# Joint Risk & Stochastics and Financial Mathematics seminar series

One should really think of Statistics as a discipline which can be used to support other disciplines

The series aims to promote communication and discussion of research in the mathematics of insurance and finance and their interface, to encourage interaction between practice and theory in these areas, and to support academic students in related programmes at postgraduate level.

LSE maps and directions

All are welcome to attend these seminars. If you are attending from outside LSE please notify Penny Montague so that we can ensure you have access to the seminar room.

## 04/04/2019 - Michael Kupper

Thursday 4th April 2019
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Michael Kupper (University of Konstanz)

Title: Computation of model-free hedging problems via penalization and neural networks

Abstract: We present a widely applicable approach to solving model-free hedging problems via neural networks. The core idea is to penalize the optimization problem in its dual formulation and reduce it to a finite dimensional one which corresponds to optimizing a neural network with smooth objective function. As an application we discuss a version of the martingale transport problem with homogeneous stock movements and illustrate the approach with several numerical examples. The talk is based on joint work with Stephan Eckstein.

## 28/03/2019 - Flavia Barsotti

Thursday 28th March 2019
NAB. 2.16 New Academic Building (54 Lincoln's Inn Fields) 12pm to 1pm
*change of location*

Flavia Barsotti

Title Behavioural modeling: contagion effects among customers' decisions and macroeconomic drivers

Abstract: The aim of the talk is to present a methodological approach suitable to model customers' behaviours by embedding i) correlation and contagion effects among their decisions and ii) the role of macroeconomic factors. The proposed approach is suitable to model both stable economic regimes and stress scenarios. As an example, the problem of lapse risk will be discussed. The mathematical setting assumes the lapse intensity following a dynamic contagion process with both endogenous and exogenous jump components. This allows to capture both correlation and contagion potentially arising among customers’ behaviours and the macroeconomic driver. The shot-noise intensity is then not constant and the resulting intensity process is not Markovian. Closed-form expressions and analytic sensitivities for the moments of the lapse intensity are provided, showing how lapses can be affected by massive copycat behaviours. Further analyses are then conducted to illustrate how the mean risk varies depending on the model’s parameters.

## 22/03/2019 - Oleksii Mostovyi

Friday 22nd March 2019
COL 6.15 Columbia House, (69 Aldwych) 12pm to 1pm
*change of location*

Oleksii Mostovyi (University of Connecticut)

Title: Optimal consumption from investment and labor income in a unifying framework of admissibility

Abstract: We consider a problem of optimal consumption from investment and labor income in an incomplete semimartingale market. We introduce a set of constraint times, i.e., a set of stopping times, at which the wealth process must stay positive, in a unifying way such that borrowing against the future income might be allowed or prohibited. Upon this, we increase dimensionality and treat as arguments of the value function not only the initial wealth but also a function that specifies the amount of labor income. Assuming finiteness of the primal and dual value functions and that the labor income is superreplicable  (these are essentially the minimal model assumptions), we establish the existence and uniqueness of a solution to the underlying problem and provide several characterizations of the optimizer and the value functions. This talk is based on joint work with Mihai Sirbu.

## 21/03/2019 -  Hyeng Keun Koo

Thursday 21st March 2019
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Hyeng Keun Koo (Ajou University)

Title: Duesenberry, Long-term Wealth Management, and Asset Pricing

I will talk about Duesenberry's theory of consumption and propose a formal model of the theory. I will show how the model can be used for  long-term investors' risk management. I will also discuss asset pricing implications of the model.

## 14/03/2019 - Lukas Gonon

Thursday 14th March 2019
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Lukas Gonon University of St. Gallen

Title: Reservoir Computing with Stochastic Inputs: Universality, Error Bounds and Financial Applications

Abstract: We study dynamic machine learning for discrete-time stochastic processes based on reservoir computing. Putting particular emphasis on echo state networks, we present results on universal approximation properties as well as error bounds for learning tasks based on these systems. Finally, we apply them to the problem of predicting realized covariances of financial time series.

The talk is based on joint works with Juan-Pablo Ortega and Lyudmila Grigoryeva.

## 28/02/2019 - Pierre-Olivier Goffard

Thursday 28th February 2019
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Pierre-Olivier Goffard (ISFA)

Title: For a few bitcoins more

Subtitle: Fraud risk assessment within blockchain transactions

Abstract: The probability of successfully spending twice the same bitcoins is considered. A double-spending attack consists in issuing two transactions transferring the same bitcoins. The first transaction, from the fraudster to a merchant, is included in a block of the public chain. The second transaction, from the fraudster to himself, is recorded in a block that integrates a private chain, exact copy of the public chain up to substituting the fraudster-to-merchant transaction by the fraudster-to-fraudster transaction. The double-spending hack is completed once the private chain reaches the length of the public chain, in which case it replaces it. The growth of both chains are modeled by two independent counting processes. The probability distribution of the time at which the malicious chain catches up with the honest chain, or equivalently the time at which the two counting processes meet each other, is studied. The merchant is supposed to await the discovery of a given number of blocks after the one containing the transaction before delivering the goods. This grants a head start to the honest chain in the race against the dishonest chain.

