fBM via Volterra kernel: where roughness comes from
A fractional Brownian motion is a Brownian integral against a one-parameter kernel $(t - s)^{H - 1/2}$. The kernel is what makes the increments correlated and the paths rough or persistent.
Fractional Brownian motion (fBM) generalises Brownian motion to processes whose variance grows like $t^{2H}$ for any $H \in (0, 1)$. The covariance-function definition (Mandelbrot-Van Ness 1968) is elegant but useless for simulation. The Volterra-kernel representation $B^H_t = c_H \int_0^t (t - s)^{H - 1/2}\,dW_s$ is what you actually compute. The kernel does all the work: a Hurst-dependent power that decays slowly for persistent processes and *spikes* near $s = t$ for rough processes. The kernel structure is also what makes simulation tractable (Bennedsen-Lunde-Pakkanen's hybrid scheme) and what the rough-Bergomi model puts under the spot dynamics.
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Worked example
The Type II Volterra fBM is defined by $B^H_t = c_H \int_0^t (t - s)^{H - 1/2}\,dW_s$ with normalisation $c_H = \sqrt{2H}$. Using the ItΓ΄ isometry, compute $\mathrm{Var}(B^H_t)$ for $H = 0.1$ and $t = 10$, and verify it matches the Hurst scaling $t^{2H}$ from the `hurst_exponent` tutorial.
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Reflection
The Volterra-kernel representation makes fBM simulable: you can discretise the integral against a Brownian increment array. Why does the BLP hybrid scheme add an Euler step near $s = t$ rather than using the bare power kernel everywhere, and what numerical pathology does that fix?