Given a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ satisfying the usual conditions, $B$ a standard one-dimensional Brownian motion and $H\in(0,1/2)$. Consider the process $X$ given by $$ X_t = \int_0^t \frac{(t-s)^{H-1/2}}{\Gamma(H+1/2)} d B_s, $$ where $\Gamma$ denotes the gamma function. The process $X$ is called fractional Brownian motion of Riemann-Liouville type with Hurst parameter $H.$ I have read in various articles that it is well-known that $X$ is not a semimartingale. I am trying to understand why this statement is true. However, I find it difficult to show this. I tried approximating the non-differentiable kernel $K(t)=\frac{t^{H-1/2}}{\Gamma(H+1/2)}$ by continuously differentiable kernels, $\varepsilon>0,$ $$ K_\varepsilon(t)= \frac{(t+\varepsilon)^{H-1/2}}{\Gamma(H+1/2)}. $$ Then, an application of the stochastic Fubini theorem shows that $$ X_t^\varepsilon = \int_0^t K_\varepsilon(t-s) dB_s $$ is a semimartingale with quadratic variation $ \langle X^\varepsilon\rangle_t = K_\varepsilon(0)^2 t $ and $\langle X^\varepsilon\rangle_t\to\infty$ as $\varepsilon\downarrow0.$ Moreover, I was able to show that $\mathbb{E}(|X_t^\varepsilon-X_t|^2)\to0$ as $\varepsilon\downarrow0.$ Now, I aim to conclude that $\langle X\rangle_t=\infty$ which would yield the claim.
I would be grateful for any suggestion on how to finish my proof or any other way that shows the assertion.
I see another way of finishing the proof. I think, we can begin to show that $\langle X,X\rangle_{1}=+\infty$ and by scaling argument, it will be proven for all $t$. We can use the quadratic variation approximation property which states that $\langle X,X\rangle_{1}$ can be approximated by $$\underset{n\rightarrow +\infty}{\lim}\sum_{j=1}^{n}\lvert X(\frac{j}{n})-X(\frac{j-1}{n})\rvert^{2} $$ Then, using that $(a^{-H}X(at)\sim X(t))$, we have that the last expression has the same distribution that $$\underset{n\rightarrow +\infty}{\lim}\sum_{j=1}^{n}\lvert X(j)-X(j-1)\rvert^{2} n^{-2H} $$ and, now we can apply ergodic theorem to state that $$\frac{1}{n}\sum_{j=1}^{n}\lvert X(j)-X(j-1)\rvert^{2} $$ converges in $\mathbf{L}^{1}$ and a.s to $\mathbf{E}[\lvert X(1)\rvert^{2}]$, and since $1-2H>0$, we can conclude that the $\langle X,X\rangle_{t}=+\infty$.
Maybe, something silly, but have you tried to use Burkholder-Davis-Gundy inequality to show that the quadratic variation was also going to $0$?
I hope it is clear enough!