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

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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).