martingale

Definition

• Probability space: a triple $(\Omega ,\Sigma ,\mathbb{P})$, where $\Omega$ is a set, $\Sigma$ is a sigma-algebra on $\Omega$, and $\mathbb{P}$ is a probability measure on $\Sigma$.

• Stochastic process: a collection of random variables indexed by a set, usually denoted as $\{X_t:t\in T\}$ or $\{X(t):t\in T\}$
  In stochastic process, $T$ often is a total ordered index set (e.g. $\mathbb{N},\mathbb{Z},\mathbb{R}$)

• Banach space: a complete normed vector space.

• Filtration: Let $(\Omega ,\Sigma ,\mathbb{P})$ be a probability space, $I$ be a total ordered index set, $\mathbb{F}=\{{\Sigma }_i:i\in I\wedge {\Sigma }_i\text{ is a sub-sigma-algebra of }\Sigma \}$, We say $\mathbb{F}$ is a filtration if ${\Sigma }_i\subseteq {\Sigma }_j$ for all $i<j$. $(\Omega ,\Sigma ,\mathbb{P},\mathbb{F})$ is called a filtered probability space.

• Adapted to the filtration: Let $(\Omega ,\Sigma ,\mathbb{P})$ be a probability space, $T$ be a total ordered index set, $\mathbb{F}=\{{\Sigma }_i:i\in T\}$ be a filtration of $\Sigma$, a stochastic process $X:T\times \Omega \to S$ is called adapted to the filtration $\mathbb{F}$ if for each $t\in T$, $X_t$ is a ${\Sigma }_t$-measurable function.

Martingale

In full generality, a stochastic process $Y:T\times \Omega \to S$ taking values in a Banach space $S$ with norm $\Vert \cdot {\Vert }_S$ is a martingale with respect to a filtration ${\Sigma }_{\ast }$ and probability measure $\mathbb{P}$ if

• ${\Sigma }_{\ast }$ is a filtration of the underlying probability space $(\Omega ,\Sigma ,\mathbb{P})$;

• $Y$ is adapted to the filtration ${\Sigma }_{\ast }$, i.e., for each $t$ in the index set $T$, the random variable $Y_t$ is a ${\Sigma }_t$-measurable function;

• for each $t$, $Y_t$ lies in the $L^p$ space $L^1(\Omega ,{\Sigma }_t,\mathbb{P};S)$, i.e.

  $${\mathbb{E}}_{\mathbb{P}}(\Vert Y_t{\Vert }_S)<+\infty ;$$

• for all $s$ and $t$ with $s<t$ and all $F\in {\Sigma }_s$,

  $${\mathbb{E}}_{\mathbb{P}}\left([Y_t-Y_s]{\chi }_F\right)=0,$$

  where ${\chi }_F$ denotes the indicator function of the event $F$.
  In Grimmett and Stirzaker’s Probability and Random Processes, this last condition is denoted as

  $$Y_s={\mathbb{E}}_{\mathbb{P}}(Y_t\mid {\Sigma }_s),$$

  which is a general form of conditional expectation.