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方面的一系列课题。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。该问题包括建模和分析由此产生的非标准随机控制问题，以充分表征最优策略。此外，还将对小比例费用进行渐近分析。第二个主题代表了金融市场半鞅模型的特征，其中共同基金定理在消费预期效用的背景下是正确的。这将为研究均衡市场的共同基金定理提供一个工具。最后，研究了基于整实线的公用事业最优投资问题的可接受策略的新定义及其对偶结果。不完全市场是一种金融模型，在这种模型中，或有债权无法通过交易复制，因此它们不是多余的。然而，在实践中，大多数模型是不完整的，这些市场中最优投资和定价的数学分析通常是非常困难的。本项目既有助于建立由不同市场摩擦引起的不完备性模型，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有望在非常一般的数学框架中更好地理解不完全市场。

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.