Topics in Financial Mathematics and Stochastic Control

该奖项资助的项目包括随机控制和金融数学的主题集合。第一个主题是最优投资问题，其中投资者向基金经理支付一定比例的利润份额。该问题涉及到对所产生的非标准随机控制问题的建模和分析，以便充分描述最优策略。此外，还将对小比例费用进行渐近分析。第二个主题描述了金融市场的半鞅模型，其中共同基金定理在预期消费效用的背景下成立。这将为研究均衡市场中的共同基金定理提供一个工具。最后一个主题是基于对整个实线上公用事业最优投资问题的允许策略的新定义以及由此产生的对偶结果。不完全市场是指不能通过交易复制未定权益的金融模型，因此它们不是多余的。虽然在实践中，大多数模型是不完整的，但在这些市场中，最优投资和定价的数学分析通常非常困难。本项目既有助于对不同市场摩擦引起的不完全性进行建模，也有助于对由这些模型产生的新的随机控制问题进行数学分析。该项目的第二个主题预计将在一个非常一般的数学框架中提供对不完全市场的更好理解。

The project funded by this award consists of a collection of topics in Stochastic Control and Financial Mathematics. The first topic is an optimal investment problem where the investor is paying a proportional share of the profit to the fund manager. The problem involves both modeling and the analysis of the resulting non-standard stochastic control problem, in order to fully characterize the optimal policy. In addition, an asymptotic analysis for small proportional fees will be performed. The second topic represents a characterization of semimartingale models of financial markets where Mutual Fund Theorems hold true in the context of expected utility from consumption. This will provide a tool for the study of mutual fund theorems for markets in equilibrium. The last topic is based on a new definition of admissible strategies for the optimal investment problem for utilities defined on the whole real line and the duality results that derive from it.Incomplete markets are financial models where contingent claims cannot be replicated by trading, so they are not redundant. While, in practice, most models are incomplete, the mathematical analysis of optimal investment and pricing in these markets is usually very difficult. The present project contributes to both modeling of incompleteness arising from different market frictions, as well as to the mathematical analysis of the new stochastic control problems resulting from such models. The second topic of the project is expected to provide a better understanding of incomplete markets in a very general mathematical framework.