Stochastic Controls, Portfolios, and Competing Particle Systems

近年来，在确定大型股票市场可观察特征的简单描述性条件方面取得了良好的进展，在这种情况下，仅基于可观察到的简单投资组合就有可能跑赢市场。然而，一个令人满意的普遍理论还没有出现：有没有一种“规范”的方法来理解现有的例子？它们是更一般结构的特例吗？在什么条件下，这种套利在任意的时间范围内是可能的？在特定的市场结构下，这种套利的“最佳可能”是什么？研究人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式的联系。他们还研究了稳定的竞争粒子系统，即多维扩散以一种产生不变度量的方式相互作用，这些不变度量与几十年来在大型股市中观察到的稳定性属性广泛一致。这类系统的随机分析提出了有趣的挑战，如实现这种扩散的随机方程的可解性；多重碰撞及其相关的碰撞局部时间、其不变分布、时间反转的研究；以及为实际实施而对所得模型中的参数进行估计。此外，它们还研究了具有不可观测参数的控制型和停止型随机优化问题，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。这项工作对变化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和实时地进行。研究人员研究随机控制、投资组合和竞争粒子系统中的问题。这些研究包括：+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这种套利的性质；+扩散过程爆炸时间分布的相关研究；+由当地时间确定的具有“奇异”(广义)漂移的随机微分方程的研究；+部分观测下的随机控制或停止问题(“自适应控制”)；研究相互竞争的布朗粒子的(基于等级的)系统，其在任何给定时间的动力学取决于对其构型的经验测量。

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.