Stochastic Control with Discretionary Stopping

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Pang, Jong-ShiFrom: Ioannis Karatzas [ik@math.columbia.edu]Sent: Friday, June 08, 2001 2:46 PMTo: Pang, Jong-ShiSubject: Re: AbstractSTOCHASTIC CONTROL WITH DISCRETIONARY STOPPINGProposal DMS-00-99690 to the National Science Foundation byIoannis Karatzas, Columbia UniversityMay 2001ABSTRACTResearch is proposed on several open questions in Stochastic Analysis andOptimization, including the following:(i) Bounded-Velocity-Follower problems, that involve filtering, absolutelycontinuous stochastic control, and optimal stopping;(ii) Bounded-Variation, Finite-Fuel Follower problems, which combinefeatures of singular stochastic control and optimal stopping;(iii) Leavable Control problems for one-dimensional diffusions, and onassociated stochastic games of the controller-and-stopper type;(iv) Leavable Utility Maximization problems, with an embedded ``retirementoption";(v) the Hedging of American Contingent Claims under portfolio constraints;and(vi) a General Probabilistic Theory for Leavable Stochastic ControlProblems, based on martingales and on equivalent changes of measure.Several of these problems share the following interesting feature: thequalitative nature of the optimal policy changes significantly, as theparameters weighing the relative importance of continuation cost, stoppingcost, and discount rate pass through certain critical values. We propose toidentify the critical parameters in problems of this type that admit exactsolutions, and to describe as explicitly as possible the associated optimalcontrol policies and stopping rules. It is expected that tools fromstochastic analysis, martingales, convex duality theory, partialdifferential equations, and variational inequa-lities, will prove crucial inthe resolution of these questions; and that valuable new tools will have tobe 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 yearsshare a common feature, in that they involve elements of both StochasticControl and of Discretionary Stopping. Such questions arise, forinstance, in target-tracking models, where one has to stay close to atarget by spending fuel, to declare when one has arrived "sufficiently close" to the target, and then to decide whether to engage the target ornot. Combined stochastic control / optimal stopping problems also arise inMathematical 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 expectedto advance significantly our understanding of stochastic optimization andthe frontiers of its applications. The strong involvement of graduatestudents in our research activities is expected to continue, and to be amajor factor in the advancement of Applied Probability and of theMathematics of Finance.------------------------------------------------------------------------------------ Original Message -----From: Pang, Jong-Shi jpang@nsf.govTo: ik@math.columbia.eduSent: 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 anyreference 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 ananswer to the salary issue and also the abstract. Best regards, Jong-Shi