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+---
+title: "Martingale"
+date: 2021-10-31
+categories:
+- Mathematics
+- Stochastic analysis
+layout: "concept"
+---
+
+A **martingale** is a type of
+[stochastic process](/know/concept/stochastic-process/)
+with important and useful properties,
+especially for stochastic calculus.
+
+For a stochastic process $\{ M_t : t \ge 0 \}$
+on a probability filtered space $(\Omega, \mathcal{F}, \{ \mathcal{F}_t \}, P)$,
+then $M_t$ is a martingale if it satisfies all of the following:
+
+1.  $M_t$ is $\mathcal{F}_t$-adapted, meaning
+    the filtration $\mathcal{F}_t$ contains enough information
+    to reconstruct the current and all past values of $M_t$.
+2.  For all times $t \ge 0$, the expectation value exists $\mathbf{E}(M_t) < \infty$.
+3.  For all $s, t$ satisfying $0 \le s \le t$,
+    the [conditional expectation](/know/concept/conditional-expectation/)
+    $\mathbf{E}(M_t | \mathcal{F}_s) = M_s$,
+    meaning the increment $M_t \!-\! M_s$ is always expected
+    to be zero $\mathbf{E}(M_t \!-\! M_s | \mathcal{F}_s) = 0$.
+
+The last condition is called the **martingale property**,
+and basically means that a martingale is an unbiased random walk.
+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:
+
+3.  For $0 \le s \le t$, the conditional expectation $\mathbf{E}(M_t | \mathcal{F}_s) \ge M_s$.
+
+Analogouly, $M_t$ is a **supermartingale**
+if the current value is an upper bound instead:
+
+3.  For $0 \le s \le t$, the conditional expectation $\mathbf{E}(M_t | \mathcal{F}_s) \le M_s$.
+
+Clearly, submartingales and supermartingales are *biased* random walks,
+since they will tend to increase and decrease with time, respectively.
+
+
+
+## References
+1.  U.H. Thygesen,
+    *Lecture notes on diffusions and stochastic differential equations*,
+    2021, Polyteknisk Kompendie.