Fractional Levy models for credit risk contagion. White noise concepts applied to strong solutions of stochastic equations.

The project proposal of this application can be split into two different pillars: The aim of the first pillar is to develop a suitable stochastic calculus for fractional Levy processes and to apply this to introduce long range dependence in the modelling of credit contagion. An important goal in the new regulatory proposal Basle II is the determination of an appropriate capital reserve for credit portfolios and, consequently, the quantification of credit risk of a given portfolio. For credit risk modellin g it is very important to understand the contagion mechanism, i.e. the propagation of economic distress from one firm to another. The aim of this project is to address this problem in a new financial setting and to introduce new appropriate mathematical t ools and structures. The second pillar deals with the application of white noise theory for Levy processes to strong solutions of S(I)(P)DE driven by Levy processes. In some recent works, white noise techniques have been employed to explicitly represent s trong solutions of stochastic differential equations driven by Levy processes. The methodology to solve S(I)(P)DE's using white noise techniques is absolutely new and innovative. Considering the explicit solution formula several very interesting questions naturally arise for future research, the most important ones being sufficient conditions for strong solutions of S(I)(P)DE with irregular coefficients and the study of strong solutions of backward S(I)DE's. The study of strong solutions of S(I)(P)DE's wi th irregular coefficients is of great interest also for a number of applications, for example in non-linear filtering. We aim at extending and solving the non-linear filtering setting for processes including jumps, and to apply this filtering setting in o rder to calibrate spot price models on electricity markets, which is of great interest for the active Norwegian electricity market.