Partially Observed Systems in Finance: Statistical Inference and Optimization

该项目关注的是分析动力系统，其组件在任何时候都不是完全可观察的。这样的系统的示例是众多的：例如，任何接收到的无线电信号都会受到噪声的影响，海洋某些部分的温度可能并非始终准确地知道，而是与其他位置的已知温度相关，房屋的价格可能在出售之前并不准确地知道，但它与类似物业的交易价格相关，等等。如何有效地估计这些系统的未知参数给定的历史数据，以及如何选择这些参数，以最大限度地提高一个给定的目标。激励这项工作的具体应用来自金融和经济学：即，从估计金融资产的未观察价格过程的参数的问题（例如，交易之间的股票价格）和最优合约设计的问题（例如，如果客户对市场拥有专有知识，经纪人应向客户收取的费用的最佳结构是什么）。此外，这项工作的结果将有助于估计和优化的部分观察到的动力系统的一般方法，因此，适用于社会和自然科学中出现的许多其他问题。参与本计画将为研究生及本科生提供良好的训练机会。本计画包含两条研究方向，研究具有部分观测值的随机动力系统。第一个是关于部分观测扩散模型中未知参数的统计推断问题。特别是，PI将研究部分观测扩散模型的最大似然估计（MLE）的大样本性质，并将在退化扩散的情况下开发近似方法，其中似然函数不能直接计算。本研究的动机是在市场微观结构的潜在价格模型的未知参数估计问题。第二类研究关注的是在委托人和代理人之间存在信息不对称的情况下最优契约的设计。PI将考虑的信息不对称与经典的道德风险类型不同，源于委托人和代理人的行为所适应的外生给定过滤的差异。这样的最优合同模型导致具有挑战性的随机控制问题的信息约束。第二条研究路线的动机，特别是在金融市场中设计最佳经纪费的问题。本文的研究成果将对理论和应用都具有重要的意义。研究的第一线（与MLE有关）预计将推进统计学的数学基础，并为部分观测系统的参数估计开发新的计算方法。这些结果将被应用于市场微观结构中的特定问题（如资产的真实未观察价格的重建和价格影响的估计）以及其他领域的推理问题（例如，计算神经科学）。第二条研究路线（关于最优合同）的结果将有助于随机控制和最优合同设计理论。这些理论成果，特别是，可以用来发展更好地了解中介（经纪人）对金融市场的影响。这个奖项反映了NSF的法定使命，并已被认为是值得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的支持。

This project is concerned with the analysis of dynamical systems whose components are not perfectly observable at all times. The examples of such systems are numerous: e.g., any received radio signal is subject to noise, the temperature in some parts of the ocean may not be known exactly at all times but is correlated with the known temperatures in other locations, a price of a house may not be known precisely before it is sold but it is correlated with the transaction prices of similar properties, etc. In this context, the PI will focus on two questions: how to efficiently estimate the unknown parameters of such systems given historical data, and how to choose these parameters in order to maximize a given objective. The concrete applications that motivate this work arise from Finance and Economics: namely, from the problem of estimating the parameters of unobserved price process of a financial asset (e.g., a stock price between transactions) and from the question of optimal contract design (e.g., what is the optimal structure of fees that brokers should charge to their clients, if the latter have proprietary knowledge about the market). In addition, the results of this work will contribute to the general methodology for estimation and optimization of partially observed dynamical systems and, hence, be applicable to many other problems arising in social and natural sciences. Participation in the project will provide good training opportunities for both graduate and undergraduate students.This project consists of two lines of research on the stochastic dynamical systems with partial observations. The first is concerned with the problem of statistical inference of unknown parameters in partially observed diffusion models. In particular, the PI will investigate the large-sample properties of maximum likelihood estimators (MLEs) for partially observed diffusion models, and will develop approximation methods in the case of degenerate diffusions where the likelihood function cannot be computed directly. This research is motivated by the problem of estimating the unknown parameters in latent price models of market microstructure. The second line of research is concerned with the design of optimal contracts in the presence of information asymmetry between the principal and the agent. The information asymmetry that the PI will consider is different from the classical moral hazard type and stems from the difference in the exogenously given filtrations to which the actions of the principal and of the agent are adapted. Such optimal contract models lead to challenging stochastic control problems with informational constraints. The second line of research is motivated, in particular, by the problem of designing optimal brokerage fees in financial markets. The results of the proposed research will make significant contributions to both theory and applications. The first line of research (concerned with MLEs) is expected to advance the mathematical foundations of Statistics and to develop new computational methods for parameter estimation in partially observed systems. These results will be applied to specific problems in market microstructure (such as the reconstruction of the true unobserved price of an asset and the estimation of price impact) as well as to inference problems in other areas (e.g., Computational Neuroscience). The results of the second line of research (concerned with optimal contracts) will contribute to the theories of stochastic control and optimal contract design. These theoretical results, in particular, can be used to develop better understanding of the effects of intermediaries (brokers) on the financial markets.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.