I am familiar with martingale pricing as primarily a notational abstraction which allows us to price contingent claims on $X_\tau$ by its conditional expectation. Usually, we interpret this to mean that $X_\tau$ follows a zero-drift stochastic process, such as a GBM. Such an interpretation then allows us to efficiently solve for the expectation using numerical methods, such as MC.
However, it seems to me that there may be cases where we may be agnostic to the process and its distribution, but still be able define the conditional value under some equivalent measure.
Does taking this conditional expectation require us to define a stochastic process and/or its terminal distribution?