk we will present a systematic extension of the general class of optimized certainty equivalent that includes the expected shortfall. We show that its representation can be simplified in many cases for efficient computations. In particular we present some result based on a probability model uncertainty derived from some Wasserstein metric and provide explicit solution for it. We further study the duality and representation of them. This talk is based on a joint work with Daniel Bartl and Ludovic Tangpi

**02.02.2018:** Peiran Jiao (Maastricht)

**Title: **Signal Processing on Social Media: Theory and Evidence from Financial Markets

**Abstract:** We analyze the processing of information from social media and news media, using a unique dataset on financial markets. We find patterns consistent with a theory of social media as an “echo chamber”: Social networks repeat information, but boundedly rational investors interpret repeated signals as new information. This is based on the empirical finding that stocks with high social media coverage experience high subsequent volatility and trading activity, while high news media coverage predicts low volatility and trading activity. Alternative mechanisms based on private information, investor disagreement, uncertainty shocks, and other behavioral biases are not consistent with the data.

**19.01.2018:** Dmitrii Lisovskii (Moscow State)

**Title: **Sequential Problems for a Brownian Bridge

**Abstract:** In the present talk we discuss two problems regarding the sequential analysis which are closely related to the optimal stopping theory. To be more specific, we present both the sequential hypothesis testing and the quickest detection problems for a Brownian bridge process, both being treated in the Bayesian setup. For the hypothesis testing problem we assume that we observe either a standard Brownian bridge process or the one with a non-zero terminal pinning point what is equivalent to the presence of the drift term. Our natural desire then is to tell apart these two cases as soon as possible with the smallest in some sense penalty. On the contrary, the quickest detection problem treats with a Brownian bridge process which changes its drift term from zero to some non-zero constant at a random time – the so-called disorder time. Our aim here is to raise an alarm as close to the disorder as possible in some appropriate sense. We will see that both problems in question can be reduced to the corresponding optimal stopping problems for time-inhomogeneous Markov processes and then solved. (The work has been fulfilled under Albert Shiryaev’s supervision.)

**12.01.2018:** Ying Hu (Rennes)

**Title: **Multidimensional (Backward) Stochastic Differential Equations with Constraints on Law

**Abstract:** In this talk, we will study reflected stochastic differential equations (as well as reflected backward stochastic differential equations) in the case where the constraint is on the law of the solution rather than on its paths. First we study such reflected (B)SDE in the multidimensional case. And then we concentrate on the propagation of chaos.

**08.12.2017:** Yiqing Lin (École Polytechnique)

**Title: **Second-order Backward SDEs with Random Terminal Time

**Abstract:** In this talk, a type of second-order backward SDEs (2BSDEs) are introduced. Precisely, backward SDEs under a non-dominated family of mutually singular probability measures are formulated, in which the terminal values are associated with stopping times. We first generalize the theory of classical BSDEs on random horizon. Then we prove the existence by considering an optimization problem of solutions of the classical BSDEs over such a family of probability measures. Meanwhile, the uniqueness for the 2BSDE solution is from the representation in terms of the solutions of classical BSDEs. We also discuss potential applications of such equations to the principal-agent problem.

**01.12.2017:** Samuel Cohen (Oxford)

**Title: ** Uncertainty in Kalman-Bucy Filtering

**Abstract:** The classical Kalman-Bucy filter is a beautiful result of practical significance. Using it, within a model for the dynamics of a signal and observation process, we can recursively find the posterior distribution of the signal given observations. In practice however, we also need to estimate the dynamics, and this introduces an additional source of uncertainty into our assessments. In this talk, we will consider a model for this uncertainty, and how it quickly leads to interesting problems in optimal stochastic control.

**17.11.2017:** Michalis Anthropelos (Piraeus)

**Title: ** Equilibrium Transactions with large investors and indifferent market makers

**Abstract:** We consider a market of financial securities where large investors trade with market makers at their indifference pricing. We adapt the model of Bank and Kramkov (2015) and, under a large class of utility functions, we give conditions that guarantee the existence and the uniqueness of an individual investor's optimal order. We then consider the case of two large investors who place together their orders to market makers. For this, we establish a notion of equilibrium transaction, where the investors' sharing of the aggregate order and price is endogenously determined. The existence of such equilibrium is proved and the indicative example of exponential utility is extensively analysed. This is a joint work with P. Bank and S. Gokay.

**27.10.2017:** Andreea Minca (Cornell)

**Title: **Systemic Risk and Central Clearing Counterparty Design

**Abstract:** We examine the effects on a financial network of multilateral clearing via a central clearing counterparty (CCP) from an ex ante and ex post perspective. The CCP is capitalized with equity and a guarantee fund. We propose a CCP design which improves aggregate surplus, increases the utility of the banks and of the CCP and reduces the expected shortfall losses on the end users. We determine the CCP's equity and guarantee fund policies as a Nash bargaining solution. A simulation study based on aggregate market data shows that under our proposed design with hybrid guarantees there exists a unique Pareto optimal equity and guarantee fund policy of the CCP, which reduces systemic risk. (Joint with: Hamed Amini and Damir Filipovic)