# Talks in Financial and Insurance Mathematics

This is the regular weekly research seminar on Insurance Mathematics and Stochastic Finance.

## Spring Semester 2011

Note: The highlighted event marks the next occurring event and events marked with an asterisk (*) indicate that the time and/or location are different from the usual time and/or location.

Date / Time Speaker Title Location
* 23 February 2011
17:15-18:15
Johannes Temme
University of Vienna
HG G 43
Abstract: Extending the work of Philip Dybvig and Yajun Wang, we consider a more risk averse agent M, a less risk averse agent L and their optimal terminal payoff X^M, X^L, respectively, for the problem of maximizing the expected utility from terminal wealth. In complete markets one can prove that X^L stochastically dominates X^M, which in turn is equivalent to X^L being distributed as X^M plus a nonnegative risk premium'' plus a noise term. This result does in general not hold true in incomplete markets, but we can prove that it follows for power utilities in stochastic volatility models of one bond and one stock when the stochastic volatility is independent of the the semimartingale driving the stock. We will also see that stochastic dominance is a very fragile property in the sense that any n-period model - no matter if complete or incomplete - can be turned by slight perturbation into an incomplete n-period model for which the stochastic dominance result fails.
24 February 2011
17:15-18:15
Antoon Pelsser
University of Maastricht
HG G 43
Abstract: Apply ideas of robustness and model uncertainty in a context of pricing derivative contracts in complete and incomplete markets. We will focus on the (simple) case with uncertainty in mean only. First, we show that in a complete market, an agent worried about model uncertainty will choose the replicating portfolio as this will eliminate the model uncertainty completely. Hence, a perfectly rational agent that is facing model uncertainty will price risks using no-arbitrage. Second, we show that in an incomplete market the agent will hedge as much of the risk as possible and will choose a market-consistent pricing operator. We apply these idea to the setting of pension funds and insurance companies: in particular the pricing of long-dated cash-flows and the pricing of longevity risk.
3 March 2011
17:15-18:15
Bruno Bouchard
Centre de Recherche en Mathématiques de la Décision, Paris, France
HG G 43
Abstract: We will discuss recent advances in the field of stochastic target problems and their application in the management of risk in finance. After reviewing various type of applications, we will focus on the problem of pricing a risk under a P&L constraint. Namely, we will show how we can provide a direct PDE characterization for the minimal initial endowment required so that the terminal wealth of a financial agent (possibly diminished by the payoff of a random claim) can match a set of constraints in probability. Such constraints should be interpreted as a rough description of a targeted profit and loss distribution. From the mathematical point of view, this is an extension of the stochastic target problem under controlled loss, studied in Bouchard, Elie and Touzi (2009), to the case of multiple constraints.
10 March 2011
17:15-18:15
Prof. Dr. Dirk Helbing
ETH Zürich
HG G 43
Abstract: In order to understand social systems, it is essential to identify the circumstances under which individuals spontaneously start cooperating or developing shared behaviors, norms, and culture. In this connection, it is important to study the role of social mechanisms such as repeated interactions, group selection, network formation, costly punishment and group pressure, and how they allow to transform social dilemmas into interactive situations that promote the social system. Furthermore, it is interesting to study the role that social inequality, the protection of private property, or the on-going globalization play for the resulting "character" of a social system (cooperative or not). It is well-known that social cooperation can suddenly break down, giving rise to poverty or conflict. The decline of high cultures and the outbreak of civil wars or revolutions are well-known examples. The more suprising is it that one can develop an integrated game-theoretical description of phenomena as different as the outbreak and breakdown of cooperation, the formation of norms or subcultures, and the occurence of conflicts.
17 March 2011
17:15-18:15
Fred Espen Benth
University of Oslo
HG G 43
Abstract: We introduce ambit processes as a flexible class of models for prices in energy markets. Both spot and forward models are presented. The focus is on modeling issues and analytical properties of the ambit processes, which in general are non-semimartingales. Some initial empirical studies of these models applied to the German EEX market are presented.
