Topics in Stochastic Control

该研究项目涉及随机控制理论以及应用概率和非线性偏微分方程的相关领域中的几个主题。一个主题是无限时间域上的风险敏感控制，其动机是非线性系统的鲁棒反馈控制器设计问题。风险敏感控制的另一个应用是在数学金融中，包括长期动态投资组合分配问题。另一个研究课题是一阶Hamilton-Jacobi-Bellman型偏微分方程。虽然这样的方程在通常意义上是非线性的，但它们相对于极大加代数运算是线性的。这允许通过最大加基展开得到近似解。最后，对国际金融中出现的经济增长和债务的随机控制模型进行了研究。随机控制为存在不确定性的动态决策的建模和分析提供了一个框架。动态规划方法提供了一种通过求解相应的非线性偏微分方程组来获得最优随机控制策略的方法。通过这笔赠款资助的研究受到工程和金融经济学中广泛应用的推动，包括稳健反馈控制器设计、非线性估计和滤波以及最优动态投资分配。在国际金融中，考虑增长/债务模型，其中的目标是选择国家投资和消费政策，这些政策优化了受强加约束的适当选择的标准。最优控制下的模型表现可以提供基准，以表明在当前政策下实际的经常账户赤字和外债水平是否可持续。

This research program concerns several topics in stochastic controltheory and related areas of applied probability and nonlinear partialdifferential equations. One topic is risk-sensitive control on aninfinite time horizon, motivated by problems of robust feedbackcontroller design for nonlinear systems. Another application ofrisk-sensitive control is in mathematical finance, including dynamicportfolio allocation problems on long time horizons. Yet anotherresearch topic concerns first order partial differential equations ofHamilton-Jacobi-Bellman type. While such equations are nonlinear in theusual sense, they are linear with respect to max-plus algebraoperations. This allows for approximate solution via max-plus basisexpansions. Finally stochastic control models for economic growth anddebt which arise in international finance are being studied.Stochastic control provides a framework for modeling and analysis ofdynamic decision making in the presence of uncertainty. The method ofdynamic programming provides a way to obtain optimal stochastic controlpolicies by solution of corresponding nonlinear partial differentialequations. The research funded through this grant is motivated by arange of applications in engineering and financial economics, includingrobust feedback controller design, nonlinear estimation and filteringand optimal dynamic investment allocation. In internationalfinance,growth/debt models are considered in which the goal is to choosenational investment and consumption policies which optimize a suitablychosen criterion subject to imposed constraints. The model performanceunder optimal control may provide benchmarks to suggest whether actualcurrent account deficits and levels of foreign debt are sustainableunder current policies.