# Abstract

**16.10.2020 Ruodu Wang **(University of Waterloo)

**Title: ****An axiomatic foundation for the Expected Shortfall**

**Abstract: **In the recent Basel Accords, the Expected Shortfall (ES, also known as CVaR and TVaR) has replaced the Value-at-Risk (VaR) as the standard risk measure for market risk in the banking sector, making it the most important risk measure in financial regulation. Although ES is, among many other nice properties, a coherent risk measure, it does not yet have an axiomatic foundation until now. We put forward four intuitive economic axioms for portfolio risk assessment - monotonicity, law invariance, prudence and no reward for concentration - that uniquely characterize the family of ES. The herein developed results, therefore, provide the first economic foundation for using ES as a globally dominating regulatory risk measure, currently employed in Basel III/IV and Swiss Solvency Test. Key to the main results, several novel notions such as tail events and risk concentration naturally arise. As a most important feature, ES rewards portfolio diversification and penalizes risk concentration in a special and intuitive way, not shared by any other risk measure. I will also discuss some uncompleted work on the implication of the axioms in insurance design.

**23.10.2020 Moris Strub **(SUSTech)

**Title: **Forward Rank-Dependent Performance Criteria: Time-Consistent Investment Under Probability Distortion

**Abstract:**We introduce the concept of forward rank-dependent performance criteria, extending the original notion to forward criteria that incorporate probability distortions. A fundamental challenge is how to reconcile the time-consistent nature of forward performance criteria with the time-inconsistency stemming from probability distortions. For this, we first propose two distinct definitions, one based on the preservation of performance value and the other on the time-consistency of policies and, in turn, establish their equivalence. We then fully characterize the viable class of probability distortion processes and provide the following dichotomy: it is either the case that the probability distortions are degenerate in the sense that the investor would never invest in the risky assets, or the marginal probability distortion equals to a normalized power of the quantile function of the pricing kernel. We also characterize the optimal wealth process, whose structure motivates the introduction of a new, distorted measure and a related market. We then build a striking correspondence between the forward rank-dependent criteria in the original market and forward criteria without probability distortions in the auxiliary market. This connection also provides a direct construction method for forward rank-dependent criteria. Finally, our results on forward rank-dependent performance criteria motivate us to revisit the classical (backward) setting. We follow the so-called dynamic utility approach and derive conditions for existence and a construction of dynamic rank-dependent utility processes. (Based on joint work with Xue Dong He and Thaleia Zariphopoulou)

**Bio: **

Moris Strub is an assistant professor at the Department of Information Systems and Management Engineering of the SUSTech College of Business. He has obtained a BSc in Mathematics and MSc in Applied Mathematics from ETH Zurich, both with distinction, and a PhD in Financial Engineering from the Chinese University of Hong Kong. Before joining SUSTech, he was a Postdoctoral Fellow at the Chinese University of Hong Kong and a Staff Associate at Columbia University. Moris main research interests are in the areas of Portfolio Selection, Behavioral Finance and Economics, Mathematical Finance, Risk Management, and Robo-Advising

**30.10.2020 GONZÁLEZ CÁZARES, JORGE** (Warwick)

**Title: **Monte Carlo methods for the extrema of Lévy models

**Abstract: **An increasing body of literature use models based on Lévy processes to model risky assets. Among the quantities of interest are the drawdown (the distance to the previous historic maximum) and its duration (how long ago was the previous historic maximum). In this talk, we consider the problem of estimating expectations related to these quantities, such as barrier options or the ulcer index. We give a brief summary of the existing Monte Carlo and multilevel Monte Carlo methods and present a new method that converges geometrically fast.

