Stochastic Representation Problems for Optimal Control and Nonlinear Models for Illiquid Financial Markets

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The project focuses on novel approaches to stochastic optimization problems arising in Mathematical Finance and Economics. A key topic is the development and analysis of new mathematical models for illiquid financial markets which account for the dependence between asset price dynamics and trading strategies, leading to new applications for the nonlinear stochastic integration theory developed by Kunita and Carmona/Nualart in the 1990s. Furthermore, we will develop new analytic and numerical methods to both one- and two-sided singular control problems, building on their relation to certain representations of stochastic processes.The vast majority of models presently studied in Mathematical Finance specify asset price dynamics as an exogenously given process, which evolves independently from the trading strategies employed by the market participants. While this idealization is appropriate for liquid markets (trading, for instance, treasury bonds or blue chips), it is largely open at present how to assess the liquidity risk in markets where asset prices are directly related to the demand generated by traders. The development and analysis of mathematical models capturing this nonlinear feedback affect between prices and trading strategies poses a challenge for stochastic optimization and control that is addressed by this project.