* 25 March 2011
14:15-15:15
Jaksa Cvitanic
California Institute of Technology
HG G 19.2
Abstract: Consider a "principal" who pays an "agent" to run a "project" whose value is modeled as a diffusion process controlled by the agent. Contract theory considers the problem of maximizing over contract payments the expected utility of the principal, while providing a required expected utility to the agent. We survey some of the methods and results on this question obtained in the last decade, mostly focusing on the case when the agent is paid only once, at the termination date of the project. We will discuss the following frameworks: (i) the principal and the agent have the same, full information; (ii) the principal cannot contract upon agent's actions (iii) the principal does not know agent's type (iv) the time of payment may be random We identify under which conditions it is optimal to pay with linear contracts (cash plus stock shares), in which it is optimal to pay by nonlinear contracts (such as options), and in which also a random benchmark value has to be used in compensation. The mathematical approach we apply is that of Stochastic Maximum Principle and Forward Backward Stochastic Differential Equations.
31 March 2011
17:15-18:15
Mark Rüegg
CelsiusPro
HG G 43
Abstract: Businesses worldwide are for an important part directly or indirectly dependent on atmospheric conditions. Companies can protect themselves against the financial consequences of unfavorable weather conditions by purchasing weather risk transfer products based on parametric triggers. Weather derivatives provide compensation for loss of revenue or additional costs. A short introduction about climate variability will be followed by an analysis of weather derivative structures and hedging strategies. Examples of how weather influences company results will illustrate that weather derivatives can benefit many industries, including the energy sector, agriculture, construction & retail.
* 7 April 2011
14:45-15:45
Prof. Dr. Walter Schachermayer
University of Vienna
HG G 19.1
Abstract: We give an elementary proof of the celebrated Bichteler-Dellacherie Theorem which states that the class of stochastic processes S allowing for a useful integration theory consists precisely of those processes which can be written in the form S = M + A, where M is a local martingale and A is a fi nite variation process. In other words, S is a good integrator if and only if it is a semi-martingale. We obtain this decomposition rather directly from an elementary discrete-time Doob-Meyer decomposition. By passing to convex combinations we obtain a direct construction of the continuous time decomposition, which then yields the desired decomposition. As a by-product of our proof we obtain a characterization of semi-martingales in terms of a variant of no free lunch, thus extending a result by Delbaen and Schachermayer (1994)
7 April 2011
17:15-18:15
Nicole El Karoui
Paris
HG G 43
Abstract: Longevity risk is a growing risk across the developed world as populations age, with a post-retirement life expectancy growing rapidly (by around 2 months a year for males and 1.5 months a year for females). Associated with this growth in an older population is a social need to develop products to allow individuals to secure lifetime income, and a business need to attract capital to support this risk class. For many underlying risks such as changes in interest rates, credit spreads, commodity prices or FX rates, financial markets provide derivative solutions for transferring risk. However, the risk of changing mortality rates is more difficult to deal with as appropriate risk management tools barely exist. Life insurances and pension funds are the most obvious, but not exclusive entities being exposed to unexpected changes in mortality. In particular, stochastic methods are necessary to assess a company's capital relief when hedging part of their longevity risk exposure. Among the many standard stochastic models for mortality, a number have been inspired by the classical credit risk and interest rate literature, as the so-called forward mortality models, which infer dynamics on the entire age/term-structure of mortality. However, these models fall to capture the so-called basic risk, due to the di erence between the exposed population, (the pensioner ar annuitant population) and the national population. An alternative is the microscopic modelling approach, which can be used for populations where individuals are characterised not only by age, but also by additional indicators that are reflective of lifestyle and living conditions. Inspired by works of Meleard, Tran, Ferriere, we propose a stochastic individual-centered particle model in continuous time, where the individuals reproduce, age, interact and die. The 'trait' is an individual heritable property (d-dimensional vector) that may influence birth and death rates and interactions between individuals, and vary by mutation. Environmental stochastic factors also influence the evolution. Simulation and asymptotic analysis provide information on the macro longevity risk. Combined with studies on demographic rates, such as fertility and immigration estimated on national population, microscopic modelling is applicable at social and political levels, o ffering guidance for strategic decision concerning, for example, immigration and retirement age policy. Such models can provide useful benefi ts for the risk analysis of a given insurance portfolio, as shown by the study of bene ts to be paid in the case of real portfolio of insured. An other application is the structuring of fi nancial products allowing to insurers to transfer only their interest rate risk in fi nancial markets.