A preprint is available on my website

## 14/02/2019 - Sara Svaluto-Ferro

Thursday 14th February 2019
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Sara Svaluto-Ferro (University of Vienna)

Title: Infinite dimensional polynomial jump-diffusions

Abstract: We introduce polynomial jump-diffusions taking values in an arbitrary Banach space B via their infinitesimal generator. We obtain two representations of the (conditional) moments in terms of solution of a systems of ODEs on (R, B∗, … , (B⊗k)∗) and (R, B∗∗, … , (B⊗k)∗∗), respectively. We illustrate how the well known moment formulas for finite dimensional polynomial jump diffusions can be deduced in this general framework. As an application, we consider probability measure-valued polynomial diffusions and polynomial forward variance curve models.

## 31/01/2019 -  Hyejin Cho

Thursday 31st January 2019
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Hyejin Cho (Université de Paris)

Title: The Order-theoretic Single Crossing Property in a One-Dimensional Screening Model

Abstract: We consider a finite one-dimensional screening of choices in monotone comparative statics (MCS). Before revealing the true state of the world, a principal sorts on actions of the agent to cause the social value of production as an informed principal. The model produces a rich order-theoretic single-crossing property according to Pick’s theorem pursuing no distortion at the top.

## 17/01/2019 - Mykhaylo Shkolnikov

Thursday 17th January 2019
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Mykhaylo Shkolnikov (Princeton University)

Title : Particles interacting through the hitting times: an application to systemic risk

Abstract: I will discuss a class of particle systems that form a natural framework for the study of systemic risk. The interaction between the particles falls into the mean field framework pioneered by McKean and Vlasov in the late 1960s, but many new phenomena arise due to the singularity of the interaction. The most striking of them is the loss of regularity of the particle density caused by the self-excitation of the system, which triggers systemic crises. Mathematically, the evolution of the system can be captured initially by a suitable Stefan problem, while the following irregular behavior necessitates a more robust probabilistic approach. Extensions to the setting where the interaction takes place on networks will be also discussed. Based on joint works with Sergey Nadtochiy.

Thursday 17th January 2019
CLM.7.02 Clement House, (99 Aldwych) 6pm to 7.30pm

Title: Volatility: Practical Options Theory

Abstract: tbc

## 13/12/2018 - Charles Bertucci

Thursday 13th December 2018
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Title : Optimal stopping in mean field games

Abstract: Mean field games are model in which a continuum of indistinguishable players face a game in which they interact between each other only through mean field terms. Such models have been introduced by Lasry and Lions in 2006. The most commonly studied case is the one in which the players control their velocity and interacts through a running/terminal cost. I will begin by recalling the main features of this seminal model as well as some results. Then I will explain how we can establish new results in a mean field game in which the players can decide to exit the game.

## 29/11/2018 - Tiziano De Angelis

Thursday 29th November 2018
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

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 study an example with linear payoffs, in which an explicit solution of the corresponding quasi-variational inequalities can be obtained.

The talk is based on https://arxiv.org/abs/1810.07674

## 15/11/2018 - Miklos Rasonyi

Thursday 15th November 2018
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Miklos Rasonyi (Central European University)

Title: Ergodic properties of certain financial models

Abstract: We consider certain Markov chains in random environments and prove their stochastic stability as well as a law of large numbers for their functionals. The results apply, in particular, to variants of rough volatility models
which can be regarded as Markov chains in a Gaussian stationary environment.

## 08/11/2018 -  Johannes Muhle-Karbe

Thursday 8th November 2018
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Johannes Muhle-Karbe (Carnegie Mellon University)

Title: A Risk-Neutral Equilibrium Leading to Uncertain Volatility Pricing

Abstract: We study the formation of derivative prices in equilibrium between risk-neutral agents with heterogeneous beliefs about the dynamics of the underlying. Under the condition that the derivative cannot be shorted, we prove the existence of a unique equilibrium price and show that it incorporates the speculative value of possibly reselling the derivative. This value typically leads to a bubble; that is, the price exceeds the autonomous valuation of any given agent. Mathematically, the equilibrium price operator is of the same nonlinear form that is obtained in single-agent settings with strong aversion against model uncertainty. Thus, our equilibrium leads to a novel interpretation of this price. (Based on joint work with Marcel Nutz)

## 01/11/2018 -  Kwok Chuen Wong

Thursday 1st November 2018
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Kwok Chuen Wong (Dublin City University)

Title:  Portfolio Optimisation with Semivariance

Abstract: In this talk, I shall investigate dynamic portfolio management using semivariance of portfolio payoff as a portfolio risk measure. Comparing with variance which is widely used in the literature, semivariance is considered to be more plausible risk measure because semivariance penalizes adverse situations only. However, in the literature, it was shown that mean-semivariance optimisation under the Black-Scholes model has no optimal solution.

