Hi we known that the quadratic variation of the one dimensional Brownian motion $B_t$ is $t$, so $d[B]_t=dt$.
Let $\sigma_t$ be a subordinator (https://en.wikipedia.org/wiki/Subordinator_(mathematics)), independent from $B$, if $B_{\sigma_t}$ is the time-changed Brownian motion $B \circ \sigma$, can I say $d[B_\sigma]_t=dt$? I think that is a diffusion, so it works.
Now, if $\sigma^{-1}$ is the inverse of $\sigma$, $\sigma^{-1}_s=\inf \{t \geq 0:\sigma_t > s \}$ for $s \geq 0$, what can I say about $d[B_{\sigma^{-1}}]_t$ ? We have that $B_{\sigma^{-1}}$ is not a Markov process, since $\sigma^{-1}$ is non-Markovian, so it's not a diffusion...
Thank you very much! If you could suggest some references, It would be a great help .