Assume a filtered probability space $(\Omega, \mathcal{G},(\mathcal{G}_t)_{t \in [0,1]}, \mathbb{P})$ and $(S_t)_{t \in [0,1]}$ an adapted stochastic process (no further path properties).
Fix $1 \leq p < \infty$. Is there an equivalent measure $\mathbb{P}' \sim \mathbb{P}$, s.t. $S_t$ is in $L^p(\Omega, \mathcal{G}, \mathbb{P}')$?
Backgound: I am reading a paper that poses this is true for a countable (but not uncountable) family of such processes, however without giving a proof. I am not even sure if that is true for the 1-dim case.
What I tried: A classical application of the Radon-Nikodym theorem. But, for a decent choice of density, I need $\mathbb{P}(\sup_t S_t < \infty) = 1$ as an assumption, what does not seem natural to me.
Thank you for taking your time.
Consider simple case, when $\Omega = [0,1]$. In this case we may consider Dirac measure $\delta_0$ and get what you want. But new process will have nothing common with the initial one.
In general case also there's such a problem.