Inspired by this non-existence result, I shall establish necessary and sufficient conditions under which the mean-semivariance optimisation possesses an optimal solution. I shall suggest the models under which such sufficient conditions are satisfied, thus, under these models, the explicit optimal solution to mean-semivariance optimisation can be obtained; such models can be applied into the themes of insurance. Besides, I shall establish that utility-semivariance optimisation possesses an optimal solution even under the Black-Scholes model. In numerical studies, among mostly encountered market values of different model parameters, it is astonishing to observe that embedding downside risk measure into utility maximization framework can significantly reduce the downside risk of the optimal portfolio payoff with an asymmetrically tiny loss in utility.

This talk is based on joint works with Paolo Guasoni, Phillip Yam, and Harry Zheng.

## 04/10/2018 - Zachary Feinstein

Thursday 4th October 2018
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Zachary Feinstein (Washington University)

Title: Pricing debt in interbank networks under comonotonic endowments

Abstract: In this talk we present formulas for the pricing of debt and equity of firms in a financial network under comonotonic endowments. We demonstrate that the comonotonic setting provides a lower bound to the price of debt under Eisenberg-Noe financial networks with consistent marginal endowments. Such financial networks encode the interconnection of firms through debt claims. The proposed pricing formulas consider the realized, endogenous, recovery rate on debt claims. Special consideration will be given to the setting in which firms only invest in a risk-free bond and a common risky asset following a geometric Brownian motion.

## 24/05/18 - Mike Ludkovski

Thursday 24 May 2018
KSW.1.04, (20 Kingsway) 12.00pm to 1.00pm

## Mike Ludkovski (UC Santa Barbara)

Title: Capacity Expansion Games: Non-zero-sum Switching/Multiple Stopping Equilibrium

Abstract: We consider non-zero-sum stochastic games of timing. Our motivation is from competitive capacity investment for a duopoly of two distinct producers. The producers are exposed to stochastically fluctuating costs and interact through aggregate supply. Capacity expansion is modeled in terms of timing strategies, i.e. repeated real options. The overall market is then described through the stochastic factor (X_t) that captures short-term fluctuations and the (relative) capacity that summarizes the current market organization.  Working in a continuous-time diffusion framework, we characterize and analyze the resulting Nash equilibrium and game payoffs using tools of optimal stopping. An example of competing green and fossil-fuel producers will be used as illustration. The second part of the talk will then consider ongoing work extending this model to a stationary switching game characterized via a sequence of switching thresholds.  Joint work with Rene Aid (Paris Dauphine, Part I) and Liangchen Li (UCSB, Part I and II).

## 30/04/18 - Albert N. Shiryaev

Monday 30 April 2018
COL 6.15 Columbia House, (69 Aldwych) 12.00pm to 1.00pm
***PLEASE NOTE: Change to usual day and venue***

Albert N. Shiryaev (Steklov Mathematical Institute)

Title: Optimal Stopping Procedures in financial models with disorder of trends ("drift-bubbles")

Abstract: We consider the financial model

X(t)=[aI(t<R)+bI(t>R)]+B(t)

where B is a Brownian motion, R is random positive variable, a and b some constants (known or unknown). Many financial models can be described by a similar way. For example, the stock AAPL (Apple computer stock) can be modelled using the above (here we consider the interval from September 1984 until the end of 2012). In the given Apple model we shall assume that the time R when the drift changes is uniformly distributed on the interval. In our talk we describe theoretical results on optimal stopping ("American options") of discovering the disorder time R (with some minimal risk) and practical results for the Apple stock.

## 26/04/18 - Lukasz Stettner

Thursday 26 April 2018
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Lukasz Stettner (Institute of Mathematics, Polish Academy of Sciences)

Title: Long Run Risk Sensitive Control and Portfolio Optimization

Abstract: I'm going to present several results concerning risk sensitive control of discrete and continuous time Markov processes over infinite time horizon and then similar results concerning risk sensitive portfolio optimization. The last problem is closely related to asymptotics of optimal power utility from terminal wealth. The talk shall consist of published and new results. The approach will be mainly probabilistic. Major problems and difficulties will be pointed out.

## 15/03/18 - Sergio Pulido

Thursday 15 March 2018
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Sergio Pulido (ENSIIE)

Title: Affine Volterra processes

Abstract: A growing body of empirical research indicates that volatility fluctuates more rapidly than Brownian motion, which is inconsistent with standard semimartingale models. Fractional volatility models and their relatives have emerged as compelling alternatives- however, their non-Markovian structure makes computations more difficult. We show that, for a large class of such models, it is nonetheless possible to compute the characteristic function by solving an integral equation similar to the Riccati equations associated with standard affine processes. Joint work with Eduardo Abi Jaber and Martin Larsson.

## 07/03/18 - Alfred Galichon

Wednesday 7 March 2018
COL 6.15 Columbia House, (69 Aldwych) 2.30pm to 3.30pm
***PLEASE NOTE: Change to usual day, time and venue***

Alfred Galichon (New York University)

Title: Topics in Equilibrium Transportation

Abstract: Motivated by problems from Economics, I will present a framework for "Equilibrium Transport", which embeds the Monge-Kantorovich "Optimal Transport" problem, but is more general, and more natural in some applications.In the discrete case, this framework allows for a unified description of Gale and Shapley's stable marriage problem, as well as Koopmans and Beckmann's optimal assignment problem.I will sketch the link with "Galois connections" and recent results by Trudinger on the local theory of prescribed Jacobian equations. I will then turn to computational issues, and will present an extension of Sinkhorn's algorithm that allows for efficient approximate computation of these problems. Finally, I will discuss the statistical estimation of these models.