**13.11.2020 Daniel Lacker **(Columbia University)

**Title: **Local stochastic volatility models and inverting the Markovian projection

**Abstract:** This talk is about a class of two-dimensional stochastic differential equations (SDEs) of McKean--Vlasov type in which the conditional distribution of the second component given the first appears in the equation for the first component. Such SDEs arise when one tries to invert the Markovian projection developed by Gyöngy (1986) and Brunick-Shreve (2013), typically to produce an Itô process with the same fixed-time marginal distributions as a given one-dimensional diffusion but with richer dynamical features. We prove the strong existence of stationary solutions for these SDEs, as well as their strong uniqueness in an important special case, by studying the associated (nonlinear, nonlocal) elliptic PDE. Variants of the SDEs discussed in this paper enjoy frequent application in the calibration of local stochastic volatility models in finance, despite the very limited theoretical understanding. Based on joint work with Mykhaylo Shkolnikov and Jiacheng Zhang.

**20.11.2020 Xiaofei Shi **(Columbia University)

**Title: Liquidity Risk and Asset Pricing**

**Abstract:** We study how the price dynamics of an asset depends on its "liquidity" - the ease with which can be traded. An equilibrium is achieved through a system of coupled forward-backward SDEs, whose solution turns out to be amenable to an asymptotic analysis for the practically relevant regime of large liquidity. We also calibrate our model to time series data of market prices and trading volume, and discuss how to leverage deep-learning techniques to obtain numerical solutions. (Based on joint works in progress with Agostino Capponi, Lukas Gonon, Johannes Muhle-Karbe and Chen Yang.

**27.11.2020 JEROME, JOE (Warwick)**

**Title: **Infinite Horizon Stochastic Differential Utility

**Abstract: **Stochastic differential utility has been widely studied since its formulation by Duffie and Epstein in 1992. It allows modelling of a much wider range of risk and intertemporal preferences and therefore provides a natural extension to the Merton problem for time-additive utility. However, whilst the finite time horizon problem is now fairly well understood, few have investigated the infinite horizon `lifetime' problem. In our paper we provide a novel formulation of the lifetime problem, highlighting and explaining the role of the transversality condition. We then discuss the parameters governing the agent's preferences, and show that certain parameter combinations considered in the literature are ill-posed over the infinite horizon.

We prove existence of a finite valued utility process for a large class of consumption streams and then show that, by considering a natural generalisation, we may assign a meaningful utility to any non-negative progressively measurable process. This means that, regardless of the choice of financial market, self-financing consumption streams are always evaluable.

Finally, we show existence and uniqueness of an optimal strategy in a Black-Merton-Scholes market.

**29.1.2021 Jean-Francois Chassagneux (Université de Paris)**

**Title**: Modeling carbon markets using Forward-Backward SDEs

**Abstract**: In this talk, I will present a mathematical modeling of carbon markets. These are cap and trade schemes, where a regulator sets a cap on the total amount of emissions of all the participants in a particular market. The models we study are risk-neutral model and are mathematically based on singular Forward-Backward SDEs. We obtain new results for markets with more than one period and, as a limit case, markets with no end date. I will also discuss some aspects of the numerical simulation of such models. This talk is based on joint works with H. Chotai (Citigroup), D. Crisan (Imperial College London), M. Muuls (Imperial College London) and M. Yang (Université de Paris).

**2.2.2021 Yufei Zhang (Oxford University)**

**Title:** Deep neural network approximations to stochastic control problems

**Abstract:** In this talk, we discuss the feasibility of algorithms based on deep artificial neural networks for the solution of high-dimensional stochastic control problems. In the first part, we show that in certain cases, including those with open loop controls, the value functions can be represented by a DNN whose complexity grows at most polynomially in both the dimension of the state equation and the reciprocal of the required accuracy, demonstrating that DNNs can break the curse of dimensionality. We then introduce a neural network based policy iteration algorithm to solve nonlinear PDEs associated with stochastic control problems, and demonstrate its super-linear convergence with any initial guess. In the second part, we analyze discrete-time approximations of mean-field control problems, whose coefficients involve mean-field interactions both on the state and actions. We show that for a large class of linear-convex mean-field control problems, the unique optimal control admits the optimal 1/2-Holder regularity in time, which can be approximated by using deep learning techniques with an order 1/2 error.

