What proves that a random process with zero diffusion is not a martingale?


Consider the process dX_t=W_t dt+0 dW_t, alternatively X_t=\int_0^t W_s ds. W_t is Brownian motion. I read a proof that X_t is a martingale that simply states “Because the diffusion of dX_t is 0, $X_t$ is not a martingale.”

By definition, a stochastic process X_t adapted to a filtration \{F_t\} is a martingale iff E(|X_t|) <\infty, t \geq 0 and E(X_t|{\cal F}_s)=X_s, 0\leq s<t

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.

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