## 01/03/18 - Jan Palczewski

Thursday 1 March 2018
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Jan Palczewski(Leeds)

Title: (Un)discounted optimal stopping problems and applications

Abstract: I will talk about an infinite horizon optimal stopping problem with a functional comprising of a running reward and a final reward. I will establish the feasibility of the stopping problem, prove the existence of optimal stopping times and a variational characterisation  (in the viscosity sense) of the value function when interest rates are not uniformly separated from 0. These results rely on certain ergodic properties of the underlying (non-uniformly) ergodic Markov process. I will further sketch how these results apply to impulse control  problems with average cost per unit time functional and, if time allows, present an example of optimal control of a battery for provision of Fast Reserve balancing service to National Grid.

## 21/02/18 - Budhi Surya

Wednesday 21 February 2018
COL 6.15 Columbia House, (69 Aldwych) 3pm to 4pm
***PLEASE NOTE: Change to usual day and venue***

Budhi Surya (Victoria University of Wellington)

Title: A Rating-Based Model of Credit Risk Under Non-Markov Chains

Abstract: In many credit risk applications, the rating based model of Jarrow-Lando-Turnbull (Journal of Finance, Vol. 50, p. 53-86, 1995) has been widely used for the pricing and hedging of corporate bonds. The model is driven by a continuous-time absorbing Markov chain. However, there have been mounting empirical evidences to suggest the contrary, see e.g. Frydman and Schuermann (Journal of Banking and Finance Vol. 32, p.1062-1075, 2008), that bonds of the same credit rating can move at different rates to other credit ratings and that the incorporation of past credit information helps improve the Nelson-Aalen estimate of cumulative default intensity. Based on these empirical findings, I propose a new rating based model of credit risk under a non-Markov chain. The model is developed based on the mixture of continuous-time absorbing Markov chains moving at different speeds, where the mixture occurs at a random time. Variety of associated distributional properties of the Markov mixture process are discussed, for example the transition matrix, the default-time distribution and forward default intensity. Identities are explicit in terms of the Bayesian update of switching probability and intensity matrices of the underlying Markov chains despite the mixture process is non Markovian. They form non-stationary function of time and have the ability to capture heterogeneity and past credit information when conditioning on available information (either full or partial) up to current time. Their availability in closed forms offers appealing features for applications in credit risk.

Reference:

B.A. Surya. (2017). Distributional properties of the mixture of continuous time absorbing Markov chains moving at different speeds. Stochastic Systems - INFORMS Applied Probability Society.

## 15/02/18 - Cristina Di Giralomi

Thursday 15 February 2018
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Cristina Di Girolami (Pescara)

Title: Path dependent stochastic calculus, an infinite dimensional PDE and financial perspectives

Abstract: This talk develops some aspects of stochastic calculus via regularization for path dependentrandom variables. After some brief reminds on stochastic calculus in a general Banach space B, main interest will be devoted to the case when B is the space of real continuous functions definedon [–T; 0], T > 0 and the process is the window process X(•) associated with a continuous realprocess X which, at time t, it takes into account the past of the process. If X is a finite quadraticvariation process (for instance Dirichlet, weak Dirichlet), it is possible to represent a large class of path-dependent random variable h as a real number plus a real forward integral in a semiexpliciteform. This representation result of h makes use of a functional solving a path dependent infinitedimensional partial differential equation of Kolmogorov type. Two recent general existence results of its classical solutions related to different classes of final conditions will be presented. The decomposition result generalizes, in some cases, the well known Clark-Ocone formula which is true when X is the standard Brownian motion W. Some examples will be given explicitly developed and discussed.

This is a joint work with Francesco Russo (ENSTA ParisTech Paris).

## 01/2/2018 - Denis Villemonais

Thursday 1 February 2018
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Denis Villemonais, École des Mines de Nancy / University of Lorraine, France

Title : Exponential convergence of conditioned processes

Abstract : In a previous work in collaboration with Nicolas Champagnat, we provided necessary and sufficient criterion for the uniform exponential convergence of Markov processes conditioned not to be killed. This criterion, and hence the uniform exponential convergence, only applies to cases where the conditioned process is uniformly exponentially ergodic. Our aim during this talk is to present applications to a new framework of assumptions that entails the non-uniform exponential convergence of conditioned Markov processes. In this new framework, most of the difficulties usually involved in proving the convergence of a conditioned Markov process to a quasi-stationnary distribution are easily overcome.

## 18/01/2018 - Christine Gruen

Thursday 18 January 2018
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Christine Gruen, Toulouse School of Economics

Title: On games with asymmetric information

Abstract: In this talk we will consider two player zero sum games where the two players have access to different informations about payoffs and the dynamics driving the game. We assume that the players can observe the actions of their opponents, which allows them to guess the private information of the other player. Optimal strategies in these games are thus much more complex than in games where both players have access to the same information. Not only do the players aim to optimise a payoff while not giving away too much information, but they choose their strategy in order to optimally manipulate the beliefs of the other players while defending themselves against the manipulation of their adversary. This typical behaviour is found in a large class of games, notably also for stopping games on which we will concentrate in this talk.

