Abstracts

## MinicoursesNizar Touzi:Hedging under constraints by face lifting and optimal stopping
We first consider the classical problem of super-replication under portfolio constraints as formulated by Cvitanic and Karatzas (1995). In the Markov framework, the value function can be characterized as the unique viscosity solution of the associated dynamic programming equation without any need to the so-called dual formulation of the problem. In the context of the Black-Scholes model, we only use the viscosity super-solution part of this result in order to derive an explicit solution of the problem. Namely, the value function is given by the (unconstrained) Black-Scholes price of a conveniently face-lifted payoff. This result was first established by Broadie, Cvitanic and Soner (1998). We next formulate a super-replication problem under Gamma constraints. This leads to a non-standard stochastic control problem which appeals to many different tools from stochastic analysis and stochastic control theory. The final result is that the value function of the super-replication problem is given by the value of an optimal stopping problem (American option) with reward defined by a convenient face-lift of the payoff. References[1] Hedging under constraints by face lifting and optimal stopping (slides of the lectures).
Hanspeter Schmidli:
## Special invited lecturesDamir Filipovic:Risk-based solvency testing for insurers
We discuss recent developments in risk-based insurance regulation. The European Solvency II framework proposes three pillars, the first of which is about risk-bearing capital requirements. A particular focus of this talk is on the Swiss Solvency Test for insurers. References[1] Risk-Based Solvency Testing for Insurers (slides of the lecture).
Marco Frittelli:
Farshid Jamshidian: ## Short lectures
Bart Oldenkamp:
Sophie Ladoucette:
The ECOMOR treaty t) is defined via the upper order statistics of a random sample.
In some sense it rephrases the largest claims treaty, another reinsurance treaty of
extreme value type. But it can also be considered as an excess-of-loss treaty with
a random retention determined by the (r + 1)th largest claim related to a specific
portfolio. Specifically, the reinsurer covers the claim amount:
t) = Σ
_{1≤i≤r}
X^{*}_{N (t)-i+1 }-
rX^{*}_{N (t)-r }
for a fixed number t) is then a function of the r+1 upper order
statistics X^{*}_{N (t)-r }≤...≤
X^{*}_{N (t)}
in a randomly indexed sample X_{1},...,X_{N(t)} of i.i.d.
claims which occur up to time epoch t ≥ 0. These claims determine the accumulated
claim amount in the random sum S :=
Σ_{N(t)}_{1≤i≤N(t)}
X. Throughout, we assume the claim
number process {_{i}N(t); t ≥ 0} to be a counting process independent of the claim size
process {X, _{i}i∈N^{*}}.
In a first part, we are interested in the asymptotic relation between the tail of the
distribution t). We get accurate
asymptotic equivalences and asymptotic bounds. Also, in the sub-exponential case, we
give a result showing the interplay between the accumulated claim amount S and the
reinsured quantity _{N(t)}R(_{r}t). In a second part, we turn to the ratio of the quantities R(_{r}t)
and the accumulated claim amount S. We give conditions that imply a dominant
influence of the quantities _{N(t)}R(_{r}t) on this sum. Finally, we touch on the question of
convergence in distribution for some quantities R(_{r}t). We get precise first and second
order large deviation results for the case where the claim size distribution F belongs
to an extremal class, eventually with remainder. The outcomes are illustrated with a
number of simulations. (joint work with J.L. Teugels)
References[1] Ladoucette S.A., Teugels J.L. (2004): Reinsurance of large claims, EURANDOM Report 2004-025 (abstract, text), Technical University of Eindhoven, The Netherlands.
David Schrager:
Alex Zilber: |

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