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$$B_H(t_2,\omega)-B_H(t_1,\omega) = \frac{1}{\Gamma(H+1/2)}\Bigg\{\int_{-\infty}^{t_2}(t-s)^{H-1/2}dB(s,\omega)-\int_{-\infty}^{t_1}(t-s)^{H-1/2}dB(s,\omega)\Bigg\}$$ It's taken from Mandelbrot & Van Ness' (1968) definition of Fractional Brownian Motion. I believe it is a definition of the difference between values of the fBm process at $t_1$ and $t_2$, but I don't understand why the RHS refers to $t$ without a subscript and $s$. I want to know this because I would like to see how the covariance: $$ E\big[B_H(t)B_H(s)\big] = \frac{1}{2}\big(|t|^{2H}+|s|^{2H}-|t-s|^{2H}\big)$$

is derived from the above definition.

Surb
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bibbly bobbly
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1 Answers1

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I agree this is very confusing. Maybe a typo in the original article ? The above definition is a notation for the well-defined ($t_2>t_1$) : $$B_{t_2}^H-B_{t_1}^H = \frac{1}{\Gamma(H+1/2)} \left\{ \int_{-\infty}^{t_1} \left( (t_2-s)^{H-1/2} -(t_1-s)^{H-1/2} \right)dB_s +\int_{t_1}^{t_2} (t_2-s)^{H-1/2} dB_s \right\} $$ where each term is a Wiener integral with finite variance (Itô isometry). Indeed if you break the first integral and assemble the parts you get $$ B_{t_2}^H-B_{t_1}^H = \frac{1}{\Gamma(H+1/2)} \left\{ \int_{-\infty}^{t_2} (t_2-s)^{H-1/2} dB_s +\int_{-\infty}^{t_1} (t_1-s)^{H-1/2} dB_s \right\} $$ which is not well-defined but way more intuitive as a definition. So $t$ is actually either $t_1$ or $t_2$. As for $s$ it is a time variable bound to the corresponding Itô integral.

Note that the definition for $t_2=t$ and $t_1=0$ is sufficient to define the above and to compute the covariance function.

vanna
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