Assume that one has constructed a probability space $(\Omega, \mathcal F, Q)$ and a cadlag process $X$ which follows some properties of interest, i.e martingale property.
Can one always construct an equivalent measure $P$?
The background and reason for my question: In various texts and articles I have seen that financial engineers construct a martingale process directly in the risk neutral world. They simply seem to assume that there exist an equivalent measure $P$ representing the physical world on which the process does not allow for arbitrage opportunities, and thus applying the fundamental theorem of asset pricing.
P.S: The process $X$ does not need to be of the same class in $P$ as in $Q$, it does not even have to still be a martingale with respect to $P$. As long as the process $X$ on $(\Omega, \mathcal F, P)$ does not allow for any arbitrage opportunities.