Multifractal Formalism for Stochastic Processes

Original definition of fractal was proposed by Mandelbrot with respect to sets. He defined fractal as a mathematical set with fractal dimension is strictly larger than its topological dimension (Mandelbrot 1975,1982). Later he extends this definition, calling a fractal any kind of self-similar structure (Mandelbrot 1985).

For the stochastic processes the notion of fractality is based on the defini­tion of self-affine processes—processes that keep statistical properties under any affine transformations. Being more strict, a stochastic process X = {X (t); t > 0; X (0) = 0} is called self-affine, if for 8 c > 0 and time moments ti,…,tk > 0, the following expression holds:

{X(ctl),…,X(c tk)} = {cHX (ti) ,…,0HX (tk)}, (3)

where H is a constant named self-affine index and symbol “=” stands for equality in distribution. For stochastic processes with stationary increments l;X (t) = X (t + l) — X (t) the self-affinity is usually defined not using the multivariate probability distribution functions as in (3) but via moments of increments:

Mq (l) = E [|liX (t) |q] = E [|X (t C l) — X (t) |q], (4)

where E […] stands for averaging over ensemble of realizations. Functional form, which describes the dependency of moments as a function of q, plays the key role in the determination of scale invariance properties. If all moments of increments Mq (l) can be represented in a power law form:

Mq (l) = Kql^q, (5)

where Kq and £q depends only on q, then the process X (t) is said to have multifractal properties. The functional dependency of scale indices £q on order q (£q = f (q)) is called a multifractal spectrum, and its form defines the self-similarity properties of the process: stochastic process X(t) is said to have monofractal properties if £q is a linear function of q (£q = qH). If £q is a nonlinear function of q then the process is said to have multifractal properties. Typical examples of monofractal processes are random walks, their genralisation of fractional Brownian motion and Levi flights. The examples of multifractal processes were discussed in previous section.

We need to mention, that multifractal properties cannot be maintained for arbitrary small or arbitrary large scales l. For strictly convex or strictly concave function £q the interval of scales is bounded either from below or above correspond­ingly (Mandelbrot et al. 1997). Alternatively, function £q may have both convex, concave and linear part (like in the QMF model (Filimonov 2010)) and there exists strictly bounded interval of scales

x ^ l ^ L, (6)

Подпись: £q > 1, Подпись: when Подпись: q > 1. Подпись: (7)

where scale index £q in (5) has nonlinear dependency with respect to q. Analogically to turbulence theory, interval (6) is called inertial. Similarly to the theory of turbulence, scale x is called scale of viscosity and scale L is an integral scale. Finally, it can be shown analytically that multifractal spectrum of the strictly nondecreasing stochastic process has to satisfy following condition (Filimonov 2010):

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