Utility Based Pricing and Hedging in Incomplete Markets with Stochastic Preferences in a Unifying Framework of Admissibility

金融工具的定价和套期保值问题无论从理论上还是从实际应用上都具有重要的意义。 在这个研究项目中，研究者开发了一种新颖的框架和方法，用于分析金融市场一般模型中的各种价格概念和对冲策略。 他的目标是为嵌入式支付流的或有债权开发一种基于效用的定价方法。在一个完整的金融市场中，每一个未定权益都可以被交易证券的投资组合复制，因此存在唯一的无仲裁价格，这是复制策略的初始值。 在一个不完全市场中，每一个未定权益都与一个无仲裁价格区间相关联（除非该权益是可复制的）。 为了克服非唯一性的问题，已经制定了替代办法（基于均衡或效用的办法）。 本项目的重点是概率过程理论方面的问题，公用事业为基础的定价和套期保值的或有债权和收入流。 这个项目的目的是双重的-调查不同的价格概念，对冲策略，以及它们之间的关系，在设置，包括随机效用（作为效用函数概念的自然概括），和一个基于效用的定价方法的建设嵌入式支付流的或有债权，在不完整的连续时间模型的金融市场。 一个新的概念，一个统一的框架的可受理性介绍了上述目标的实现。 统一的框架允许考虑潜在的效用最大化问题与不同类型的预算约束，在一个配方。 在技术层面上，分析依赖于现有的和新的技术，半鞅，随机最优控制和凸分析的一般理论。 该方法为扩展基于公用事业的定价技术的适用范围以及基于公用事业的价格和对冲策略的渐进扩展提供了坚实的基础。

The problems of pricing and hedging of financial instruments are of fundamental importance from both theoretical and practical sides of mathematical finance. In this research project, the investigator develops a novel framework and an approach for analysis of various notions of prices and hedging strategies in general models of financial markets. He aims to develop a utility-based pricing method for contingent claims with embedded payment streams. In a complete financial market every contingent claim can be replicated by a portfolio of the traded securities and therefore admits a unique arbitrage-free price, which is an initial value of the replicating strategy. In an incomplete market, to every contingent claim is associated an interval of arbitrage-free prices (unless the claim is replicable). In order to overcome the issue of non-uniqueness, alternative (equilibrium- or utility-based) approaches have been developed. The present project focuses on the probabilistic process-theoretic aspects of the problems of utility-based pricing and hedging of contingent claims and income streams. The aim of this project is twofold -- investigation of different notions of prices, hedging strategies, and relationships between them, in the settings that include stochastic utility (as a natural generalization of the utility function concept), and a construction of a utility-based pricing methodology for the contingent claims with embedded payment streams, in incomplete continuous-time models of financial markets. A novel concept of a unified framework of admissibility is introduced for the accomplishment of the goals described above. The unified framework permits consideration of the underlying utility maximization problems with different types of budget constraints, in one formulation. On the technical level, the analysis relies on the existing and new techniques drawing on the general theory of semimartingales, stochastic optimal control, and convex analysis. The approach provides a firm ground for an extension of the scope of applicability of the utility-based pricing techniques and for asymptotic expansions of the utility-based prices and hedging strategies.