Research is proposed on several open questions in Stochastic Analysis and Optimization, including the following:

(i) Bounded-Velocity-Follower problems, that involve filtering, absolutely continuous stochastic control, and optimal stopping; (ii) Bounded-Variation, Finite-Fuel Follower problems, which combine features of singular stochastic control and optimal stopping; (iii) Leavable Control problems for one-dimensional diffusions, and on associated stochastic games of the controller-and-stopper type; (iv) Leavable Utility Maximization problems, with an embedded ``retirement option"; (v) the Hedging of American Contingent Claims under portfolio constraints; and (vi) a General Probabilistic Theory for Leavable Stochastic Control Problems, based on martingales and on equivalent changes of measure.

Several of these problems share the following interesting feature: the qualitative nature of the optimal policy changes significantly, as the parameters weighing the relative importance of continuation cost, stopping cost, and discount rate pass through certain critical values. We propose to identify the critical parameters in problems of this type that admit exact solutions, and to describe as explicitly as possible the associated optimal control policies and stopping rules. It is expected that tools from stochastic analysis, martingales, convex duality theory, partial differential equations, and variational inequa-lities, will prove crucial in the resolution of these questions; and that valuable new tools will have to be developed, in order to deal with the non-standard issues that will arise.

The optimization questions that we plan to study over the next five years share a common feature, in that they involve elements of both Stochastic Control and of Discretionary Stopping. Such questions arise, for instance, in target-tracking models, where one has to stay close to a target by spending fuel, to declare when one has arrived "sufficiently close" to the target, and then to decide whether to engage the target or not. Combined stochastic control / optimal stopping problems also arise in Mathematical Finance: in . the context of computing the upper- and lower- hedging prices of American contingent claims under portfolio constraints; in . portfolio/consumption optimization with an embedded "retirement option"; and in . the study of dynamic measures for managing risk. The resolution of such problems, as suggested in this proposal, is expected to advance significantly our understanding of stochastic optimization and the frontiers of its applications. The strong involvement of graduate students in our research activities is expected to continue, and to be a major factor in the advancement of Applied Probability and of the Mathematics of Finance.


Original Message From: Pang, Jong-Shi To: Sent: Thursday, June 07, 2001 11:06 AM

> Professor Karatzas, > > Did you see the following email of mine sent May 30, 2001? Please reply > promptly so that I can process my recommendation. > > Looking forward to hearing from you. > > Jong-Shi > %%%%%%%%%%%%%%%%%%%%% > > > Dear Professor Karatzas, > > I am ready to recommend an award to your NSF proposal. Before I prepare > the paperwork, I need to clarify one thing about your salary. Specifically, > are you drawing 1 month salary from your current grant, which expires > 07/31/01? > If you are, then I will recommend a start date of 08/01/01 for your new > grant. > Otherwise, we can keep your requested 07/01/01 start date. I plan to > recommend > a continuing grant to fund the requested amount of $358,529 for 60 months. > > I will need an abstract for the project, to be sent to me by email (do not > send attachments). > This abstract shall be no more than one page in > length, and shall consist of two paragraphs. There shall be no special > symbols or equations. The first paragraph shall > be a technical description of the project, aimed at professional peers. > Often the proposal summary is an appropriate start, > phrased in the third person. The second paragraph will be a nontechnical > description that presents the work, its motivation, > and its significance. Think of the audience as a Congressman who asks "What > are you doing?", "Why would you do that?" > and "What does it mean?". > > The abstract is put in a public database, and may be read (and they have > been read in the past!) by Congressmen and > their staffers, so the second paragraph is important. Include any reference > to areas of important federal interest, such > as training or applications of strategic Federal interest. > > I will initiate the paperwork for my recommendation once I receive an answer > to the salary issue and also > the abstract. > > Best regards, > > Jong-Shi > >

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Henry A. Warchall
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Columbia University
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