## 07/12/2017 - Thomas Kruse

Thursday 7 December 2017
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Thomas Kruse (Duisburg-Essen)

Title: Multilevel Picard approximations for high-dimensional nonlinear parabolic partial differential equations

Abstract: In this talk we present a family of new approximation methods for high-dimensional PDEs and BSDEs.

A key idea of our methods is to combine multilevel approximations with Picard fixed-point approximations. Thereby we obtain a class of multilevel Picard approximations.
Our error analysis proves that for semi-linear heat equations, the computational
complexity of one of the proposed methods is bounded by $O(d\,\eps^{-(4+\delta)})$ for any $\delta > 0$, where $d$ is the dimensionality of the problem and $\eps\in(0,\infty)$ is the prescribed accuracy.

We illustrate the efficiency of one of the proposed approximation methods by means of numerical simulations presenting approximation accuracy against runtime for
several nonlinear PDEs from physics (such as the Allen-Cahn equation) and financial engineering (such as derivative pricing incorporating default risks) in the case of $d=100$ space dimensions.

The talk is based on joint work with W. E, M. Hutzenthaler, and A. Jentzen.

## 23/11/2017 - Charles-Albert Lehalle

Thursday 23 November 2017
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Charles-Albert Lehalle (Capital Fund Management)

Title: Closing The Loop of Optimal Trading: a Mean Field Game of Controls

Abstract: This talk explains how to formulate the now classical problem of optimal liquidation (or optimal trading) inside a Mean Field Game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader in front of a " background noise " (or " mean field "). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price. Here the trader faces the uncertainty of fair price changes too but not only. He has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically. Our MFG formulation of this problem belongs to the class of " extended MFG ", we hence provide generic results to address these " MFG of controls ", before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of " heterogenous preferences " (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can " learn " it day after day, observing others' behaviors.

## 09/11/2017 - Thibaut Mastrolia

Thursday 9 November 2017
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Thibaut Mastrolia (CMAP)

Title: An overview of contract theory: from the Holmström-Milgrom paradigm to multi-agent systems

Abstract: In this talk, we investigate a situation in which two economical entities interact to maximize their own payoff. One of them is called the Principal (she) and proposes to the second one, named the Agent (he), some incentives to modify her wealth. The main difficulty comes from the fact that the Principal observes the result of the work of her Agent without observing his work directly. This kind of situation coincides exactly with a moral hazard problem in which the Principal has to design an employment contract given to her Agent to maximize her utility without observing directly his work. We identify this paradigm with a Stackelberg equilibrium that can be explicitly solved in several examples. We begin to study the example of Holmström and Milgrom dealing with one Principal and one Agent in the continuous case. We then extend this investigation to N-interacting Agents hired by one Principal and to a model with one Agent hired by several Principals.

## 26/10/2017 - Joaquin Narro

Thursday 26 October 2017

CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Joaquin Narro (Alcazar Investment Management Ltd / Bainbridge Partners LLP)

Title: Forecasting Prices of Electricity Futures: Practice vs. Theory

Abstract: Electricity plays a major role in people’s lives. The reliability and affordability of electricity, together with the need to address climate change, are major issues for our society at large. The world-wide drive to support low-carbon electricity generation is based on a myriad of legislative initiatives with implications far beyond traditional power systems. In this complex environment, it is vital for all stakeholders involved in electricity generation, from consumers to producers, from investors to regulators, to understand the pricing of electricity. This presentation explores several examples of forecasting of electricity prices using different mathematical models, highlighting the practical benefits and hindrances of different approaches.

## 12/10/2017 - Saul Jacka

Thursday 12 October 2017
CLM.7.02 Clement House, (99 Aldwych) 12pm to 1pm

Saul Jacka, Professor, Warwick

Title: Optimal stopping, the dual problem and smooth pasting

Abstract: Let $G$ be a semimartingale, and $S$ its Snell envelope. Under the assumption that $G\in\mathcal{H}^1$, we show that the finite-variation part of $S$ is absolutely continuous with respect to the decreasing part of the finite-variation part of $G$. In the Markovian setting, this enables us to identify sufficient conditions for the value function of the optimal stopping problem to belong to the domain of the extended (martingale) generator of the underlying Markov process. We then show that the dual of the optimal stopping problem is a stochastic control problem for a controlled Markov process, and the optimal control is characterised by a function belonging to the domain of the martingale generator. Finally, we give an application to the smooth pasting condition.

## 28/09/2017 - Christoph Belak

Thursday 28 September 2017
CLM 7.02, Clement House (99 Aldwych) 12pm to 1pm

Christoph Belak, TU Kaiserslautern

Title: Utility Maximization with Constant Costs

Abstract: We study the problem of maximizing expected utility of terminal wealth for an investor facing a mix of constant and proportional transaction costs. While the case of purely proportional transaction costs is by now well understood and existence of optimal strategies is known to hold for very general class of price processes, the case of constant costs remains a challenge since the existence of optimal strategies is not even known in tractable models (such as, e.g., the Black-Scholes model). In this talk, we present a novel approach which allows us to construct optimal strategies in a multidimensional diffusion market with price processes driven by a factor process and for general lower-bounded utility functions.

