Consider the process , alternatively . is Brownian motion. I read a proof that is a martingale that simply states “Because the diffusion of is 0, $X_t$ is not a martingale.”
By definition, a stochastic process adapted to a filtration is a martingale iff and
Question: What exactly about either of these conditions establishes that if a random process has 0 diffusion, it is not a martingale?
I am asking because I see the 0-diffusion condition used often for this purpose, but in the above example, of a process which is still random even though it has a zero diffusion, I don’t get it.