Optimal prediction problems driven by Lévy processes

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Optimal prediction problems driven by Lévy processes

Group Probability and Stochastic Analysis

An optimal prediction problem is a type of optimal control problem very similar to classic optimal stopping problems, however with the crucial difference that the gains process is not adapted to the filtration generated by the driving process. Consider for instance the problem of stopping a Brownian motion at a time closest (in some suitable metric) to the time the Brownian motion attains its ultimate supremum. This is an optimal prediction problem: the gains are determined by the difference between the chosen stopping time and the time at which the Brownian motion actually does attain its ultimate supremum, and as the latter quantity is not known until the whole path of the process is revealed the gains is indeed not adapted.

Besides being mathematically very appealing, optimal prediction problems have many applications, for instance in mathematical finance, insurance, medicine and engineering, to name a few. They were relatively recently introduced, and the bulk of the work done so far is for diffusions only. In this project, several types of optimal prediction problems driven by Lévy processes rather than diffusions will be studied. Lévy processes play a role in mathematical finance as an extension of the classic Black & Scholes model and in insurance as an extension of the classic Cramer-Lundberg model for instance.

Note that this project requires a very strong background in probability theory, in particular stochastic processes, and mathematical analysis. Preferably the student should already be familiar with Lévy processes. Experience with computer programming and implementing numerical schemes is very helpful.

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