Studying time-changed Levy processes, I ended up writing the following equation:
$$W\left(\int_0^t v(s) \, ds\right) = \int_0^t \sqrt{v(s)} \, d W(s)$$
where $v(s)$ can be either a deterministic function or a stochastic processes.
The problem is that my argument leading to that equation is completely heuristic and very much related to the paper I was reading.
If $v(t)$ is deterministic, what I wrote should be true at least in terms of distribution. Indeed, both quantities should be normally distributed with mean $0$ and variance $\int_0^t v(s) ds$.
Apart from this, I can't go further.
What do you think about that equation? Is it really true? If so, in which sense and why?
Does it really matter whether $v$ is deterministic or stochastic?
Thanks a lot for all your help!!