Stochastic Portfolios, Controls, and Interacting Particles

该奖项将集中于金融数学中的三个重要领域：(1)随机投资组合理论，(2)随机控制，(3)熵梯度流。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数学框架。它是描述性的，而不是规范性的，与实际投资组合股票市场的可观察特征一致，是一种对实际应用有用的理论工具。此外，它还提供了对市场结构稳定性的优异表现和套利问题的洞察，以及导致机构(例如，基金会、捐赠基金、养老金)投资的长期可控行为的投资组合构建问题。带部分观测的随机控制出现在顺序检测变点、信号处理、金融、学习未知参数的其他环境中，以及必须同时和实时进行动态优化的情况下。例如，在跟踪模型中，出现了同时涉及控制和停止功能的优化问题，在这种模型中，人们必须通过消耗燃料来尽可能接近某个目标，宣布自己何时已经足够接近，然后决定是否与目标交战。它还出现在必须解决勘探(了解不可观测的数量)和开采(采取现在付出代价，但以后会带来好处的行动)之间的紧张关系的情况下。这一项目的第三个重点是研究熵梯度流动，它补充了热力学第二定律，并与以下现象有关：当前构型的熵相对于稳态不仅随着系统接近平衡而减小，而且以尽可能有效的方式这样做--根据相空间上的适当距离，状态遵循熵可能最陡峭的下降路径。这对于神经网络的训练、随机优化中的梯度下降算法以及最优投资组合清算都有重要的意义。该奖项将为研究生提供研究经验的机会。PI将发展出一种投资组合理论，该理论不依赖于等价鞅度量的存在，并允许一个投资组合的表现优于另一个投资组合。这一理论将基于局部鞅平减指数、可选分解定理和适当的泛函分析工具。它将只排除通常被称为套利的非常恶劣的形式，并帮助开发一个简单、精简的金融数学框架，对对冲和投资组合优化等基本问题提供完整的答案。这一理论将涵盖资产数量任意的市场(如债券市场，或有拆分和合并的市场)、公开市场(有固定、有限数量的公司，但构成随市值变化，如S)、投资组合约束、模型不确定性，以及基于扩散系统相互作用的模型的市场稳定性。PI将处理随机优化问题，这些问题结合了最优控制、停止和对不可观测的参数或信号进行过滤的特点。众所周知，此类问题很难明确解决，而且几乎没有相关理论。最后，该项目将利用时间反转和最佳传输技术探索最近开发的一种工具的全部适用性：奥托微积分的轨迹方法。PI将研究它与McKean-Vlasov类型的扩散、深层神经网络的训练以及相对熵沿马尔可夫过程流动的最陡峭下降的相关性，包括借助适当加权的Sobolev规范在离散结构上的相关性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This award will focus on three areas of importance in financial mathematics: (1) stochastic portfolio theory, (2) stochastic control, and (3) entropic gradient flows. Stochastic portfolio theory is a mathematical framework for analyzing portfolio behavior and equity market structure. It is descriptive as opposed to normative, consistent with observable characteristics of actual portfolios equity markets, and a theoretical tool useful for practical applications. In addition, it provides insights into questions of outperformance and arbitrage of market-structure stability, and of portfolio construction that results in controlled behavior over long-time horizons for purposes of institutional (for example, foundation, endowment, pension) investing. Stochastic control with partial observations arises in the sequential detection of change-points, in signal processing, in finance, in other contexts where learning about unknown parameters, and where dynamic optimization must take place simultaneously and in real time. Optimization problems that involve features of both control and stopping arise, for instance, in 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. It also arises in situations where one has to resolve the tension between exploration (learning about unobservable quantities) and exploitation (taking an actions that costs now, but yields benefits later). The third focus of this project, the study of entropic gradient flows, complements the second law of thermodynamics and pertains to phenomena where the entropy of the current configuration, relative to the steady-state, not only decreases as the system approaches equilibrium, but does so in the most efficient manner possible – the state follows a path of steepest possible descent for the entropy, in terms of an appropriate distance on phase space. This has important implications for the training of neural networks, for gradient descent algorithms in stochastic optimization, and for optimal portfolio liquidation. The award will provide graduate students with the opportunity for research experiences. The PI will develop a theory of portfolios that does not rely on the existence of equivalent martingale measures and allows the outperformance of one portfolio by another. This theory will be based on local martingale deflators, the optional decomposition theorem, and appropriate functional-analytic tools. It will exclude only very egregious forms of what is commonly called arbitrage, and help develop a simple, streamlined mathematical framework for finance with complete answers to the basic questions of hedging and portfolio optimization. This theory will cover markets with an arbitrary number of assets (such as bond markets, or those with splits and mergers), open markets (with a fixed, finite number of companies but of a composition varying with capitalization, such as S&P 500), portfolio constraints, model uncertainty, and market stability via models based on systems of diffusions interacting through their ranks. The PI will work on stochastic optimization problems that combine features of optimal control, stopping, and filtering of unobservable parameters or signals. Such problems are notoriously hard to solve explicitly and little theory exists for them. Finally, the project will explore the full range of applicability for a tool the PI developed recently, using time-reversal and optimal transport techniques: the trajectorial approach to the Otto calculus. The PI will study its relevance for diffusions of McKean-Vlasov type, the training of deep neural networks, and the steepest descent of the relative entropy along flows of Markov processes, including on discrete structures with the help of appropriately weighted Sobolev norms.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.