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Modeling fractal dynamical systems in the complex space. From macro-observation to micro-observation. (English) Zbl 1022.37029

For suitable modeling of fractional Markovian processes one should consider complex-valued stochastic processes following 3 axioms (strip modeling, centered small imaginary parts and consistency principle). This is the main message of this paper under consideration. For this purpose, the author treats ordinary stochastic differential equations as dynamical systems in the complex plane. He describes the dynamics of fractal systems driven by multidimensional fractional Brownian motions with independent increments. A brief summary on facts related to fractional Brownian motion modeled as rotating white noise is given. Eventually, the presented approach is applied to the stock market dynamics for which Mandelbrot (1999) has already pointed out its fractional properties (which the standard models of mathematical finance do not have so far!). It turns out that modeling of micro-dynamics in the complex plane is a suitable way to describe zooming from macro- to micro-type of observations.

MSC:

37F05 Dynamical systems involving relations and correspondences in one complex variable
37H10 Generation, random and stochastic difference and differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93E03 Stochastic systems in control theory (general)
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References:

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