Research on Stochastic Optimization and Applications

随机优化及其应用研究惠望项目摘要许多随机优化问题都假定决策者有完全的干预系统的自由。也就是说，控制策略可以连续、瞬时地调整，也可以在某一时间间隔内任意停止状态过程。然而，这一假设在实践中经常被违反。该项目的第一部分是关于随机优化问题的一些新公式，以适应这些实际约束。这些模型有一个共同特征：由外生信号过程决定的随机干预时间。在最优控制的背景下，只有当外部过程给决策者开了绿灯时，才允许决策者调整控制水平；例如，当泊松过程跳跃时。在最优停止区域中，状态进程仅在接收到来自外部进程的特定信号时才被允许停止。这两个公式都允许显式解，并且可以用来对控制策略或停止时间的实际约束进行建模。项目的第二部分研究了一类带跳的扩散过程。主要的努力放在它们在经济和金融方面的应用上。引入这种过程的原因是，经典的股价扩散模型无法解释许多经验难题。例如，在期权定价中经常观察到一种称为波动率微笑的异常现象，金融资产的收益分布呈现尖峰特征。我们打算解释这些现象，使用跳跃过程来模拟金融市场中的股票价格或投资问题中经济项目的价值。潜在过程中的不连续行为需要新的技术来获得某些定价和财富优化问题的显式解。研究项目包括常用的奇异期权定价、效用最大化分析和投资项目评价。随机优化是现代应用数学的主要主题之一，在工程、生物、经济和金融等学科中有许多应用。本研究项目的目的是系统地发展更忠实于现实生活问题的数学理论，使从中得出的结论能够更有信心地使用。例如，更好地理解股票价格的不确定性可以帮助代理人降低金融实践中的风险。然而，在开发更现实的模型时，人们必须意识到复杂性和数学可控性之间的微妙平衡。一个现实但过于复杂的模型在数学上可能难以驾驭。因此，该项目的目标是开发包含实际问题本质的解析或数值可解模型。为此，对于一般的随机优化问题，考虑了一些新的公式，以适应一些实际的约束；例如，控制器只能在外部过程发出特定信号的时候才能干预系统。还考虑了一些新的经济和金融模型，这些模型包含了股票价格在短期内的剧烈变化。这些新模型可以用来解释经典模型未能解决的许多经验难题，并帮助代理人开发更好的投资组合以降低风险。

Research on Stochastic Optimization and ApplicationsHui WangProject AbstractMany stochastic optimization problems assume that the decision maker has the total freedom to intervene the system. That is, the control policies can be adjusted continuously and instantaneously, or the state processes can be stopped arbitrarily within some time interval. However, this assumption is often violated in practice. The first part of the project is concernedwith some new formulations for stochastic optimization problems in order to accommodate these practical constraints. These models share a common feature: random intervention times determined by exogenous signal processes. In the context of optimal control, the decision maker is allowed to adjust the level of control only at times when an exogenous process gives him a green light; for example, when an Poisson process makes a jump. In the area of optimal stopping, the state process is allowed to be stopped only at times when it receives certain signals from an exogenous process. Both formulations admit explicit solutions and can be applied to model practical constraints on control policies or stopping times. The second part of the project is concerned with a class of diffusion processes with jumps. The major effort is put on their applications to economics and finance. The reason for the introduction of such processes is that the classical diffusion model for stock prices cannot explain many empirical puzzles. For example, an abnormality called volatility smile is often observed in option pricing, and the return distributions of financial assets exhibit a leptokurtic feature. We intend to explain these phenomena, using the jump processes to model stock prices in a financial market or the values of economic projects in investment problems. The discontinuous behavior in the underlying processes requires new techniques to obtain explicit solutions to certain pricing and wealth optimization problems. The research project includes pricing commonly traded exotic options, analysis of utilitymaximization, and evaluation of investment projects.Stochastic optimization is one of the main topics in modern applied mathematics, with many applications in disciplines like engineering, biology, economics and finance, etc. The purpose of this research project is to systematically develop mathematical theories that are more faithful to real life problems, so that the conclusions drawn from them can be used with more confidence. For example, a better understanding for the uncertainty of stock prices can help agents reduce the risk in financial practice. However, in developing more realistic models, one must be aware of the subtle balance between complexity and mathematical tractability. A realistic yet too complicated model can be mathematically untractable. The goal of the project, therefore, is to develop analytically or numerically solvable models that embrace the essence of the practical problems. To this end, some new formulations are considered for general stochastic optimization problems in order to accommodate some practical constraints; for example, the controller can intervene the system only at times when an exogenous process sends out a certain signal. Also considered are some new models for economics and finance that incorporate the drastic changes of stock prices over short periods of time. These new models can be used to explain many empirical puzzles that the classical models fail to do, and to help agents develop better portfolios to reduce risk.