Disputas: Krzysztof Jarosław Paczka
M. Sc. Krzysztof Jarosław Paczka ved Matematisk institutt vil forsvare sin avhandling for graden ph.d.:
Stochastic calculus and optimal control under model uncertainty
Krzysztof Jarosław Paczka
Tid og sted for prøveforelesning
Professor Nizar Touzi, Ecole Polytechnique
- Professor Marie-Claire Quenez, Université Denis Diderot
- Professor Tom Lindstrøm, Universitet i Oslo
Leder av disputas
Instituttleder Arne Huseby, Matematisk institutt, Universitet i Oslo
- Professor Bernt Øksendal, Matematisk institutt, Universitet i Oslo
- Professor Giulia Di Nunno, Matematisk institutt, Universitet i Oslo
The financial markets use the mathematical models to price complicated instruments or to hedge financial positions. However, these models depend on some parameters, which are unknown and need to be estimated, and that can lead to incorrect pricing or hedging. In the thesis we propose a mathematical framework to deal with such an uncertainty. Applying our framework to valuation or hedging problems gives prices and hedges that are robust, i.e. they don’t explicitly depend on the chosen parameters.
The thesis concentrates mainly on the jump models, which have three sources of uncertainty: drift (i.e. long term trend), volatility (a measure for variation of a process around its trend) and jumps (i.e. sudden changes of a process). The process, which incorporates such an uncertainty, is called a G-Levy process. We generalize some classical results (stochastic calculus, martingale decomposition etc.) to that process. Our results are the foundation for complicated models driven by a G-Levy process. These models might be used to represent, for example, a price of a stock, and using our theory one can price exotic derivatives of that stock.
We also consider models without jumps and introduce some strongly robust stochastic optimal control theory. As an application of this theory, we consider a problem of choosing an optimal portfolio of stocks and a savings account. Usually such a portfolio depends on a set of parameters. However, we have proved using out theory that in some situations we may find a portfolio, which is optimal for every possible set of parameters.
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