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Towards a general noise calculus with applications to finance science and technology

Obiettivo

This proposal has its focus on developing/broadening the white noise theory with applications to financial mathematics and stochastic partial differential equations (SPDEs), to allow for non-Gaussian or even non-Levy, processes.

The following five research topics will be considered.
1) Solution of general hyperbolic SPDEs perturbed by fractional Brownian noise. This will add greatly to the knowledge of how an important class of dynamical systems behave in a noisy environment. Creating a stochastic LP-theory, giving information on how roughness of the noise is transferred to the solution, to aid in the computation of convergence rates for numerical methods.
2) Development of numerical methods and numerical analysis for use on SPDEs perturbed by non- Gaussian noise. There are currently no methods available in this situation while the demands for practical methods from the applied scientific community are clearly seen. The results will definitely be beneficial for the scientific community involved in stochastic modelling.
3) and 4) Two extensions in minimal variance hedging. Calculating the change of value of processes, perturbed by intensity processes such as the doubly stochastic Poisson process (the Cox process), requires extension of the Malliavin calculus to non-Levy processes and the use of a Clark-Haussmann-Ocone theorem for this setting which is the goal of the first part of this section. The second has a similar aim but deals with applying the Donsker's delta function to compute hedging strategies for more general contingent claims. The key results should be explicit, easy-to-use, formulas so as to avoid the imprecise and time consuming use of Monte-Car lo simulation

5) Anticipating calculus. To deal with price dynamics of stocks, which are not adapted t o a given filtration (market information), it is essential to develop a calculus for non-adapted processes. An anticipative Ito formula for Levy processes has recently been proved. This will be used to study portfolios under non-adaptedness. The situation of non-adaptedness will occur e.g. if an insider hedges a portfolio.

Invito a presentare proposte

FP6-2002-MOBILITY-5
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Coordinatore

UNIVERSITETET I OSLO
Contributo UE
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Indirizzo
Problemveien 5-7
OSLO
Norvegia

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