Topics in Stochastic Analysis and Optimization

该项目侧重于随机分析和优化中的几个主题，其中包括：(i)在新制度特征不确定的情况下，对不能直接观察到的结构变化点进行自适应顺序检测；（ii）具有任意停止的随机控制，以及同时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，在对最佳停止的理解方面取得的进展，使大量问题的明确解决成为可能。在这个项目中，设想在自适应变化点检测的重要和日益现实的问题上取得类似的进展-当一个人必须了解不可观察的参数并同时实时地优化系统性能时。此外，计划包括将最优停车与随机控制和过滤理论“融合”，并考虑在存在部分观测的情况下随机优化与任意停车的组合问题的精确解决方案。这一努力将首次使该领域达到相当明确地解决几个零和非零和类型的组合控制和停止的随机博弈的门槛。在目标跟踪模型的研究中出现了涉及随机控制和最优停止特征的优化问题，其中一个必须通过消耗燃料来保持尽可能接近某个目标，然后宣布何时到达足够接近目标，然后决定是否与目标接触。组合最优随机控制/停止问题也出现在数学金融中，在投资组合约束下计算美国或有债权的上下限对冲价格，嵌入退休期权的投资组合/消费问题，管理风险的动态措施研究，以及委托/代理类型的随机博弈。对变化点的自适应顺序检测的工作对几个应用领域（信号处理、金融、无线通信、制造、图像或语音识别）具有明确的影响，在必须同时实时地学习未知参数和动态系统优化的环境中。本项目提出的问题的解决有望促进我们对随机优化的理解，并扩大其应用的前沿。研究生对研究活动的参与预计将继续保持强劲的步伐，并成为应用概率论和金融数学发展的主要因素。

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.