What follows, up to the horizontal line, is taken from Rogers "Arbitrage with fractional Brownian motion".
Consider an interval $[0,T]$ on which is defined the fractional Brownian motion $B$, and consider its partitions $\pi_n = \{t^n_k = \frac{kT}{n} : 0\le k\le n\},\ n\in\mathbb N$.
Let $p\ge1$, the $p$-variation of $B$ is
$$
V_p(B) = \lim_{n\to\infty} \sum_{k=0}^{n-1} |B(t^n_{k+1})-B(t^n_k)|^p =
\begin{cases}
\infty, & \text{if }\ pH < 1, \\
0, & \text{if }\ pH > 1.
\end{cases}
$$
If $H>1/2$ we can choose $p\in(1,\frac1H)$ so that $pH<1$, then the $p$-variation is infinite, hence the quadratic variation of $B$ is infinite too.
If $H<1/2$ we can choose $p>2$ so that $pH<1$, then again we obtain that the $p$-variation and the quadratic variation of $B$ are infinite.
In both cases the quadratic variation of $B$ is not finite, hence the fBm is not a semimartingale for $H\ne1/2$.
Could somebody further explain the above reasoning? I don't fully get what has to be proved, is it related to the fact that a semimartingale has to have finite variation? But which variation: quadratic, p-variation or another one?
Moreover, I don't understand how to deduce what the quadratic variation is, given that we know the p-variation. Is it related to the fact that given $p_1<p_2$ then $V_{p_2}\le V_{p_1}$?
Fianlly, what about the case $H=1/2$, in which $B$ is the usual Brownian motion? If we take $p\in(1,2)$ then we are still in the case $pH<1$ and so the $p$-variation is infinite hence the quadratic variation of $B$ is infinite too, contradicting the fact that B is a martingale.
Best Answer
Assume $B$ is a semimartingale, then it has finite quadratic variation.
Recall that if $s < b$ then $V_b \le V_s$.
If $H<1/2$ we can choose $p>2$ s.t. $pH<1 \implies V_p = \infty \implies \infty\le V_2 \implies V_2 = \infty$, i.e. the quadratic variation ($p=2$) is infinite too: contradiction.
If $H>1/2$ we can choose $p\in(\frac1H,2)$ s.t. $pH>1 \implies V_p = 0 \implies V_2 \le 0 \implies V_2 = 0 \implies B$ must have finite variation. But on the other hand, for $p\in(1,\frac1H)$ we have $V_p = \infty$, hence $B$ cannot have finite variation: contradiction.
Either way, if $H\ne\frac12$, the fBm is not a semimartingale.