Coefficients of Mandelbrot - van Ness integral representation of fractional Brownian motion

Question

There are several integral representations of fractional Brownian motion (Hurst parameter $H$) with respect to standard Brownian motion. One of the most commonly used one is Mandelbrot-van Ness. However, I've seen two slightly different versions of Mandelbrot-van Ness and am not sure how they are consistent with each other:

(1) Mandelbrot and van Ness, 1968: $t>0$, $$ W_t^{H} = \frac{1}{\Gamma(1/2+H)}\left\{ \int_{-\infty}^0 \left[ \frac{1}{(t-s)^{1/2-H}} - \frac{1}{(-s)^{1/2-H}} \right] dW_s + \int_0^t \frac{dW_s}{(t-s)^{1/2-H}} \right\}. $$

(2) Mandelbrot - van Ness, alternative form (see Jost 2008): $t\in \mathbb{R}$, $$ W_t^{H} = C_H \left\{ \int_{-\infty}^t \frac{dW_s}{(t-s)^{1/2-H}} - \int_{-\infty}^0 \frac{dW_s}{(-s)^{1/2-H}}\right\}, $$ where $C_H = \sqrt{\frac{2H\times \Gamma(3/2- H)}{\Gamma(1/2 + H)\times \Gamma(2-2H)}}$.

I noticed that the domain of $t$ are slightly different in the above two versions, one for $t>0$ and one for $t\in\mathbb{R}$. Also, the terms are arranged slightly differently in the above two versions. Supposedly, both versions should satisfy the same correlation function: $$ \mathbb{E}\left[ W_{t'}^{H} W_t^{H} \right] = \frac{1}{2}\left\{ t^{2H} + t'^{2H} - |t-t'|^{2H} \right\}. $$ My question is that why are the normalizing coefficients in the two versions so different? $$ \frac{1}{\Gamma(1/2+H)} \quad \text{ .vs. } \quad \sqrt{\frac{2H\times \Gamma(3/2- H)}{\Gamma(1/2 + H)\times \Gamma(2-2H)}}\quad ??? $$

Answer 1

If you let $B_{0}^{H}=0$, then $W_t^{H} = C_H \left\{ \int_{-\infty}^t \frac{dW_s}{(t-s)^{1/2-H}} - \int_{-\infty}^0 \frac{dW_s}{(-s)^{1/2-H}}\right\}$ with $C_H = \sqrt{\frac{2H\times \Gamma(3/2- H)}{\Gamma(1/2 + H)\times \Gamma(2-2H)}}$.

In Definition 2.1. of Mandelbrot and Van Ness (1968), $B_{0}^{H}=b_0$.

Answer 2

The fractional Brownian motion as defined by the Mandelbrot Van Ness Representation actually defines a processes $W^H$ which has the correlation structure $$\mathbb{E}W^H_s W^H_t = \frac{V_H}{2}\{|t|^{2H} + |s|^{2H} - |t-s|^{2H}\},$$ with $$V_H = \left(\frac{1}{\Gamma(H+\frac{1}{2})}\right)^2\left\{\int_{0}^{\infty}\left((1+s)^{H-\frac{1}{2}} - s^{H-\frac{1}{2}}\right)^2ds + \frac{1}{2H}\right\}.$$ See Corollary 3.4. in [Mandelbrot and Van Ness 1968].

Now if we assume that one can show that $C_H^{-2} = \Gamma(H+\frac{1}{2})^{2} V_H$ then the to representations are consistent.

To the question: Why did Mandelbrot and Van Ness chose $\Gamma(H+\frac{1}{2})^{-1}$ as a scaling factor? I can only suspect that they were motivated to take the same scaling factor as for the Riemann-Liouville Brownian motion, which in turn comes from the scaling of the fractional integration operator.