14 April 2011
17:15-18:15
Dr. Albina Danilova
LSE
HG G 43
Abstract: Abstract: Given a deterministically time-changed Brownian motion Z starting from 1, whose time-change V (t) satisfies V (t) > t for all t\geq 0, we perform an explicit construction of a process X which is Brownian motion in its own filtration and that hits zero for the first time at V (s), where s := inf {t > 0 : Z_t = 0}. We also provide the semimartingale decomposition of X under the filtration jointly generated by X and Z. Our construction relies on a combination of enlargement of filtration and filtering techniques. The resulting process X may be viewed as the analogue of a 3-dimensional Bessel bridge starting from 1 at time 0 and ending at 0 at the random time V (s). We call this a dynamic Bessel bridge since V (s) is not known in advance. Our study is motivated by insider trading models with default risk. (this is a joint work with Luciano Campi and Umut Cetin)
* 5 May 2011
14:45-15:45
Prof. Dr. Terry Lyons
University of Oxford
HG G 19.1
Abstract: How can one describe a probability measure of paths? And how should one approximate to this measure so as to capture the effect of this randomly evolving system. Markovian measures were efficiently describes by Strook and Varadhan through the Martingale problem. But there are many measures on paths that are not Markovian and a new tool, the expected signature provides a systematic ways of describing such measures in terms of their effects. We explain how to calculate this expected signature in the case of the measure on paths corresponding to a Brownian motion started at a point x in the open set and run until it leaves the same set. A completely new (at least to the speaker) PDE is needed to characterise this expected signature. Joint work with Ni Hao.
5 May 2011
17:15-18:15
Prof. Dr. Fabrizio Durante
Free University of Bozen-Bolzano
HG G 43
Abstract: In a seminal paper (AOS, 1978), Kimeldorf and Sampson showed that any joint distribution function of a random pair (X,Y) can be approximated by the joint distribution function of another such pair, say (U,V ), where each of U, V is now almost surely an invertible function of the other. In the present talk, we aim at revisiting these results by using a measure-theoretic approach. Firstly, we present a different proof of Kimeldorf and Sampson's result based on ergodic techniques. Then we use the previous idea in order to introduce and study a new class of multivariate dependence structures, named shuffles of copulas. The presented investigations are also used in order to approximate copulas by means of elements from a given basis set.
12 May 2011
17:15-18:15
Ronnie Sircar
Princeton University
HG G 43
Abstract: The problem of diminishing oil reserves from known sources is of long-standing importance; at the same time, there were over thirty new discoveries during 2009. We study the stochastic effect of resource exploration in dynamic Cournot game-theoretic models of exhaustible resources. We first treat the case of a monopolist who may undertake costly exploration to replenish his diminishing reserves. We then consider a stochastic game between such an exhaustible producer and a green'' producer that has access to an inexhaustible but relatively expensive source, such as solar power. The effort control variable is taken to be either continuous or discrete (switching control). In both settings, we assume that new discoveries occur according to a jump process with intensity given by the exploration effort. This leads to a study of systems of nonlinear first order delay ODEs. We derive asymptotic expansions for the case of a small exploration success rate and present some numerical investigations. Joint work with Michael Ludkovski (UC Santa Barbara).
* 19 May 2011
14:15-15:15
Enno Veerman
University of Amsterdam
HG G 19.1
Abstract: In this talk we consider affine processes on a general state space. We show that an affine process on an arbitrary state space is a Feller process and characterize it as the solution of a martingale problem with affine parameters. Existence of an affine process is established using the positive maximum-principle. For this we deduce necessary and sufficient parameter restrictions, so-called admissibility conditions, which we obtain in a rather general form. We work these out more explicitly for a general polyhedral state space and for a general quadratic state space, including the parabolic state space and the Lorentz cone. (joint work with Peter Spreij)
19 May 2011
17:15-18:15
Enkelejd Hashorva
University of Lausanne
HG G 43
Abstract: We motivate this talk by first presenting diverse practical problems from insurance and finance related to conditional excess distributions and aggregation of multivariate risks. Based on available results and methodology we suggest to model the multivariate risk dependence, the key model specification, by some elliptically symmetric copula. We derive explicit asymptotic solutions for approximation of (joint) conditional excess distribution and aggregated risks by addressing the tail asymptotic behaviour of such elliptical risks. We shall further discuss related problems in the framework of Brownian Motion and Brownian Pillow.
