martingale

Martingales definition

Definition. Let $(\\Omega,\\mathcal{F},(\\mathcal{F}_{t})_{t\\in\\mathbb{T}},\\mathbb{P})$ be a filtered probability space and $(X_{t})$ be a stochastic process such that $X_{t}$ is integrable (http://planetmath.org/Integral2) for all $t\\in\\mathbb{T}$ . Then, $X=(X_{t},\\mathcal{F}_{t})$ is called a submartingale if

\\mathbb{E}^{\\mathbb{P}}[X_{t}|\\mathcal{F}_{s}]\\geq X_{s},\\,\\mbox{for every $s<%t$, a.e.[$\\mathbb{P}$],}

and a supermartigale if

\\mathbb{E}^{\\mathbb{P}}[X_{t}|\\mathcal{F}_{s}]\\leq X_{s},\\,\\mbox{for every $s<%t$, a.e.[$\\mathbb{P}$].}

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

\\mathbb{E}^{\\mathbb{P}}[X_{t}|\\mathcal{F}_{s}]=X_{s},\\,\\mbox{forevery $s<t$, a.e.[$\\mathbb{P}$].}

Similarly, if the $\\{\\mathcal{F}_{t}\\}$ form a decreasing collection of $\\sigma$ -subalgebras of $\\mathcal{F}$ , then $X$ is called a reverse submartingale if

\\mathbb{E}^{\\mathbb{P}}[X_{s}|\\mathcal{F}_{t}]\\geq X_{t},\\,\\mbox{for every $s<%t$, a.e.[$\\mathbb{P}$],}

and a reverse supermartingale if

\\mathbb{E}^{\\mathbb{P}}[X_{s}|\\mathcal{F}_{t}]\\leq X_{t},\\,\\mbox{for every $s<%t$, a.e.[$\\mathbb{P}$].}

Remarks

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The martingale property captures the idea of a fair bet, where theexpected future value is equal to the current value.
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The submartingale property is equivalent to
$\\int_{A}X_{t}\\,d\\mathbb{P}\\geq\\int_{A}X_{s}\\,d\\mathbb{P}\\,\\,\\,\\mbox{for every %$A\\in\\mathcal{F}_{s}$ and $s<t$}$
and similarly for the other definitions. This is immediatefrom the definition of conditional expectation.