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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.

The 2017-18 Joint Risk and Stochastics and Financial Mathematics Seminar Series starts in October 2017.

**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.**

Thursday 1 March 2018

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

Title & Abstract to be confirmed

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*.

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).

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.

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.

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.

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.

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.

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.

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.

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).

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.

Thursday 14 September 2017

OLD 2.22, Old Building 12pm to 1pm

Chiara Donnini**, **Assistant 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.

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

**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.

**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.

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.

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

Chair of Stochastic Analysis and Financial Mathematics, Humboldt-Universität zu Berlin

**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.

*****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

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.

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

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

Professor, Department of Applied Mathematics,

Wrocław University of Science and Technology

**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.

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

Postdoctoral Researcher, Faculty of Mathematics,Technical University of Vienna

**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.

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.

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.

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.

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.

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).

Thursday 13 October 2016: Room CLM.7.02, Clement House (Aldwych), 12pm to 1pmLSE 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.

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