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authorPrefetch2021-11-14 17:54:04 +0100
committerPrefetch2021-11-14 17:54:04 +0100
commitc0d352dd0f66b47ee91fb96eaf320f895fa78790 (patch)
tree961eb3f1c6afcd418b0319aa2ec4c2c51b84f92a /content/know/concept/martingale
parentf2970c55894b3c8d5fd2926a8918d166988109fe (diff)
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@@ -37,6 +37,14 @@ Accordingly, the [Wiener process](/know/concept/wiener-process/) $B_t$
(Brownian motion) is an example of a martingale,
since each of its increments $B_t \!-\! B_s$ has mean $0$ by definition.
+Martingales are easily confused with
+[Markov processes](/know/concept/markov-process/),
+because stochastic processes will often be both,
+e.g. the Wiener process.
+However, these are distinct concepts:
+the martingale property says nothing about history-dependence,
+and the Markov property does not say *what* the future expectation should be.
+
Modifying property (3) leads to two common generalizations.
The stochastic process $M_t$ above is a **submartingale**
if the current value is a lower bound for the expectation: