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.

1 Answer

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

$$E\Big[E(X_{t+1}|X_1,\dots,X_t)\Big]=E(X_{t+1}) \geq E(X_t)$$

So a sub-martingale has also its unconditional expected value non-decreasing. Therefore any economic variable that appears to consistently increase in value is a candidate for being a sub-martingale, since the defining property here conditions only on the variable's past, not on any other factors that may influence it. National Output/GDP could be reasonably modeled as a sub-martingale, since it historically "always" increases, bar catastrophic events.

In general, any economic variable that can be modeled as a random walk with non-negative drift,

$$X_{t+1} = \alpha + X_{t} + u_t,\;\; \alpha \geq 0$$

is a sub-martingale.