Topics in Mathematical Finance and Stochastic Control

本文考虑了不完全市场中最优投资问题中价值函数的二次可微性，以及在整条实线上定义效用函数的情况下效用价格的敏感性分析。研究的重点是数学命题成立的充分必要条件。特别是，我们正在寻找可以在这个框架中分析的最大类别的或有索赔。风险承受能力财富过程被证明是定价的合适数字。此外，还研究了一类多维随机对策。在像布莱克和斯科尔斯模型这样的模型所描述的完全市场中，价格是由“无套利”考虑因素唯一定义的。在更现实的不完整模型中，或有索取权的价格只能根据特定投资者的偏好和财富来确定。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。该定价技术可应用于大量不完整模型，包括非交易资产期权和能源衍生品。

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.