# Economic examples of (sub)martingales

A (discrete time) martingale is a stochastic process $\{X_t\}_{t\in\mathbb N}$ that satisfies, for all $t\in\mathbb N$, $$E(|X_t|)<\infty$$ and $$E(X_{t+1}|X_1,\dots,X_t）=X_t.$$ And a submartingale is one with $E(X_{t+1}|X_1,\dots,X_t）\ge X_t$.

I know the classic example of the gambler's payoff in a fair game. But I'm hoping to find other real world examples of martingales and submartingales, particularly those pertaining to economics. Ideally, these examples would have some sort of empirical backing.

Applying the Law of Iterated Expectations on the defining property of a sub-martingale $E(X_{t+1}|X_1,\dots,X_t）\ge X_t$ we have that
$$E\Big[E(X_{t+1}|X_1,\dots,X_t)\Big]=E(X_{t+1}) \geq E(X_t)$$
$$X_{t+1} = \alpha + X_{t} + u_t,\;\; \alpha \geq 0$$