I am self-studying stochastic calculus in the context of mathematical finance, and the following is stated in my text without justification:
An Itô process is a random process $X(t)$ whose differential can be expressed as $$dX(t) = \xi\left(t, X(t)\right)dt + \sigma\left(t, X(t)\right)dZ(t).$$
A martingale is a process $X(t)$ for which $\textbf{E}[X(t + s) | X(t)] = X(t).$
An Itô process is a martingale if and only if the coefficient of dt, the drift, is identically zero.
Is there a proof of the above that one with only a background in elementary probability and measure theory and knowledge of the above definitions could understand?
This may appear to be a duplicate of the following: show that the solution is a local martingale iff it has zero drift
The above solution seems to require a more advanced understanding of stochastic processes. Is there a less rigorous, more general reasoning that I could use to convince myself?