Topics in Stochastic Analysis and Optimization

该项目关注随机分析和优化中的几个主题，包括：(I)在存在关于新制度特征的不确定性的情况下，对不能直接观察到的结构变化点的自适应顺序检测；(Ii)具有自由停止的随机控制，以及具有停止和控制特征的随机对策；(Iii)部分观测下的随机控制，也称为“自适应控制”。最近，对最优停止的理解的进步使一大类问题的显式解决成为可能。在这个项目中，设想在重大的和日益现实的自适应变点检测问题上取得类似的进展--当一个人必须了解不可观测的参数时，同时实时地优化系统性能。此外，计划还包括将最优停止与随机控制和过滤理论相结合的工作，并考虑在存在部分观测的情况下，对随机优化和任意停止的组合问题的准确解决。这一努力将首次将该领域带入相当明确地解决几个组合控制和停止的随机对策的门槛，既有零和类型的，也有非零和类型的。在目标跟踪模型的研究中，出现了同时涉及随机控制和最优停止特征的优化问题，其中一个人必须通过消耗燃料来尽可能接近某个目标，以便在到达足够接近目标的时候宣布，然后决定是否与目标交战。在计算组合约束下美式未定权益的上下限套期保值价格时，在嵌入退休期权的投资组合/消费问题中，在动态风险管理措施的研究中，以及在委托/代理型随机博弈中，组合最优随机控制/停止问题也出现在数学金融学中。在未知参数的学习和动态系统优化必须同时和实时进行的环境中，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)具有明显的影响。本课题所提出的问题的解决，有望促进我们对随机优化问题的理解，拓展其应用领域。研究生对研究活动的参与预计将继续以强劲的步伐进行，并将成为推动应用概率和金融数学进步的主要因素。

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.