# What is the extension of fractional Brownian motion to describe statistical multiscaling?

A random variable $$X(t)$$ is said to be monoscaling if $$X(t) = a^{-H}X(at).$$ $$H$$ is called the Hurst exponent, and $$a$$ is a scaling factor. A key model of statistical monoscaling is the fractional Brownian motion.

More generally, a variable can be multiscaling, meaning each moment changes to a different degree upon scaling $$t$$ to $$at$$.

How does one model statistical multiscaling?

• The terms monoscaling and multiscaling are new to me. By multi-scaling, do you mean that the variance, skewness, and kurtsosis also have the sort of fractionally autoregressive properties?
– John
Oct 11, 2019 at 17:40
• Yes, exactly. The $q$th order moments at aggregation scale $a$ are $M(q,a) = \langle X(at)^q \rangle$. If $M(q,a) \propto a^{qH}$, the random variable $X(t)$ is said to be monoscaling, while if $M(q,a) \propto a^{\tau(q)}$, where $\tau(q)$ is some non-linear function, $X$ is said to be multiscaling. "multi" denotes the need for multiple Hurst exponents to describe the scale dependence of the random variable, if I understand properly. I have some experimental data that exhibit multiscaling and I don't know how to model them. Oct 11, 2019 at 18:53
• I can't say that I can help much. You might try starting with Mandelbrot et al's "A Multifractal Model of Asset Returns." (Cowles #1164).
– John
Oct 11, 2019 at 20:09