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

分形建模的新方向：估计、过滤和应用

• 批准号: RGPIN-2015-06749
• 负责人: Fisher, Adlai
• 金额: $ 1.02万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=650786
• 关键词: New; Directions; Fractal; Modeling; Estimation

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. **

分形学和多重分形学在数学、工程和自然科学的许多领域都具有重要意义。在地质学(De Wijs，1951)和湍流建模(Kolmogorov，1962；Mandelbrot，1972,1974)中开发并应用了第一个多重分形法。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。在金融学中，我自己的研究开发了第一个基于时间变形的布朗运动的多重分形随机过程(Calvet，Fisher和Mandelbrot，1997)。随后的研究提供了基于矩的推断(Calvet和Fisher，2002)、精确过滤和最大似然估计(Calvet和Fisher，2001)；粒子过滤(Calvet，Fisher和Thompson，2006)；使用平衡理论的扩展，包括多重分形跳跃扩散(Calvet和Fisher，2007,2008)；以及利率(Calvet，Fisher和Wu，2013)和期权定价(CalveFisher，Fisher，Fearnley和Leippold，2014)的应用。*拟议的研究促进了多重分形工具的发展，并开发了新的应用。到目前为止，基于离散值状态变量的马尔可夫切换多重分形(MSM)一直是多重分形在金融中应用的关键构件。最近的粒子滤波技术允许考虑更广泛的具有扩散分量的模型集。这在期权定价应用中尤其有利，相对于标准基准显示出良好的结果。这些粒子过滤方法的应用和进一步发展将是一个关键的研究领域。该研究计划还将开发更简单的计算方法，使混合数据抽样方法(“Midas”、Ghysel、Santa-Clara和Valkanov，2006)适应多重分形过程。*该研究计划还将寻求通过调查股票之间的信息传输来确定金融数据多重分形的原因，股票被建模为一个网络。我和合著者已经证明了每日股票回报中信息传播缓慢的证据，最好的例子是双曲线衰减(Boguth，Carlson，Fisher，and Simutin，2014)。这项拟议的研究将使用一个相对较新的名为RavenPack的新闻数据库，检查以更高频率传输的信息。新闻来自专业博客、报纸和新闻通讯社，时间戳到毫秒。为了处理这个新闻数据库，比如为一家特定公司提取多年的新闻，由于数据的大小，需要并行数据处理。这项研究将使人们能够准确跟踪股票之间的新闻传播，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性。**