Stochastic Portfolios, Controls, and Interacting Particles

该奖项将集中在金融数学的三个重要领域：（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相一致，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见解，以及为机构（例如，基金会，捐赠基金，养老金）投资而在长期范围内导致受控行为的投资组合构建。具有部分观测的随机控制出现在变点的顺序检测中，在信号处理中，在金融中，在学习未知参数的其他环境中，以及动态优化必须同时且在真实的时间中发生。例如，在跟踪模型中出现了涉及控制和停止特征的优化问题，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近的位置，然后决定是否与目标交战。它也出现在人们必须解决探索（了解不可观测的量）和开发（采取现在需要成本，但以后会产生收益的行动）之间的紧张关系的情况下。该项目的第三个重点是熵梯度流的研究，它补充了热力学第二定律，并涉及到当前配置的熵相对于稳态不仅随着系统接近平衡而减少，而且以最有效的方式减少的现象-状态遵循熵的最陡可能下降的路径，在相空间上的适当距离。这对神经网络的训练、随机优化中的梯度下降算法和最优投资组合清算都有重要意义。该奖项将为研究生提供研究经验的机会。PI将开发一种投资组合理论，该理论不依赖于等价鞅测度的存在，并允许一个投资组合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，和适当的功能分析工具。它将只排除那些通常被称为套利的非常令人震惊的形式，并帮助开发一个简单的、流线型的金融数学框架，对对冲和投资组合优化的基本问题提供完整的答案。这一理论将涵盖具有任意数量资产的市场（如债券市场，或具有拆分和合并的市场），开放市场（具有固定的有限数量的公司，但其组成随资本化而变化，如标准普尔500），投资组合约束，模型不确定性，以及通过基于通过其等级相互作用的扩散系统的模型的市场稳定性。PI将工作在随机优化问题，结合联合收割机的最佳控制，停止，和不可观测的参数或信号的过滤功能。众所周知，这些问题很难明确解决，而且几乎没有理论存在。最后，该项目将探索PI最近开发的一种工具的全方位适用性，使用时间反演和最佳传输技术：奥托演算的矢量方法。PI将研究其与McKean-Vlasov型扩散的相关性，深度神经网络的训练，以及马尔可夫过程沿着流的相对熵的最陡下降，包括在适当加权的Sobolev范数的帮助下的离散结构。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估而被认为值得支持。

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.