Stochastic Controls, Games and Portfolios

这个项目涉及与随机控制、博弈和投资组合有关的主题，包括随机投资组合理论中相对套利的研究，其中一个人寻找允许相对于大型股票市场进行套利的简单的描述性条件，然后试图描述最有效的这种套利的性质；具有任意停止的随机控制问题，以及具有停止和控制的特征的随机博弈；有界变化的随机控制问题或“奇异”类型的随机控制问题；以及部分观测下的随机控制和/或停止问题。近年来，研究人员和他的合作者在识别简单的、关于可观察特征的描述性条件方面取得了相当大的进展，例如多样性或足够的内在波动性，这些条件允许构建简单的(仅基于可观察到的数量)投资组合，其表现可以超过大型股票市场。这个项目试图以这些结果为基础，以了解最优套利的性质，即“最低成本”，以及它们与随机分析(上鞅、退化扩散的退出度量)、抛物型偏微分方程(凸性等定性性质的传播)和金融市场的精细结构(套利机会或“泡沫”的开始，及其对定价和对冲的影响)的联系。此外，研究人员对随机停止的随机控制问题有了相当大的了解，当“控制者”(影响游戏动态的玩家)和“阻止者”(决定游戏持续时间的玩家)合作以最小化预期成本时。这个项目开始努力理解这种游戏的非合作版本，既有零和背景下的，也有非零和背景下的。一种丰富的理论似乎正在涌现，我们打算完全按照提案中的描述来追求其发展。此外，研究将集中在存在不可观测参数的情况下的随机优化问题，通过具有已知先验分布和连续更新的随机变量在贝叶斯框架中建模。众所周知，这些问题很难明确解决，但我们在这个方向上有雄心勃勃的计划和一些初步成果。随机投资组合理论是一种相对新颖的分析投资组合行为和股票市场结构的数学框架；它是描述性的，而不是规范的，与实际投资组合和真实市场的可观察特征一致，并提供了一种理论工具(对套利问题、具有受控行为的投资组合的构建等)。这在实际应用中也是有用的。例如，在目标跟踪模型的研究中，出现了同时涉及随机控制和最优停止特征的优化问题，在该模型中，人们必须通过消耗燃料来尽可能地接近某个目标，以宣布何时已经到达“足够近”，然后决定是否与目标交战。组合最优随机控制/停止的问题也出现在数学金融中：在计算投资组合约束下美国或有债权的上下限套期保值价格的背景下；在带有“退休”期权的投资组合/消费问题中；在管理风险的动态措施的研究中；在动态一致效用的背景下；以及在委托/代理型的随机博弈中。具有部分观测值的随机控制对于变点的自适应顺序检测、信号处理、金融和其他应用领域具有重要意义，在这些领域中，未知参数的学习和动态系统优化必须同时且实时地进行。

This project addresses topics related to stochastic controls, games and portfolios, including the study of relative arbitrage in stochastic portfolio theory, where one seeks simple, descriptive conditions that allow for arbitrage relative to a large equity market, and then tries to describe the nature of the most efficient such arbitrage; problems of stochastic control with discretionary stopping, and stochastic games with features of both stopping and control; problems of stochastic control of bounded variation or "singular" type; and problems of stochastic control and/or stopping under partial observations. In recent years, the investigator and his collaborators have made considerable progress in identifying simple, descriptive conditions on observable characteristics, such as diversity or sufficient intrinsic volatility, which allow the construction of simple (based solely on observable quantities) portfolios that can outperform a large equity market. This project seeks to build on these results in order to understand the nature of optimal, that is, "least-expensive", arbitrages of this type, and their connections to stochastic analysis (exit measures for supermartingales, degenerate diffusions), parabolic partial differential equations (propagation of qualitative properties such as convexity), and the fine structure of financial markets (the onset of arbitrage opportunities, or of "bubbles", and its implications for pricing and for hedging). Additionally, the investigator has gained considerable understanding of stochastic control problems with discretionary stopping, when a "controller" (a player who affects the dynamics of the game) and a "stopper" (a player who decides the duration of the game) cooperate to minimize an expected cost. This project embarks on an effort to understand non-cooperative versions of such games, both in zero-sum and in non-zero-sum contexts. A rich theory seems to emerge, and we intend fully to pursue its development as described in the proposal. Furthermore, research will focus on such stochastic optimization problems in the presence of unobservable parameters, modeled in a Bayesian framework by means of random variables with known prior distributions and continuous updating. Such problems are notoriously hard to solve explicitly, but we have ambitious plans in this direction and some preliminary results. Stochastic portfolio theory is a relatively novel mathematical framework for analyzing portfolio behavior and equity market structure; it is descriptive as opposed to normative, is consistent with observable characteristics of actual portfolios and real markets, and provides a theoretical tool (with insights into questions of arbitrage, construction of portfolios with controlled behavior, etc.) which is also useful for practical applications. Optimization problems that involve features of both stochastic control and optimal stopping arise, for instance, 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", and then to decide whether to engage the target or not. Problems of combined optimal stochastic control/stopping 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 problems with an embedded "retirement" option; in the study of dynamic measures for managing risk; in the context of dynamically consistent utilities; and in stochastic games of the principal/agent type. Stochastic control with partial observations has implications for the adaptive sequential detection of change-points, for signal processing, for finance, and for other fields of application where learning about unknown parameters and dynamic system optimization have to take place simultaneously, and in real time.