Some Mathematical Finance Problems Under Model Uncertainty

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

The study of financial questions under model uncertainty has recently attracted renewed attention. From a practical point of view, no single predetermined model can fully describe a complicated financial market; it is more reasonable to take into account all probabilistic models that are compatible with market data. Another reason to consider model uncertainty lies in the fact that different investors may have different beliefs or predictions about how the market would evolve, reacting differently to the same stimuli. In this research project, the investigator studies several stochastic optimization problems under model uncertainty that are related to differential and Nash equilibrium in game theory, quantile hedging, and derivative pricing with transaction costs. These studies aim to illuminate some aspects of model uncertainty, which can be applied not only to financial markets but also to other dynamical systems. Mathematically, model uncertainty is usually represented by a non-dominated set of mutually singular probabilities (a negligible set under one probability can have positive mass under another probability). The investigator analyzes four stochastic optimization problems when the uncertain evaluation criterion varies in the nondominated probability set: (1) to determine whether a robust Dynkin game has a value and admits an optimal triplet; (2) to determine whether a robust non-zero-sum game among many players has a Nash equilibrium; (3) to find an optimal strategy for a robust quantile-hedging problem; and (4) to find a no-arbitrage condition for a robust continuous-time form of the fundamental theorem of asset pricing with transaction costs. The lack of a reference probability in the nondominated probability set makes difficult the use of classic tools, such as the dominated convergence theorem and Komlos separation lemma, to analyze the nonlinear expectation associated to the probability set. This project aims to develop new methods to handle the much more complicated probabilistic situations under model uncertainty. These are expected to be useful also in the more general subjects of stochastic control and optimization.