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。

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).