Stochastic Controls, Portfolios, and Competing Particle Systems

随机控制、组合和竞争粒子系统

• 批准号: 1405210
• 负责人: Ioannis Karatzas
• 金额: $ 59.03万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405210&HistoricalAwards=false
• 关键词: Stochastic; Controls; Portfolios; Competing; Particle

In recent years good progress has been made in identifying instances of simple, descriptive conditions on observable characteristics of large equity markets, under which it is possible to outperform the market using simple portfolios based solely on observables. Nevertheless, a satisfactory general theory has not emerged yet: Is there a "canonical" way to understand the existing examples? Are they special cases of a much more general construction? Under what conditions is such arbitrage possible over arbitrary time horizons, and what is the "best possible" such arbitrage under specific market structures? The investigator and his collaborators study these issues and their connections to stochastic analysis and partial differential equations. They also study stable competing particle systems, that is, multidimensional diffusions interacting through their ranks in a manner giving rise to invariant measures that are in broad agreement with stability properties observed in large equity markets over decades. The stochastic analysis of such systems presents interesting challenges, such as the solvability of the stochastic equations that implement such diffusions; the study of multiple collisions, of their associated collision local times, of their invariant distributions, of time reversal; and the estimation of parameters in the resulting models for practical implementation. Furthermore, they work on stochastic optimization problems of the control and stopping type in the presence of unobservable parameters, modeled in a Bayesian framework by means of random variables with known prior distributions and continuous updating. This work 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. The investigator studies problems in stochastic controls, portfolios, and competing particle systems. These include: + 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; + The related study of the distribution of the time-to-explosion for diffusion processes; + The study of stochastic differential equations with "singular" (generalized) drift, determined by local time; + Problems of stochastic control or stopping under partial observations ("adaptive control"); and + The study of (rank-based) systems of competing Brownian particles, whose dynamics at any given time depend on the empirical measure of their configuration.

近年来，在确定大型股票市场的可观察特征的简单描述性条件方面取得了良好进展，在这种情况下，使用仅基于可观察特征的简单投资组合可以超越市场。 然而，一个令人满意的一般理论还没有出现：有没有一个“规范”的方式来理解现有的例子？ 它们是一个更普遍的结构的特例吗？ 在什么条件下，这种套利在任意的时间范围内是可能的，什么是“最好的可能”，这种套利在特定的市场结构？ 调查员和他的合作者研究这些问题及其与随机分析和偏微分方程的联系。 他们还研究了稳定的竞争粒子系统，即多维扩散通过它们的行列相互作用的方式产生不变的措施，在广泛的协议与稳定性在大型股票市场观察了几十年。 这样的系统的随机分析提出了有趣的挑战，如可解性的随机方程，实现这种扩散;多个碰撞的研究，其相关的碰撞当地时间，其不变的分布，时间反转;和估计的参数在实际执行的模型。 此外，他们的工作在控制和停止类型的随机优化问题中存在不可观测的参数，在贝叶斯框架中建模的随机变量与已知的先验分布和连续更新。 这项工作具有影响的自适应顺序检测的变化点，信号处理，金融，和其他领域的应用，学习未知参数，动态系统优化，必须同时发生，并在真实的时间。研究者研究随机控制、投资组合和竞争粒子系统中的问题。 其中包括：+随机投资组合理论中的相对套利研究--人们寻找简单的描述性条件，允许相对于大型股票市场进行套利，然后试图描述最有效的套利性质; +扩散过程爆炸时间分布的相关研究;带“奇异”随机微分方程的研究（广义）漂移，由当地时间决定; +随机控制或部分观测下的停止问题（“自适应控制”）;和+研究（基于秩）系统的竞争布朗粒子，其动态在任何给定的时间取决于经验措施，他们的配置。