Stochastic Control under Model Uncertainty

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The problem of model uncertainty has recently received widespread attention in applied mathematics, especially after being identified as a cause for the recent credit crisis. The investigator seeks to develop probabilistic tools for stochastic control problems under uncertainty, in particular in mathematical finance and related game theoretic problems. The focus is on situations of severe model uncertainty, where no reference probability measure can be constructed and, therefore, standard tools from stochastic analysis cannot be applied. The first part of this project studies the problem of superhedging for a European, possibly path-dependent option under model uncertainty as well as the fundamental duality between trading strategies and (possibly singular) risk-neutral measures. The second part is concerned with stochastic differential games of control and stopping and, in a special case, the pricing of American options under uncertainty. The third part investigates a nonlinear version of the Doob-Meyer decomposition, which is a step towards a theory of semimartingales and stochastic integration under uncertainty. This project seeks a better mathematical understanding of model uncertainty. Its results help to resolve important questions of pricing, risk measurement, and hedging in mathematical finance. Moreover, the project leads to methodological developments in the theory of stochastic analysis and control.