# Recent Developments in Financial and Insurance Mathematics and the Interplay with the Industry

### Soren Asmussen

University of Aarhus, Denmark### Nicole Bäuerle

Universität Karlsruhe, Germany### Ralf Korn

Universität Kaiserslautern, Germany

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## Abstract

The workshop \emph{Recent Developments in Financial and Insurance Mathematics and the Interplay with the Industry}, organised by S{\o}ren Asmussen (Aarhus), Nicole B\"{a}uerle (Karlsruhe) and Ralf Korn (Kaiserslautern) was held February 18th--February 24th, 2006. The participants came from all over the world, including Hong Kong and the US. The workshop was attended by 25 mathematicians, most of them were leading experts in mathematical finance and actuarial mathematics and a few researchers from the industry. In total, there were 21 talks distributed over five days with a long lunch break between 12:30 and 16:00 which made it possible to discuss the latest results and open problems posed in the morning sessions. The major themes of the talks had been \begin{itemize} \item Optimization problems in finance and insurance \item Multidimensional modeling in finance and insurance \item Valuation of insurance products, credit risk and electricity markets \item Risk measures and distributions with heavy tails. \end{itemize} We proceed with a brief overview of the four subjects above and a summary of the principal results of those areas that were reported on in the workshop: \noindent {\em Optimization problems in finance and insurance:} Quite a large number of the talks investigated stochastic dynamic portfolio problems in finance as well as in insurance. {\em Tomas Bj\"ork} who gave the first talk considered a general investment problem where the local rate of return is unobservable. Using non-linear filtering techniques and the martingale method he derived fairly explicit results. {\em Thaleia Zariphopoulou} introduced a new way to quantify performance measurement in asset allocation. Motivated by the martingale optimality principle she showed how optimal portfolios in this setup can be obtained by solving a fast diffusion equation posed inversely in time. {\em Ralf Korn} gave a review on old and new results for worst-case portfolio problems. He reported that he now found a HJB-system linking the indifference approach to classical stochastic control theory. {\em Martin Schweizer's} talk was on utility indifference valuation of contingent claims in an incomplete market driven by two Brownian motions where the traded and non-traded assets are stochastically correlated. He showed explicit formulas for the indifference value of in case of exponential utility. {\em Xin Guo} established a new theoretical connection between singular control of finite variation and optimal switching problems. This approach provides a novel method for solving explicitly high-dimensional singular control problems. {\em Jostein Paulsen} considered two optimal dividend problems with and without reinvestment (which could prevent from ruin) and where dividend payments and reinvestment are subject to fixed and proportional cost. He gave an explicit solution in case the expected discounted payout minus reinvestment has to be maximized. Finally {\em Mogens Steffensen} studied optimal consumption and insurance payment streams in a multistate Markovian framework seen from the individuals's perspective. He derived a general solution in case of power utility maximization. \noindent {\em Multidimensional modeling in finance and insurance:} It is obvious from the workshop that multidimensional (correlated) stochastic processes have become a very important topic recently. In particular as far as L\'evy processes are concerned.In his talk, {\em Filip Lindskog} extended (under some conditions) the classical Cram\'er-Wold results on projections and convergence of probability measures to measures with a singularity. This result can also be applied to L\'evy processes. {\em Thomas Mikosch} showed how to combine functional regular variation with heavy-tailed large deviations for partial sums. As an application he derived asymptotic results for ruin probabilities of a multivariate random walk with regularly varying step sizes. {\em Claudia Kl\"uppelberg} presented a multivariate model for operational risk processes. She discussed in particular the influence of the L\'evy copula on the Value-at-risk of the summed process. Finally, {\em Nicole B\"auerle} showed how to characterize dependence properties and comparison results for L\'evy processes with the help of the L\'evy measure and the L\'evy copula. Some applications in insurance and finance were reported. \noindent {\em Valuation of insurance products, credit risk and electricity markets:} Several talks demonstrated that the mixture between financial and actuarial aspects of problems still is increasing. {\em Thomas M{\o}ller} told the audience that a new market for so-called mortality derivatives is now appearing due to the systematic risk in life insurance portfolios. He showed how insurers can reduce their risk by trading e.g. survivor swaps. {\em Andrew Cairns} discussed some new stochastic models for mortality, in particular the Olivier-Smith model which borrows ideas from interest rate modeling. {\em Hailiang Yang} investigated the valuation of insurance liabilities of equity-indexed annuities and participating life insurance policies where the equity price process is given by a Markov regime switching model. {\em Alfred M\"uller} reported on challenges in modeling electricity price processes and introduced a three-factor model for the spot market price which captures seasonal effects. {\em Uwe Schmock} presented a one-period model for dependent risks which generalizes both the standard collective risk model and CreditRisk and where the portfolio loss distribution can be computed with a numerically stable algorithm. {\em Holger Kraft} introduced a unified framework for modeling credit risks with bankruptcy and contagion and showed how to compute prices of derivatives in the setup. \noindent {\em Risk measures and distributions with heavy tails:} It is clear from the workshop that heavy tails still are an important topic as far as applications are concerned. {\em Christian Hipp} explained in his talk how third order asymptotic expansions for the tail probability of -fold convolutions of claim sizes with regular varying tails can be computed. He ended with some conjectures about Weibull type and Lognormal type claim sizes. {\em Jens Perch Nielsen} talked about new approaches to regression which can determine the full distribution and where errors can have heavy tails. {\em Hansj\"org Furrer} presented the evolution of the regulatory framework from Solvency 0 to Solvency II and posed some open questions concerning the risk measurement in a multi-period framework. {\em Dirk Tasche} investigated in his talk how kernel estimation can be combined with importance sampling to obtain efficient estimations of Value-at-risk contributions. \bigskip In addition to the excellent scientific program, there were two scheduled social activities: Due to the splendid weather, the traditional hike to St.\ Roman on Wednesday afternoon was a true pleasure despite the muddy short-cut at the beginning. On Thursday night there was a piano concert given by Mogens Steffensen with a lot of entertaining Elton John songs. For some of the participants this was their first trip to Oberwolfach and they were very impressed by this experience. We, the organizers, would like to thank the ''Mathematisches Forschungsinstitut Oberwolfach'' for providing such an excellent environment and for the technical support. The participants encouraged the idea of organizing a similar workshop in about three years. \hfill S{\o}ren Asmussen \hfill Nicole B\"{a}uerle \hfill Ralf Korn