Stochastic Controls, Games and Portfolios

该项目涉及与随机控制、博弈和投资组合相关的主题，包括随机投资组合理论中相对套利的研究，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套利的性质；具有任意停止的随机控制问题及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随机控制和/或局部观测下的停止问题。近年来，研究者和他的合作者在确定可观察特征（如多样性或足够的内在波动性）的简单描述性条件方面取得了相当大的进展，这些条件允许构建简单（仅基于可观察数量）的投资组合，这些投资组合可以超越大型股票市场。本项目旨在以这些结果为基础，以理解这种类型的最优套利（即“最便宜”套利）的本质，以及它们与随机分析（超鞅的退出措施、退化扩散）、抛物型偏微分方程（诸如凸性等定性性质的传播）和金融市场精细结构（套利机会或“泡沫”的开始）之间的联系。以及它对定价和对冲的影响)。此外，研究者已经对随机控制问题有了相当大的理解，当“控制者”（影响游戏动态的玩家）和“停止者”（决定游戏持续时间的玩家）合作以最小化预期成本时。该项目致力于理解这类游戏在零和和非零和情境下的非合作版本。一种丰富的理论似乎正在出现，我们打算充分按照建议所描述的那样发展它。此外，研究将集中在存在不可观测参数的随机优化问题，在贝叶斯框架中通过具有已知先验分布和持续更新的随机变量建模。众所周知，这些问题很难明确地解决，但我们在这个方向上有雄心勃勃的计划，并取得了一些初步成果。随机投资组合理论是分析投资组合行为和股票市场结构的一个较新的数学框架；它是描述性的，而不是规范性的，与实际投资组合和实际市场的可观察特征一致，并提供了一个理论工具（对套利问题的见解，具有控制行为的投资组合的构建等），对实际应用也很有用。涉及随机控制和最优停止特征的优化问题出现了，例如，在目标跟踪模型的研究中，一个人必须通过消耗燃料保持尽可能接近某个目标，宣布何时到达“足够接近”，然后决定是否与目标接触。组合最优随机控制/停止问题也出现在数学金融学中：在组合约束下计算美国或有债权的上对冲价格和下对冲价格的背景下；嵌入“退休”选项的投资组合/消费问题；在风险管理动态措施研究中；在动态一致的实用程序环境中；在主/代理类型的随机博弈中。部分观测的随机控制对变化点的自适应顺序检测、信号处理、金融以及其他需要同时实时地学习未知参数和动态系统优化的应用领域具有重要意义。

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.