martingaleMartingale2003-04-05 12:40:172008-12-27 12:16:58Definitionmartingalesupermartingalesubmartingalereverse submartingalereverse supermartingalemartingalesupermartingalesubmartingale% this is the default PlanetMath preamble. as your knowledge
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\textbf{Definition}. Let $(\Omega, \F,(\F_t)_{t\in\mathbb{T}},\Prob)$ be a filtered probability space and $(X_t)$ be a stochastic process such that $X_t$ is \PMlinkname{integrable}{Integral2} for all $t\in\mathbb{T}$. Then, $X=(X_t, \F_t)$ is called a \emph{submartingale} if
$$\mathbb{E}^{\Prob}[X_t|\F_s] \geq X_s,\, \mbox{for every $s < t$, a.e.[$\Prob$],}$$
and a \emph{supermartigale} if
$$\mathbb{E}^{\Prob}[X_t|\F_s] \leq X_s,\, \mbox{for every $s < t$, a.e.[$\Prob$].}$$
\noindent A submartingale that is also a supermartingale is called a
\emph{martingale}, i.e., a martingale satisfies
$$\mathbb{E}^{\Prob}[X_t|\F_s] = X_s,\, \mbox{for
every $s < t$, a.e.[$\Prob$].}$$
\noindent Similarly, if the $\{\F_t\}$ form a decreasing collection of $\sigma$-subalgebras of $\F$, then $X$ is called a \emph{reverse submartingale} if
$$\mathbb{E}^{\Prob}[X_s|\F_t] \geq X_t,\, \mbox{for every $s < t$, a.e.[$\Prob$],}$$
and a \emph{reverse supermartingale} if
$$\mathbb{E}^{\Prob}[X_s|\F_t] \leq X_t,\, \mbox{for every $s < t$, a.e.[$\Prob$].}$$
\medskip
\textbf{Remarks}
\begin{itemize}
\item
The martingale property captures the idea of a fair bet, where the
expected future value is equal to the current value.
\item
The submartingale property is equivalent to $$\int_A X_t \, d\Prob \geq
\int_A X_s \, d\Prob \,\,\, \mbox{for every $A \in \F_s$ and $s <
t$}$$ and similarly for the other definitions. This is immediate
from the definition of conditional expectation.
\end{itemize}