The main idea is to characterize the value function associated with the optimization problem as the pointwise infimum V of a suitable set of superharmonic functions. The advantage of this approach is that the pointwise infimum inherits the superharmonicity property, which in turn allows us to prove a verification theorem for candidate optimal strategies under mild regularity assumptions on V. Indeed, for the verification procedure based on superharmonic functions to be applicable, it suffices that the pointwise infimum V is continuous.

In order to establish the continuity of V, we adapt the stochastic Perron's method to our situation to show that V is a discontinuous viscosity solution of the associated quasi-variational inequalities. A comparison principle for discontinuous viscosity solutions then closes the argument and shows that V is continuous. With this, the verification theorem becomes applicable and it follows that the pointwise infimum V coincides with the value function and that the candidate optimal strategies are indeed optimal.

This talk is based on joint work with Sören Christensen (University of Hamburg) and Frank T. Seifried (University of Trier).

## 20/09/2017 - Keita Owari

Wednesday 20 September 2017
CLM 3.04, Clement House (99 Aldwych) 12pm to 1pm

Keita Owari, Assistant Professor, Ritsumeikan University

Title: A Komlós-Type Theorem in Dual Orlicz Spaces

Abstract: We give a Komlós type result for bounded sequences in dual Orlicz spaces (i.e. Orlicz spaces which are the duals of ($\Delta_2$) Orlicz spaces; e.g. the space of random variables with some exponential moments). Its "utility grade" version asserts that any bounded sequence in such a  space has an (a.s. convergent) sequence of forward convex combinations whose  supremum remains in the same space. If the probability space is atomless, this type of Komlós theorem characterises the class of dual Orlicz spaces, or their preduals, the $\Delta_2$-Orlicz spaces. Some consequences in convex duality in finance are also given.

## 14/09/2017 - Chiara Donnini

Thursday 14 September 2017
OLD 2.22, Old Building 12pm to 1pm

Chiara DonniniAssistant Professor, University of Naples

Title: Strictly fair allocations in economies with atoms

Abstract: We investigate the fairness property of Walrasian allocations in mixed exchange markets that are economies having both atoms (large or non-negligible traders) and an atomless sector (small or negligible traders). We provide two sufficient conditions under which equal-income Walrasian equilibria are the only efficient allocations which are also strictly equitable as defined by Zhou. A further characterization is obtained without extra assumptions via a weaker notion of envy-freeness based on the Aubin approach. Our analysis proceeds with a discussion on the problems arising in differential information economies.

## Density of probability measures with the martingale representation property

### Sergio Pulido

Thursday 16 March 2017: Room CLM.7.02, Clement House (Aldwych), 12pm to 1pm

Associate Professor, ENSIIE

Title - Density of probability measures with the martingale representation property

Abstract - Using the theory of analytic maps, we prove density results for measures satisfying a backward formulation of the martingale representation property. These results are useful to study equilibrium-based mechanisms of pricing. This is joint work with Dmitry Kramkov.

## Asymptotic static hedge via symmetrization

### ***PLEASE NOTE: Change to usual day and venue***

Monday 13 March 2017 CLM.4.02, Clement House (Aldwych), 12pm to 1pm

Assistant Professor, School of Management,Tokyo University of Science

Title - Asymptotic static hedge via symmetrization

Abstract - In the talk, we are interested in the risk to cover (some portion of) the price of the option at a default time. The risk, which we call timing risk, is a risk of uncertain dividend, especially of its payment time. Credit derivatives typically are exposed to the risk. We will discuss how it could be hedged by a static position of European path-independent options, generalizing P. Carr and J. Picron (1999) where they applied the semi-static hedging formula of barrier options to hedge a payment at a stopping time in a Black-Scholes environment. We will give an exact hedging formula in an multi-dimensional general diffusion setting.

## Robust optimal investment in discrete time for unbounded utility function

### Romain Blanchard

Thursday 9 March 2017: Room CLM.7.02, Clement House (Aldwych), 12pm to 1pm

Title - Robust optimal investment in discrete time for unbounded utility function

We investigate the problem of maximising  worst case expected terminal utility in a discrete time financial model with a finite horizon under non-dominated model uncertainty. We use dynamic programming framework together with measurable elections arguments to prove that under mild integrability assumption, an optimal portfolio exists for unbounded utility function defined on the half-real line. We revisit also the non-arbitrage condition in the robust framework.

## Stochastic illiquidity

### Dirk Becherer

Thursday 2 March 2017: Room CLM.7.02, Clement House (Aldwych), 12pm to 1pm

Title - Stochastic illiquidity

Abstract - In classical models from math. finance, dynamic trading strategies are executed against price processes which are exogenously given, and are not affected by the strategies. Economically, this means to assume that liquidity is unlimited or investors are 'small'. We discuss optimal control problems from mathematical finance in models for large investors, whose trading strategies have an intertemporal effect on the prices, against which they are executed. An original feature of the model, that we discuss, is that the transient price impact due to illiquidity in our model is stochastic and multiplicative, instead of being additive and deterministic (as a function of the strategy).

