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

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

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.


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