Stochastic Controls, Games and Portfolios

本项目涉及与随机控制、博弈和投资组合有关的主题，包括研究随机投资组合理论中的相对套利，其中寻求允许相对于大型股票市场套利的简单描述性条件，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，同时具有停止和控制特征的随机对策，有界变差或“奇异”型随机控制问题;以及部分观测下的随机控制和/或停止问题。 近年来，研究者和他的合作者在识别可观察特征的简单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（仅基于可观察量）的投资组合，这些投资组合可以超越大型股票市场。 这个项目试图建立在这些结果，以了解最佳的性质，也就是说，“最便宜的”，这种类型的套利，和他们的连接到随机分析（上鞅的出口测度，退化扩散），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”的出现及其对定价和对冲的影响）。 此外，调查人员已经获得了相当大的理解随机控制问题与酌情停止，当一个“控制器”（玩家谁影响的动态游戏）和“塞子”（玩家谁决定的持续时间的游戏）合作，以尽量减少预期成本。 这个项目开始努力了解这种游戏的非合作版本，无论是在零和非零和的情况下。一个丰富的理论似乎正在出现，我们打算按照建议中所述，充分发展这一理论。此外，研究将集中在这样的随机优化问题中存在的不可观察的参数，建模在贝叶斯框架中的随机变量与已知的先验分布和连续更新。众所周知，这些问题很难明确解决，但我们在这方面有雄心勃勃的计划，并取得了一些初步成果。随机投资组合理论是一个相对新颖的数学框架，用于分析投资组合行为和股票市场结构;它是描述性的，而不是规范性的，与实际投资组合和真实的市场的可观察特征相一致，并提供了一个理论工具（对套利问题的见解，具有受控行为的投资组合的构建等）。这对于实际应用也是有用的。涉及随机控制和最优停止特性的优化问题出现在例如目标跟踪模型的研究中，其中必须通过消耗燃料尽可能接近某个目标，以宣布何时到达“足够接近”，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：在投资组合约束下计算美国或有债权的上下对冲价格的背景下;在嵌入“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;在动态一致的效用的背景下;以及在委托人/代理人类型的随机博弈中。部分观测的随机控制对变点的自适应顺序检测、信号处理、金融以及其他需要同时学习未知参数和动态系统优化的应用领域都有影响，并且是真实的。

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.