Stochastic Control with Discretionary Stopping

Pang，Jong-Shi发件人：Ioannis Karatzas [ik@math.columbia.edu]发送人：Friday，June 08，2001 2：46 PM To：Pang，Jong-Shi Subject：Re：Abstract随机控制与离散停止Proposal DMS-00-99690 to the National Science Foundation by Ioannis Karatzas，哥伦比亚大学2001年5月摘要研究随机分析与优化中的几个开放性问题，包括：（i）有界速度跟随器问题，涉及滤波、绝对连续随机控制和最优停止;（ii）有界变差，燃料跟随器问题，它结合了奇异随机控制和最优停止的特征;（iii）一维扩散的可离开控制问题，以及相关的随机博弈的破坏者和停止者类型;（iv）可离开效用最大化问题，带有嵌入的"退休选项”;（v） 投资组合约束下美式未定权益的套期保值;（vi）基于鞅和等价测度变化的可离开随机控制问题的一般概率理论。其中几个问题具有以下有趣的功能：最优策略的定性性质发生了显著变化，因为衡量继续成本，停止成本，和折现率通过一定的临界值。我们建议识别这类问题的关键参数，承认exactsolutions，并尽可能明确地描述相关的optimalcontrol政策和停止规则。可以预期，随机分析、鞅、凸对偶理论、偏微分方程和变分不等式等工具在解决这些问题时将是至关重要的;为了处理即将出现的非标准问题，我们必须开发有价值的新工具。我们计划在未来五年内研究的优化问题有一个共同的特点，因为它们涉及随机控制和自由裁量权停止的元素。例如，在目标跟踪模型中，这样的问题就出现了，在这个模型中，一个人必须通过消耗燃料来接近目标，宣布何时到达目标的“足够近”，然后决定是否与目标交战。结合随机控制/最优停止问题也出现在数学金融：在。 计算美国国债套期保值价格上限和下限的背景 投资组合约束下的或有债权; 投资组合/消费优化与嵌入式“退休期权”;和。 风险管理的动态措施的研究。这些问题的解决方案，在这个建议中建议，预计将大大推进我们的理解随机优化及其应用的前沿。 研究生对我们的研究活动的强烈参与预计将继续下去，并将成为应用概率和金融数学进步的一个主要因素。原始信息-发件人：Pang，Jong-Shi jpang@nsf. gov收件人：ik@math.哥伦比亚. edu发送时间：2001年6月7日，星期四上午11时06分Karatzas教授，您看到我2001年5月30日发送的以下电子邮件了吗？ 请尽快回复，以便我处理我的建议。期待您的回音。尊敬的Karatzas教授，我准备向您的NSF提案推荐一个奖项。 在我准备文件之前，我需要澄清一件关于你的薪水的事情。具体地说，你是从你目前的补助金中提取一个月的薪水，该补助金于2001年7月31日到期。如果你是，那么我会建议开始日期08/01/01为您的新赠款。否则，我们可以保留您要求的07/01/01开始日期。 我计划建议继续提供赠款，为所要求的358 529美元供资，为期60个月。我需要一个项目的摘要，通过电子邮件发送给我（不发送附件）。本摘要篇幅不超过一页，由两段组成。不应有特殊符号或公式。第一段应是针对专业同行的项目技术说明。通常，提案摘要是一个适当的开始，用第三人称表达。第二段将是一个非技术性的描述，介绍工作，其动机和意义。把观众想象成一个国会议员，他问：“你在做什么？””、“你为什么要那样做？“和“这是什么意思？".摘要被放在一个公共数据库中，并且可以被读取（并且它们在过去已经被读取了！）国会议员和他们的工作人员，所以第二段很重要。包括任何涉及重要联邦利益的领域，如战略联邦利益的培训或应用。一旦我收到关于工资问题的答复和摘要，我将开始写推荐信。最好的问候，钟诗

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