* 26 May 2011
16:15-17:15
Maxim Bichuch
Princeton University
HG G 19.2
Abstract: Abstract: We consider an agent who seeks to optimally invest and consume in the presence of proportional transaction costs. The agent can invest in two types of futures contracts, and in a money market account. She may also consume and get utility $U(c)\stackrel{\triangle}=}\frac{c^p}{p},~ c\ge 0$, where $p\in(0,1)$ and $c$ is the rate of consumption. The agent can control the rate of consumption and influence the evolution of wealth by controlling the number of futures contracts held. Proportional transaction costs $\lambda_i=\alpha_i\lambda$ are charged for every trade in future contracts of type $i,~i=1,2$. The agent wishes to maximize the expected discounted integral over $[0,\infty)$ of the utility of consumption. We compute an asymptotic expansion of the value function in powers of $\lambda^{\frac13}.$ This is a joint work with Steven E. Shreve.
26 May 2011
17:15-18:15
Jens Perch Nielsen
Cass Business School, City University London, und Festina Lente
HG G 43
Abstract: Reserving in general insurance is often done using chain-ladder-type methods. We propose a method aimed at situations where there is a sudden change in the economic environment affecting the policies for all accident years in the reserving triangle. It is shown that methods for forecasting nonstationary time series are helpful. We illustrate the method using data published in Barnett and Zehnwirth (2000). These data illustrate features we also found in data from the general insurer RSA.
9 June 2011
17:15-18:15
Dr. Thorsten Rheinländer
LSE
HG G 19.1
Abstract: Recently semi-static hedging of barrier options has received much attention since it allows to replicate the payoff of a path-dependent option by purchasing a non-vanilla European option at the hitting time of the barrier. This method, however, requires a certain quasi self-dual (QSD) property of the underlying asset price process to hold. We aim to characterize QSD processes in the framework of both continuous martingales as well as exponential Lévy processes. In the former case, in particular we extend recent work by M. Tehranchi to characterize Ocone martingales in terms of certain associated stochastic exponentials. In the latter case, we derive new conditions for QSD building up on work by I. Molchanov and M. Schmutz, and show that QSD holds for many popular classes of Lévy models. This is joint work with Michael Schmutz, Universität zu Bern.
* 23 June 2011
16:15-17:15
Prof. Dr. Zoltan Sasvari
TU Dresden
HG G 19.1
Abstract: In the present talk we will speak about some selected topics from the theory of positive definite functions, their generalizations and their applications to probability and statistics. The topics include the extension problem, covariance functions of stationary processes, functions with a finite number of negative squares and their connection to intrinsically stationary processes.
23 June 2011
17:15-18:15
Goran Peskir
University of Manchester
HG G 19.1
Abstract: We show that in the absence of any information about the hidden' target in terms of the observed sample path, and irrespectively of the distribution law of the observed process, the median' rule is optimal in both the space domain and the time domain. While the fact that the median rule minimises the spatial expectation can be seen as a direct extension of the well-known median characterisation dating back to R. J. Boscovich, the fact that this also holds for the temporal expectation seems to have stayed unnoticed until now. We will present a review of the recent results building on this observation. The talk is dedicated to the tercentenary of the birth of R. J. Boscovich (1711-1789).
* 29 June 2011
15:15-16:15
Prof. Dr. Thorsten Schmidt
TU Chemnitz
HG G 19.1
Abstract: We consider defaultable models under the additional information given by ratings. The proposed framework generalizes existing models available in the literature and studies the implications which can be drawn purely from the assumption about absence of arbitrage. A given example is based on a filtering approach with incomplete information. This is joint work with Jacek Jakubowski and Mariusz Nieweglowski.

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