PlanetMath: martingale

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martingale

(Definition)

Definition 1 Let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space, $\{\mathcal{F}_t\}$ a filtration of $\sigma$ -subalgebras of $\mathcal{F}$ and $\{X_t,t \in \mathbb{R} \}$ a stochastic process. $(X_t,\mathcal{F}_t)$ is called a submartingale (resp. supermartingale) if

$\displaystyle \mathbb{E}^{\mathbb{P}}[X_t\vert\mathcal{F}_s] \geq (\leq) X_s,\,$ for every $0\leq s < t$ , a.e.[ $\mathbb{P}$ ].

A submartingale that is also a supermartingale is called a martingale, i.e., a martingale satisfies

$\displaystyle \mathbb{E}^{\mathbb{P}}[X_t\vert\mathcal{F}_s] = X_s,\,$ for every $0\leq s < t$ , a.e.[ $\mathbb{P}$ ].

Similarly, if the $\{\mathcal{F}_t\}$ form a decreasing collection of $\sigma$ -subalgebras of $\mathcal{F}$ and

$\displaystyle \mathbb{E}^{\mathbb{P}}[X_s\vert\mathcal{F}_t] \geq (\leq) X_t,\,$ for every $0\leq s < t$ , a.e.[ $\mathbb{P}$ ].

$(X_t,\mathcal{F}_t)$ is called a reverse submartingale (resp. reverse supermartingale)

Remark 1 The martingale property captures the idea of a fair bet, where the expected future value is equal to the current value.

Remark 2 The submartingale property is equivalent to

$\displaystyle \int_A X_t \, d\mathbb{P}\geq \int_A X_s \, d\mathbb{P}\,\,\,$ for every $A \in \mathcal{F}_s$ and $0\leq s < t$

and similarly for the other definitions. This is inmediate from the definition of conditional expectation

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"martingale" is owned by CWoo. [ full author list (4) | owner history (1) ]

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Also defines:

martingale, supermartingale, submartingale

Keywords:

martingale, supermartingale, submartingale

Attachments:: gale (Definition) by skubeedooo
Doob's optional sampling theorem (Theorem) by skubeedooo

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Cross-references: conditional expectation, definitions, equivalent, current, future value, property, collection, decreasing, stochastic process, filtration, probability space
There are 13 references to this entry.

This is version 19 of martingale, born on 2003-04-05, modified 2007-07-10.
Object id is 4157, canonical name is Martingale.
Accessed 12813 times total.

Classification:

AMS MSC:	60G42 (Probability theory and stochastic processes :: Stochastic processes :: Martingales with discrete parameter)
	60G44 (Probability theory and stochastic processes :: Stochastic processes :: Martingales with continuous parameter)
	60G46 (Probability theory and stochastic processes :: Stochastic processes :: Martingales and classical analysis)

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