Broken Brownian motion

The fractional Brownian motion or fractional Brownian motion is a class of centered Gaussian processes , represented by the following covariance are characterized: ${\displaystyle (X^{H}(t))_{t\geq 0}}$

${\displaystyle E[X^{H}(t)X^{H}(s)]={\frac {1}{2}}(|t|^{2H}+|s|^{2H}-|t-s|^{2H}),}$

where H is a real number in (0, 1). H is often called the Hurst parameter. For H = 1/2, the broken Brownian motion is a one-dimensional Brownian motion .

properties

Self-likeness

${\displaystyle X^{H}}$is self-similar . More precisely, the processes and have the same distribution for every fixed c> 0 . ${\displaystyle (X^{H}(ct))_{t\geq 0}}$ ${\displaystyle (c^{H}X^{H}(t))_{t\geq 0}}$

Stationary increments

The relationship follows directly from the representation of the covariance function

${\displaystyle E[(X^{H}(t)-X^{H}(s))^{2}]=|t-s|^{2H},\quad t,s\geq 0.}$

In particular, the increments are therefore stationary . In addition:

• if H = 1/2 the process has independent increments;
• if H > 1/2, the increments are positively correlated ;
• if H <1/2, the increments are negatively correlated.

Path properties

The paths of the broken Brownian motion with the Hurst parameter H are Hölder continuous with an index for each . ${\displaystyle \alpha }$ ${\displaystyle \alpha <H}$

Stochastic integration

It is possible to define stochastic integrals with respect to the Broken Brownian motion.

See also

• Anomalous diffusion

Web links

• Lecture video about the Broken Brownian Movement

swell

• Francesca Biagini, Yaozhong Hu, Bernt Øksendal, Tusheng Zhang: Stochastic Calculus for Fractional Brownian Motion . Springer, London 2010, ISBN 1-84996-994-9 (Softcover reprint of hardcover 1st ed. 2008).