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| 5 juin 2015
An Introduction to the Benchmark Approach
Eckhard Platen (University of Technology, Sydney, Australia)
14:00-15:00, salle Extranef 126
This lecture introduces into the benchmark approach, which provides a generalized framework for financial and insurance modelling. It allows for a unified treatment of derivative pricing, portfolio optimization and risk management. It extends beyond the classical asset pricing theories, with significant differences emerging for extreme maturity contracts relevant to pensions and insurance. The Law of the Minimal Price will be presented for derivative pricing. A Naïve Diversification Theorem allows forming a proxy for the numeraire portfolio. The richer modelling framework of the benchmark approach leads to the derivation of tractable, realistic models under the real world probability measure. It will be explained how the approach differs from the classical risk neutral approach. Examples on long term and extreme maturity derivatives demonstrate the important fact that a range of contracts can be less expensively priced and hedged than suggested by classical theory.
| 2 juillet 2015
Some fractional extensions of the Poisson process
Enzo Orsingher (Sapienza University of Rome, Italy)
In this talk we present three different fractional generalizations of the classical homogeneous Poisson process. The first one is the time-fractional Poisson process where the governing difference-differential equations involve Dzerbayshan-Caputo derivatives. In this case we obtain explicit probability distributions and we show also that the time-fractional Poisson process is a time-changed classical Poisson process. We analyze also the renewal structure of the time fractional Poisson process with Mittag-Leffler distributed intertimes. Another generalization is the so called space-fractional Poisson process which is a process with independent , homogeneous increments (this is the main difference with respect to the time-fractional Poisson process). We give its explicit distribution and analyze some of its properties. A third version of the fractional Poisson process is constructed as a weighted sum of homogeneous processes and discuss some of its applications to random flights. At the end we present a large class of time-rescaled Poisson processes with independent increments and present in detail some particular cases.
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