Topics in Stochastic Analysis and Optimization

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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.