New Directions in Fractal Modeling: Estimation, Filtering, and Applications

分形和多重分形在数学、工程和自然科学的许多领域具有重要意义。第一个多重分形度量被发展并应用于地质学（de Wijs, 1951）和湍流建模（Kolmogorov, 1962; Mandelbrot, 1972, 1974）。随后的应用包括天文学、遗传学、水文学、气象学、医学、网络流量建模和地震学。在金融领域，我自己的研究开发了第一个基于时间变形布朗运动的多重分形随机过程（Calvet, Fisher, and Mandelbrot, 1997）。随后的研究提供了基于矩的推理（Calvet and Fisher, 2002）、精确滤波和最大似然估计（Calvet and Fisher, 2001）；粒子滤波（Calvet, Fisher, and Thompson, 2006）；利用平衡理论的扩展，包括多重分形跳跃扩散（Calvet and Fisher, 2007 - 2008）；以及对利率（Calvet, Fisher, and Wu, 2013）和期权定价（Calvet, Fisher, Fearnley, and Leippold, 2014）的应用。***提出的研究推进了多重分形工具并开发了新的应用。迄今为止，基于离散值状态变量的马尔可夫开关多重分形（MSM）已成为多重分形在金融领域应用的关键组成部分。最近的粒子滤波技术允许考虑具有扩散成分的更广泛的模型集。这在期权定价应用中尤其有利，因为相对于标准基准，期权定价应用的效果更好。这些粒子过滤方法的应用和进一步发展将是一个重点研究领域。该研究项目还将通过将混合数据采样方法（“MIDAS”，Ghysels, Santa-Clara, and Valkanov, 2006）应用于多重分形过程，开发更简单的计算方法。***研究项目还将通过调查股票之间的信息传递来确定金融数据中多重分形的原因，其中股票被建模为一个网络。我和合著者已经在每日股票回报中展示了缓慢的信息扩散的证据，最好用双曲线衰减来捕捉（Boguth, Carlson, Fisher, and Simutin, 2014）。拟议中的研究将使用一个相对较新的名为RavenPack的新闻数据库，以更高的频率检查信息传输。新闻来源于专业博客、报纸和通讯社，时间戳以毫秒为单位。要处理这个新闻数据库，例如提取特定公司多年的新闻，由于数据的大小，需要并行数据处理。这项研究将允许精确跟踪新闻在股票中的传播，识别相关的传播网络，识别与波动性的联系，以及资产回报的分形属性。**

Fractals and multifractals are of major importance in mathematics and many areas of engineering and the natural sciences. The first multifractal measures were developed and applied in geology (de Wijs, 1951) and the modelling of turbulence (Kolmogorov, 1962; Mandelbrot, 1972, 1974). Subsequent applications include astronomy, genetics, hydrology, meteorology, medicine, network traffic modeling, and seismology.  In finance, my own research developed the first multifractal stochastic processes, based on time-deformed Brownian motions (Calvet, Fisher, and Mandelbrot, 1997). Subsequent research provides moment-based inference (Calvet and Fisher, 2002), exact filtering and maximum-likelihood estimation (Calvet and Fisher, 2001); particle filtering (Calvet, Fisher, and Thompson, 2006); extensions using equilibrium theory, including multifractal jump-diffusions (Calvet and Fisher, 2007, 2008); and applications to interest rates (Calvet, Fisher, and Wu, 2013) and option pricing (Calvet, Fisher, Fearnley, and Leippold, 2014).***The proposed research advances multifractal tools and develops new applications. To date, the Markov-switching Multifractal (MSM), based on discrete-valued state variables, has been a key building block for multifractal applications in finance.  Recent particle filtering techniques permit consideration of a broader set of models with diffusive components. This should in particular prove advantageous in option pricing applications, which show good results relative to standard benchmarks.  Application and further development of these particle filtering methods will be a key area of research. The research program will also develop simpler computational methods by adaptating Mixed Data Sampling methods ("MIDAS", Ghysels, Santa-Clara, and Valkanov, 2006) to multifractal processes.***The research program will also seek to determine the causes of multifractality in financial data by investigating the transmission of information across stocks, where stocks are modeled as a network. Coauthors and I have shown evidence of slow information diffusion, best captured by a hyperbolic decay, in daily stock returns (Boguth, Carlson, Fisher, and Simutin, 2014). The proposed research will examine information transmission at much higher frequencies, using a relatively recent news database called RavenPack. News are sourced from professional blogs, newspapers and newswires time stamped to the milliseconds. To process this news database such as extracting news for a specific firm on multiple years requires parallel data processing because of the size of the data. This research will permit precise tracking of the transmission of news across stocks, identification of the relevant transmission network, identification of the link to volatility, and to fractal properties in asset returns. **