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在数学金融的理论和实践方面都具有重要的基础意义。在这个研究项目中，研究者开发了一个新的框架和方法来分析金融市场一般模型中的各种价格和对冲策略的概念。他的目标是开发一种基于效用的定价方法，用于具有嵌入式支付流的或有索赔。在一个完整的金融市场中，每一项或有债权都可以被交易证券的投资组合复制，因此允许一个唯一的无套利价格，这是复制策略的初始值。在不完全市场中，每一项或有索赔都有一个无套利价格区间（除非该索赔是可复制的）。为了克服非唯一性问题，已经开发了替代方法（基于均衡或基于效用）。本项目侧重于基于效用的定价和或有债权和收入流的套期保值问题的概率过程理论方面。该项目的目的是双重的——在包括随机效用（作为效用函数概念的自然概括）在内的设置中，调查不同的价格概念、对冲策略以及它们之间的关系，以及在金融市场的不完全连续时间模型中，为带有嵌入式支付流的或有债权构建基于效用的定价方法。为了实现上述目标，引入了一个统一的可采性框架的新概念。统一的框架允许在一个公式中考虑不同类型预算限制的潜在效用最大化问题。在技术层面上，利用半鞅一般理论、随机最优控制和凸分析等已有的新技术进行分析。该方法为基于效用的定价技术的适用范围的扩展以及基于效用的价格和对冲策略的渐近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.