<p>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</p>

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

<p>So a sub-martingale has also its <em>unconditional</em> 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.</p>

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

<p>$$X_{t+1} = \alpha + X_{t} + u_t,\;\; \alpha \geq 0$$</p>

<p>is a sub-martingale.</p>

CC BY-SA 3.0
2015-06-30T00:27:13.340Z
2015-06-30T10:16:57.783Z
2015-06-30T10:16:57.783Z
Alecos Papadopoulos
<p>A (discrete time) martingale is a stochastic process $\{X_t\}_{t\in\mathbb N}$ that satisfies, for all $t\in\mathbb N$, 
$$
E(|X_t|)&lt;\infty
$$
and 
$$
E(X_{t+1}|X_1,\dots,X_t）=X_t.
$$
And a submartingale is one with $E(X_{t+1}|X_1,\dots,X_t）\ge X_t$.</p>

<p>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.</p>

<reference-request><empirical-evidence>
Economic examples of (sub)martingales
Herr K.
# Economic examples of (sub)martingales

(Asked by: username_0 on 2015-06-29)

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.

**Tags:** reference-request, empirical-evidence

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## 1 answers

### Answer by username_1 on 2015-06-30. Score: 2 | Accepted 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.


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