Topics in Mathematical Finance and Stochastic Control

本文考虑了不完全市场下最优投资问题中价值函数的二次可微性，以及效用函数定义在整个实线上的情况下基于效用的价格的灵敏度分析。研究的重点是数学陈述成立的充要条件。特别是，我们正在寻找可以在这个框架内分析的最大类别的或有债权。事实证明，风险容忍度财富过程是定价的合适数字。此外，还研究了一类多维随机对策，在完全市场中，价格是由“无套利”考虑唯一定义的。在不完整的更现实的模型中，或有债权的价格只能在考虑到特定投资者的偏好和财富的情况下确定。研究了可以从货币市场(无限信用额度)借入任意数量现金的投资者的价格对或有债权数量的依赖关系。定价技术可以应用于大量不完整的模型，包括非交易资产的期权和能源衍生品。

In this project, the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets and the sensitivity analysis of utility-based prices for the case of utility functions defined on the whole real line is considered. The research focuses on necessary and sufficient conditions under which the mathematical statements hold. In particular, we are looking for the largest class of contingent claims that can be analyzed in this framework. The risk-tolerance wealth process turns out to be the appropriate numeraire for pricing. In addition, a class of multi-dimensional stochastic games is studied.In complete markets, described by models like the Black and Scholes model, prices are uniquely defined by "no-arbitrage" considerations. In more realistic models, which are incomplete, prices of contingent claims can only be defined taking into account the preferences and wealth of specific investors. The dependence of prices on the number of contingent claims for investors who can borrow any amount of cash from the money market (infinite credit line) is studied. The pricing techniques can be applied to a large number of incomplete models, including options on non-traded assets and energy derivatives.