**12.2.2021 Said Hamadene, LMM, Le Mans University**

**Title:** Mean-field reflected backward stochastic differential equations

**Abstract:** In this talk, we study a class of reflected backward stochastic differential equations (BSDEs) of mean-field type, where the mean-field interaction in terms of the expected value $\E[Y]$ of the $Y$-component of the solution enters both the driver and the lower obstacle. We consider the case where the lower obstacle is a deterministic function of $(Y,\E[Y])$. Under mild Lipschitz and integrability conditions on the coefficients, we obtain the well-posedness of such a class of equations. Under further monotonicity conditions we show convergence of the standard penalization scheme to the solution of the equation. This class of models is motivated by applications in pricing life insurance contracts with surrender options. With Boualem Djehiche (KTH, Stockholm) and Romuald Elie, UMLV, Marne La Vallée, France.

**19.2.2021 Eduardo Davila, Yale University **

**Title:** Optimal Financial Transaction Taxes

**Abstract:** This paper characterizes the optimal transaction tax in an equilibrium model of financial markets.

If investors hold heterogeneous beliefs unrelated to their fundamental trading motives and the planner

calculates welfare using any single belief, a positive tax is optimal, regardless of the magnitude of

fundamental trading. Under some conditions, the optimal tax is independent of the planner’s belief.

The optimal tax can be implemented by adjusting its value until total volume equals fundamental

volume. Knowledge of i) the share of non-fundamental trading volume and ii) the semi-elasticity of

trading volume to tax changes is sufficient to quantify the optimal tax.

**26.2.2021 Bahman Angoshtari (University of Miami)**

**Title:**Optimal Consumption under a Habit-Formation Constraint

**Abstract**: We consider an infinite-horizon optimal consumption problem for an individual who forms a consumption habit based on an exponentially-weighted average of her past rate of consumption. The novelty of our approach is in introducing habit formation through a constraint, rather than through the objective function, as is customary in the existing habit-formation literature. Specifically, we require that the individual consume at a rate that is greater than a certain proportion $\alpha$ of her consumption habit. Our habit-formation model allows for both addictive ($\al=1$) and non-addictive ($0<\al<1$) habits. Assuming that the individual invests in a risk-free market, we formulate and solve a deterministic control problem to maximize the discounted CRRA utility of the individual's consumption-to-habit process subject to the said habit-formation constraint. We derive the optimal consumption policies explicitly in terms of the solution of a nonlinear free-boundary problem, which we analyze in detail. Impatient individuals (those with sufficiently large utility discount rates) always consume above the minimum rate; thus, they eventually attain the minimum wealth-to-habit ratio. Patient individuals (those with small utility discount rates) consume at the minimum rate if their wealth-to-habit ratio is below a threshold, and above it otherwise. By consuming patiently, these individuals maintain a wealth-to-habit ratio that is greater than the minimum acceptable level. Joint work with Erhan Bayraktar and Virginia Young.

**5.3.2021 Dylan Possamai (ETH)**

**Title:**Time-inconsistent control and backward integral Volterra SDEs

**Abstract**: This paper is the first attempt at a general non-Markovian theory of time-inconsistent stochastic control problems in continuous-time. We consider sophisticated agents who are aware of their time-inconsistency and take into account in future decisions. We prove here that equilibria in such a problem can be characterised through a new type of multi-dimensional system of backward SDEs, for which we obtain wellposedness. Unlike the existing literature, we can treat the case of non-Markovian dynamics, and our results go beyond verification type theorems, in the sense that we prove that any equilibrium must necessarily arise from our system of BSDEs. If time permits, an application to contract theory will be presented, thus highlighting a potential fruitful link with BSVIEs. This is a joint work with Camilo Hernández, Columbia University.