CAREER: Stochastic Control Problems in Financial Engineering

这项早期的教师职业发展(Career)研究的目标是理解和解决当代金融学中的随机控制问题。有两个主要的研究方向。第一类是当标的资产价格模型中存在随机参数和不连续性时，最优投资和套期保值投资组合的结构性质分析。这个问题将使用随机控制、随机微分方程和随机分析的方法来研究。将探索数值方法，可能使用模拟和利用最优解的结构的有限维近似。第二个目标是分析多期投资政策，这些政策对于基本金融模型中的错误是稳健的。这种分析将使用凸优化和随机控制的结果来进行。这项研究的主要目标是了解稳健投资组合的结构属性，即即使用于决策的基础模型存在错误，但在现实世界中表现良好的投资组合，并开发计算这些错误的数值方法。更广泛地说，目标是开发分析方法来理解一般动态优化问题的稳健决策的结构性质。如果成功，这项研究的结果将有助于更好地理解与涉及随机参数、跳跃和违约风险的金融模型相关的最优投资和对冲决策的结构属性。实证结果表明，这些都是不可忽视的重要风险来源，成功完成这项研究将使我们更全面地了解对这些风险来源的适当反应，以及计算这些决策的有效数值方法。对于稳健投资的问题，如果成功，这项研究将使我们更好地理解在现实世界中表现良好的投资组合的结构，即使潜在的数学模型存在错误，更广泛地说，将更深入地了解模型不确定性和建模错误对一般多时期问题决策的影响。所提出的研究还将有助于计算稳健最优决策策略的数值方法。

The objective of this Early Faculty Career Development (CAREER)research is to understand and solve stochastic control problems of contemporary interest in finance. There are two broad directions of research. The first concerns the analysis of structural properties of optimal investment and hedging portfolios when there are stochastic parameters and discontinuities in the underlying asset price models. This issue will be studied using methods from stochastic control, stochastic differential equations, and stochastic analysis. Numerical methods, possibly using simulation together with a finite dimensional approximation that exploits the structure of the optimal solution, will be explored. The second objective concerns the analysis of multi-period investment policies that are robust with respect to errors in the underlying financial model. This analysis will be carried out using results from convex optimization and stochastic control. The primary goal of this research is to understand the structural properties of robust portfolios, that is, investment portfolios that perform well in the real world even when there are errors in the underlying model being used to make decisions, and to develop numerical methods for computing them. More generally, the objective is to develop analytical methods for understanding the structural properties of robust decisions for general dynamic optimization problems. If successful, the results of this research will lead to an improved understanding of the structural properties of optimal investment and hedging decisions associated with financial models involving stochastic parameters, jumps, and default risk. Empirical results indicate that these are significant sources of risk that should not be ignored, and successful completion of this research will lead to a more complete understanding of the appropriate response to such sources of risk, and efficient numerical methods for computing these decisions. For the problem of robust investment, this research will lead, if successful, to a better understanding of the structure of investment portfolios that perform well in the real world, even when there are errors in the underlying mathematical model, and more generally, to greater insight into the impact of model uncertainty and modeling errors on decision making for general multi-period problems. The proposed research will also contribute to numerical methods for computing robust optimal decision policies.