What is the fractal dimension of Brownian motion?
A d-dimensional Brownian motion is known to be re- current, i.e., the particle returns to the origin, for d ≤2 and escapes to infinity for d >2. It is also known that the fractal (Hausdorff) dimension of the graph of a Brownian motion is equal to 3/2 for d =1, and 2 for d ≥2.
Is Brownian motion a fractal?
Brownian motion is a simple concept. A particle making random jumps traces out a trail which, if one steps back, has structure on all scales – it is a fractal.
Is fractional Brownian motion stationary?
Fractional Brownian motion (fBm) is the only Gaussian self-similar process with stationary increments.
What is rough volatility?
Rough volatility is a relatively new concept originating from the empirical observation that log-volatility essentially behaves as a fractional Brownian motion at any reasonable timescale.
Which is an example of a fractional Brownian motion?
Theoretical approach. Since R H is a p ositive definite operator, the is a fractional Brownian motion. In particular, for H = 0, W 0 and for H = 1 / 2, W 1 / 2 is the standard Brownian motion. is a fBm (H).
Who is the fractional Brownian noise named after?
For n = 1, n-fBm is classical fBm. Like the Brownian motion that it generalizes, fractional Brownian motion is named after 19th century biologist Robert Brown; fractional Gaussian noise is named after mathematician Carl Friedrich Gauss .
Is the increment of fBm independent of Brownian motion?
Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a continuous-time Gaussian process BH ( t) on [0, T ], that starts at zero, has expectation zero for all t in [0, T ], and has the following covariance function :
What is the Hurst index of fractional Brownian motion?
where H is a real number in (0, 1), called the Hurst index or Hurst parameter associated with the fractional Brownian motion. The Hurst exponent describes the raggedness of the resultant motion, with a higher value leading to a smoother motion. It was introduced by Mandelbrot & van Ness (1968) .