Joint work with Todor Bilarev, Peter Frentrup, HU Berlin, some related papers are on arxiv.

## Bail-ins and bail-outs: incentives, connectivity, and systemic stability

### Agostino Capponi

***PLEASE NOTE: Change to usual day and venue***

Friday 24 February 2017: Room  32L.G.15 (32 Lincoln's Inn Fields) 12pm to 1pm

Assistant Professor, Industrial Engineering & Operations Research, Columbia University

Title - Bail-ins and bail-outs: incentives, connectivity, and systemic stability

Abstract - We analyze the stability of an interbank network, in which rescues in the form of subsidized bail-ins or public bailouts can be coordinated to stop financial contagion. The coordination of a rescue consortium between a benevolent social planner and the banks is modeled as a sequential game. We show that the equilibrium welfare losses are generically unique, depending heavily on the network structure, which influences whether or not the social planner's threat to not intervene is credible. We provide conditions under which the threat is credible and characterize the optimal intervention plan.

Our analysis shows that sparsely connected networks may enhance financial stability in two ways: (i) a smaller amplification of the shock without intervention may enhance credibility of the social planner's threat and (ii) because default resolution costs are concentrated, the creditors of defaulting banks can be incentivized to make large contributions to a subsidized bail-in. This may make a sparsely connected network socially preferable over a more densely connected network, even if the densely connected network is financially more stable in the absence of any intervention.

Based on joint work with Benjamin Bernard and Joseph Stiglitz

## Sequential testing problems for bessel processes

### Peter Johnson

Thursday 16 February 2017: Room CLM.7.02, Clement House (Aldwych), 12pm to 1pm

Postdoctoral researcher, University of Manchester

Title - Sequential testing problems for bessel processes

Abstract - Consider the motion of a Brownian particle that takes place either in a two- dimensional plane or in the three-dimensional space. Given that only the distance of the particle to the origin is being observed, the problem is to detect the true dimension as soon as possible and with minimal probabilities of the wrong terminal decisions. This talk will discuss the solution to this problem in the Bayesian formulation under any prior probability of the true dimension when the passage of time is penalised linearly.

This is a nice example of tackling an optimal stopping problem for a 2-dimensional coupled Markov process. The solution uses a measure change, a stochastic time-change, Mayer and Lagrange reformulations, and allowing for negative initial times, which could help provide ideas for solving other 2-dimensional optimal stopping problems.

## Ruin probabilities: exact and asymptotic results

### Zbyszek Palmowski

***PLEASE NOTE: Change to usual day and venue***

Monday 13 February, Room COL 6.15 Leverhulme Library, Columbia House (Aldwych), 3pm to 4pm

Title - Ruin probabilities: exact and asymptotic results

Abstract - Ruin theory concerns the study of stochastic processes that represent the time evolution of the surplus of a stylized non-life insurance company. The initial goal of early researchers of the field, Lundberg (1903) and Cramer (1930), was to determine the probability for the surplus to becomenegative. In those pioneer works, the authors show that the ruin probability decreases exponentially fast to zero with initial reserve tending to infinity when the net profit condition is satisfied and clam sizes are light-tailed.
During lecture we explain when and why we can observe this phenomenon
and discuss also the heavy-tailed case.We demonstrate main techniques and results related with the asymptotics of the ruin probabilities: Pollaczek-Khinchin formula, Lundberg bounds, change of measure, Wiener-Hopf factorization, principle of one big jump and theory of scale functions of Levy processes.

## Existence of extremal diffusions matching a continuum of marginal and applications

### Julio Backhoff

Thursday 2 February 2017: Room CLM.7.02, Clement House (Aldwych), 12pm to 1pm

Title - Existence of extremal diffusions matching a continuum of marginal and applications

Abstract - Given the law of a diffusion process, we consider the problem of adjusting its drift via change of measure in a cost-optimal way so as to meet a prescribed continuum of marginals. When the cost criterion is the relative entropy, the optimizer is a singular diffusion (the so called critical Nelson process), extensively studied in the literature typically through approximation or large deviation techniques. In this talk we will consider different optimality criteria, and using convex duality as well as stochastic control techniques, obtain the existence of a singular optimal diffusion. As an application, we will discuss the link between this problem and imperfect hedging with static portfolios.This is work in progress with J. Fontbona.

## High-order filtered schemes for time-dependent second order HJB equations

### Athena Picarelli

Thursday 19 January 2017: Room CLM.7.02, Clement House (Aldwych), 12pm to 1pm

Research Fellow, Mathematical Institute, University of Oxford

Title - High-order filtered schemes for  time-dependent second order HJB equations

Abstract - In this work, we present and analyse a class of "filtered" numerical schemes for second order Hamilton-Jacobi-Bellman equations.

Our approach follows the ideas recently introduced in B.D. Froese and A.M. Oberman, "Convergent filtered schemes for the Monge-Ampère partial differential equation" (SIAM J. Numer. Anal., 2013) and more recently applied by other authors to stationary or time-dependent first order Hamilton-Jacobi equations.

