# Stochastic Analysis in Finance and Insurance

### Nizar Touzi

Ecole Polytechnique, Palaiseau, France### Dmitry Kramkov

Carnegie Mellon University, Pittsburgh, United States### Martin Schweizer

ETH Zürich, Switzerland

## Abstract

The workshop \emph{Stochastic Analysis in Finance and Insurance}, organised by Dmitry Kramkov (Pittsburgh), Martin Schweizer (Z\"{u}rich) and Nizar Touzi (Paris) was held January 27th -- February 2nd, 2008. The meeting had a total of 44 participants from all over the world with a good blend of more experienced researchers and many younger participants.

During the five days, there were a total of 29 talks with many lively interactions and discussions. The organisers had to exercise some constraint on the participants in order not to overload the programme, and this prompted many discussions and collaborations during the long lunch breaks and in the evenings.

The topics presented in the talks covered a very wide spectrum. There were some major areas with several talks as well as other more individual contributions pointing towards new developments in mathematical finance. The overall tendency went towards more sophisticated and more realistic models of financial markets; the first generation models with frictionless classical semimartingale prices seem largely understood, and one of the major trends is now towards what might be termed second generation modelling. This includes as major topics \emph{transaction costs} and \emph{large investors and liquidity issues} as well as other nonstandard models or ideas related to such developments. A second major topic revolved around \emph{risk measures or monetary utility functions}, and there were several talks on \emph{option pricing}, on \emph{optimisation problems from finance} and on \emph{credit risk}.

We now give a short overview of the topics covered in the talks, roughly ordered into the themes listed above.

\emph{Transaction costs:} The classical Merton problem of optimal investment under transaction costs was reconsidered by \emph{Jan Kallsen} who presented a new approach via shadow prices leading to a simpler way of finding the optimal strategy. \emph{Yuri Kabanov} extended the classical hedging theorem under transaction costs from European to the case of American contingent claims.

\emph{Large investors and liquidity issues:} A model for the optimal liquidation of a large portfolio position was presented by \emph{Alexander Schied}; his results showed that the market impact of such a trader can lead to some unexpected effects. \emph{Mete Soner} studied the problem of superreplicating options in an illiquid market by means of PDE and stochastic control techniques. \emph{Thorsten Rheinl\"ander} developped a new model for utility maximisation by a large trader and showed how this can be modelled via nonlinear stochastic integration theory. A partial equilibrium model for a large investor interacting with other market participants was presented by \emph{Peter Bank}, who emphasised the importance of appropriate financial modelling of gains from trade in continuous time.

\emph{Risk measures or monetary utility functions:} This topic had the largest number of talks. \emph{Michael Kupper} introduced divergence utilities and showed that this fairly large class can be very well manipulated and leads to explicit solutions for optimisation and risk sharing problems. \emph{Damir Filipovi\'c} studied extensions of convex risk measures from the space of all bounded random variables and showed that under law-invariance, the canonical model space consists even of all integrable random variables. \emph{Walter Schachermayer} proved a very new result: He showed that the only time-consistent law-invariant dynamic convex risk measure is the entropic risk measure. \emph{Freddy Delbaen} gave a detailed study of the representation for the penalty function of time-consistent dynamic monetary utility functions with the help of backward stochastic differential equations. Another extension of risk measures to Orlicz hearts was studied by \emph{Patrick Cheridito}, and \emph{Gordan ${\it\check Z}\!$itkovi\'c} introduced maturity-independent risk measures and pointed out some nontrivial existence problems related to this concept.

\emph{Option pricing:} For a class of stochastic volatility models, \emph{David Hobson} showed how to obtain option price comparisons by means of time changes and other purely probabilistic arguments. \emph{Ludger R\"uschendorf} gave a broad overview of methods to obtain comparison results for option prices in large classes of processes and showed several techniques to achieve this goal. \emph{Semyon Malamud} presented a new approach for deriving indifference prices for contingent claims under power utility in discrete-time models having a certain new structure, and pointed out several appealing properties of this class of models. \emph{Vicky Henderson} considered perpetual American options in an incomplete market and determined the optimal exercise strategy when the number of options one owns is infinitely divisible.

\emph{Optimisation problems from finance:} An overview and some new developments for risk-sensitive portfolio optimisation were presented by \emph{Jun Sekine}. \emph{Paolo Guasoni} studied the problem of finding optimal portfolios and risk premia explicitly in the limit of a long time horizon. \emph{Bruno Bouchard} started with quantile hedging and related problems and embedded these into a general stochastic target problem with controlled probability or controlled losses.

\emph{Credit risk:} Valuation of credit-sensitive instruments often involves first passage times, and \emph{Tom Hurd} presented new ideas on how these can be handled more explicitly for a fairly large class of jump-diffusion processes. \emph{Ronnie Sircar} gave an asymptotic analysis of multiscale models for multiname credit risk derivatives and illustrated that his approach leads to computationally tractable and yet fairly accurate results when calibrated to market data.

In addition to the above roughly thematically grouped talks, there were presentations that did not fall readily into a particular area; this illustrates the diversity and multiple facets of the field of mathematical finance. \emph{Yannis Karatzas} started the workshop with a very stimulating talk on so-called diverse financial markets and the idea of finding there optimal arbitrage strategies. \emph{Thaleia Zariphopoulou} presented a new way to look at performance measurement in financial markets and formulated this as a novel and intriguing mathematical problem involving a stochastic partial differential equation. \emph{Jak$\it\check s\!$a Cvitani\'c} gave an overview of recent developments and results on contract theory in continuous time. Motivated by the question of how to model the influence of information on financial markets, \emph{Kostas Kardaras} introduced a topology on $\sigma$-fields and on filtrations and presented some first continuity results. \emph{Josef Teichmann }explained how one can compute moments of affine processes in a very easy way; this was motivated by many examples arising in the valuation of financial derivatives. With the goal of modelling both stock prices and the infinite family of all call options in a joint model, \emph{Ren\'e Carmona} studied dynamic local volatility models and derived the corresponding drift restrictions arising from absence of arbitrage. Another very thought-provoking talk was given by \emph{Denis Talay} who presented some mathematical models and problems connected to so-called technical analysis in financial markets. \emph{Mihai S\^\i rbu} considered a general semimartingale model and gave some necessary and some sufficient conditions for the validity of a mutual fund theorem.

In contrast to the picture shown on the institute homepage, there was no snow around the institute, as remarked (and regretted) by several participants. On the other hand, this allowed on Wednesday a very pleasant excursion to St.~Roman with a very good participation rate.

For us as the organisers, it was (as always) a great pleasure to be at Oberwolfach and to benefit from the excellent infrastructure, support and scientific environment. We thank the Mathematisches Forschungsinstitut Oberwolfach for making this possible, and we are very happy to report that this sentiment was also expressed by all the participants both during and after the conference in many ways. The idea of having a similar workshop in about three years met with very enthusiastic reactions.

## Cite this article

Nizar Touzi, Dmitry Kramkov, Martin Schweizer, Stochastic Analysis in Finance and Insurance. Oberwolfach Rep. 5 (2008), no. 1, pp. 173–240

DOI 10.4171/OWR/2008/04