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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=614337
• 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.

分形和多重分形在数学、工程和自然科学的许多领域都具有重要意义。第一个多重分形测度被开发并应用于地质学（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多重分形随机过程，基于时间变形的布朗运动（Calvet，Fisher，and Mandelbrot，1997）。随后的研究提供了基于矩的推理（Calvet和Fisher，2002年），精确滤波和最大似然估计（Calvet和Fisher，2001年）;粒子滤波（Calvet，Fisher和Thompson，2006年）;使用平衡理论的扩展，包括多重分形跳跃扩散（Calvet和Fisher，2007年，2008年）;利率应用（Calvet，Fisher和Wu，2013年）和期权定价（Calvet，Fisher，Fearnley和Leippold，2014年）。 提出的研究进展多重分形工具和开发新的应用。迄今为止，马尔可夫切换多重分形（MSM），离散值的状态变量的基础上，已成为一个关键的构建块的多重分形在金融中的应用。最近的粒子滤波技术允许考虑一组更广泛的模型与扩散组件。这应该特别证明在期权定价应用中是有利的，相对于标准benchmarks. Application和这些粒子滤波方法的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，通过适应混合数据采样方法（“MIDAS”，Ghysels，Santa-Clara和Valkanov，2006）多重分形过程。 该研究计划还将通过调查股票之间的信息传输来确定金融数据中多重分形的原因，其中股票被建模为网络。合著者和我已经证明了信息扩散缓慢的证据，最好用双曲线衰减来捕捉，在每日股票收益中（Boguth，Carlson，Fisher和Simplified，2014）。这项拟议中的研究将使用一个名为RavenPack的相对较新的新闻数据库，以更高的频率检查信息传输。新闻来源于专业博客、报纸和新闻专线，时间戳精确到毫秒。为了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的消息在股票之间的传输，识别相关的传输网络，识别的链接到波动性，并在资产收益的分形属性。