For high order approximation schemes (where "high"  stands  for  greater than  one), the inevitable loss  of  monotonicity prevents  the  use  of  the classical theoretical results for convergence to viscosity solutions.

The work introduces a suitable local modification of these schemes by "filtering" them with a monotone scheme, such that they can be proven convergent and still show an overall high order behaviour for smooth enough solutions.

We give theoretical proofs of these claims and illustrate the behavior with numerical tests from mathematical finance, focusing also on the use of backward differencing formulae for constructing the high order schemes.

## Strict local martingales and initial expansions of filtrations

Thursday 24 November 2016: Room CLM.7.02, Clement House (Aldwych), 12pm to 1pm

Postdoctoral Researcher, ETH Zurich

Title - Strict local martingales and initial expansions of filtrations

Abstract - Beginning with a non negative model following a stochastic differential equation with stochastic volatility, we show how a strict local martingale might arise from a true martingale as a result of an enlargement of the underlying filtration. More precisely, we implement a particular type of enlargement, an "initial expansion" of the filtration, for various kinds of stochastic differential equation models, and we provide sufficient conditions such that this expansion can turn a martingale into a strict local martingale. Applications of our work include the modeling and detection of financial bubbles. For example, one might postulate that a bubble arises as a result of the arrival of new information, which we can model via an enlargement of the filtration.

## Conditional infimum and recovery of monotone processes

Assistant Professor of Mathematical Finance, ETH Zurich

Title - Conditional infimum and recovery of monotone processes

Abstract - Monotone processes, just like martingales, can often be reconstructed from their final values. Examples include the running maximum of supermartingales, of fractional Brownian motion, and more generally, running maxima and local times of sticky processes. An interesting corollary is that any positive local martingale can be reconstructed from its final value and its global maximum.

These results are derived from a simple no-arbitrage principle for monotone processes on certain complete lattices, analogous to the fundamental theorem of asset pricing in mathematical finance. The framework of complete lattices is sufficiently general to handle also more exotic examples, such as the process of convex hulls of multidimensional diffusions, and the process of sites visited by a random walk. The notion of conditional infimum is at the center of all of these results.

## Leveraged funds: robust replication and performance evaluation

### Paolo Guasoni

Thursday 10 November 2016: Room CLM.7.02, Clement House (Aldwych), 12pm to 1pm

Professor, School of Mathematical Sciences, Dublin City University

Title - Leveraged funds: robust replication and performance evaluation

Abstract - Leveraged and inverse exchange-traded funds and certificates seek daily returns equal to a multiple of an index' return. The trading costs implied by the frequent portfolio adjustments required create a tension between tracking error, which reflects short-term correlation with the index, and excess return, the long-term deviation from the leveraged index' performance.

With proportional trading costs, the optimal replication policy is robust to the index' dynamics. Overall fund performance depends on the implied spread, the product of tracking error and excess return, rescaled for leverage and volatility. The implied spread is insensitive to the risk premia and allows to compare funds tracking different factors of the same index.

## Systemic influences on optimal investment in stocks and credit default swaps

### Christoph Frei

Thursday 27 October 2016: Room CLM.7.02, Clement House (Aldwych), 12pm to 1pm

Associate Professor of Mathematical Finance, University of Alberta

Title - Systemic influences on optimal investment in stocks and credit default swaps

Abstract - Recent events have shown that the dependence structure of financial markets is more complex than what is captured by classical models. For example, during the 2008 financial crisis, the financial instability of some companies spread out to affect other companies.

The goal of this talk is to analyze how such systemic influences are reflected in optimal investment decisions. To this end, we introduce a model with dependence structure between market risk and default risk of the companies. An investor can use stocks and credit default swaps (CDSs) to participate in the market.

We derive an explicit expression for the optimal investment strategy in stocks and CDSs. This allows us to analyze the mechanisms driving the optimal investment decisions. We then develop a novel calibration procedure so that we can fit the model to historical time series of stock and CDS data. An empirical analysis reveals the critical role of systemic risk in portfolio monitoring.

This talk is based on joint work with Agostino Capponi (Columbia University).

## Epstein-zin stochastic differential utility: foundations, properties, and portfolio optimization

### Frank Seifried

Thursday 13 October 2016: Room CLM.7.02, Clement House (Aldwych), 12pm to 1pm
LSE maps and directions

Professor for Probability Theory, University of Trier

Title - Epstein-zin stochastic differential utility: foundations, properties, and portfolio optimization

Abstract - This talk presents some recent contributions to the theory and applications of Epstein-Zin (EZ) stochastic differential utility.

First, we provide novel results on existence, uniqueness and concavity as well as a utility gradient inequality for EZ utility in a general semimartingale setting. In the second part, I would like to address consumption-portfolio choice with EZ utility. We develop a new approach to solve such problems in a large class of incomplete market models, based on fixed point arguments and the associated FBSDE system. Finally, using an asymptotic analysis we show how small proportional transaction costs influence optimal consumption and investment decisions of an agent with EZ utility.