Topics in Stochastic Analysis and Optimization

本计画主要研究随机分析与最佳化的几个主题，包括：（i）在新型态特性不确定的情况下，无法直接观测的结构变点的自适应顺序侦测;（ii）具有任意停止的随机控制，以及具有停止与控制特性的随机游戏;（iii）部分观测下的随机控制，也称为“自适应控制”。 在理解最佳停止的进展最近有可能明确解决的一大类问题。 在这个项目中，可以预见的是，类似的进步显着的，越来越现实的，自适应变点检测的问题-当一个人必须了解不可观察的参数，同时，在真实的时间，以优化系统性能。 此外，计划包括工作的最佳停止与随机控制和过滤理论的“融合”，并考虑在部分观测的存在下，随机优化与任意停止相结合的问题的确切解决方案。 这一努力将首次使该领域的门槛相当明确地解决几个随机游戏相结合的控制和停止，零和非零和类型。 涉及随机控制和最优停止特征的优化问题出现在目标跟踪模型的研究中，其中一个人必须通过消耗燃料尽可能接近某个目标，宣布何时到达足够接近目标，然后决定是否与目标交战。 组合最优随机控制/停止的问题也出现在数学金融的背景下计算的上限和下限对冲价格的美国或有债权投资组合约束下，在投资组合/消费问题与嵌入式退休期权，在研究动态措施管理风险，并在随机游戏的委托人/代理人类型。 在学习未知参数和动态系统优化必须同时和在真实的时间中进行的情况下，关于变点的自适应顺序检测的工作对于几个应用领域（信号处理、金融、无线通信、制造、图像或语音识别）具有明确的含义。 该项目中提出的问题的解决方案有望促进我们对随机优化的理解，并扩大其应用的前沿。研究生参与研究活动预计将继续以强劲的步伐，并在应用概率和金融数学的进步的一个主要因素。

This project focuses on several topics in stochastic analysis and optimization that include: (i) Adaptive Sequential Detection of a structural change-point which is not directly observable, in the presence of uncertainty regarding the characteristics of the new regime; (ii) Stochastic Control with Discretionary Stopping, as well as Stochastic Games with features of both stopping and control; (iii) Stochastic Control under Partial Observations, also known as "adaptive control". Progress in the understanding of optimal stopping has recently made possible the explicit resolution of a large class of problems. In this project, it is envisioned that similar advances in significant, and increasingly realistic, questions of adaptive change-point detection -- when one has to learn about unobservable parameters and simultaneously, in real time, to optimize system performance. In addition, plans include work on a 'fusion' of optimal stopping with stochastic control and filtering theories, and contemplate the exact resolution of combined problems of stochastic optimization with discretionary stopping in the presence of partial observations. This effort will, for the first time, bring the field to the threshold of solving fairly explicitly several Stochastic Games of combined Control and Stopping, of both zero- and non-zero-sum type. Optimization problems that involve features of both stochastic control and optimal stopping arise in the study of target-tracking models, where one has to stay as close as possible to a certain target by spending fuel, to declare when one has arrived sufficiently close to the target, then to decide whether to engage the target or not. Problems of combined optimal stochastic control/stopping also come up in Mathematical Finance in the context of computing the upper- and lower-hedging prices of American contingent claims under portfolio constraints, in portfolio/consumption problems with an embedded retirement option, in the study of dynamic measures for managing risk, and in stochastic games of the principal/agent type. The work on adaptive sequential detection of change-points has clear implications for several fields of application (signal processing, finance, wireless communication, manufacturing, image or speech recognition), in contexts where learning about unknown parameters and dynamic system optimization have to be made simultaneously and in real time. The resolution of the problems suggested in this project is expected to advance our understanding of stochastic optimization and expand the frontiers of its applications. The involvement of graduate students in the research activities is expected to continue at a strong pace, and to be a major factor in the advancement of Applied Probability and of the Mathematics of Finance.