Martingale

Discrete Time Filtration

A (discrete time) filtration on the probability space $(\Omega ,\Sigma ,\mathbb{P})$ is a sequence of sigma-algebras on $\Omega$, $({\mathcal{F}}_n{)}_{n\in \mathbb{N}}$, with the property that ${\mathcal{F}}_0\subset {\mathcal{F}}_1\subset \cdots \subset \Sigma .$

Discrete Time Stochastic Process

Let $(\Omega ,\Sigma ,\mathbb{P})$ be a probability space with filtration $({\mathcal{F}}_n{)}_{n\in \mathbb{N}}$. A sequence of random variables $(X_n{)}_{n\in \mathbb{N}}$ is called a discrete time stochastic process. $(X_n{)}_{n\in \mathbb{N}}$ is said to be adapted to the filtration $({\mathcal{F}}_n{)}_{n\in \mathbb{N}}$ if for each $n\in \mathbb{N}$, $X_n$ is ${\mathcal{F}}_n$-measurable.

Martingale

Let $(\Omega ,\Sigma ,\mathbb{P})$ be a probability space with filtration $({\mathcal{F}}_n{)}_{n\in \mathbb{N}}$. Let $(M_n{)}_{n\in \mathbb{N}}$ be an adapted stochastic process with $M_n\in L^2(\Omega ,{\mathcal{F}}_n,\mathbb{P})$ for each $n\in \mathbb{N}$. $(M_n{)}_{n\in \mathbb{N}}$ is called a martingale with respect to the filtration $({\mathcal{F}}_n{)}_{n\in \mathbb{N}}$ if $M_n=E(M_{n+1}\mid {\mathcal{F}}_n)$ for all $n\in \mathbb{N}$. $(M_n{)}_{n\in \mathbb{N}}$ is called a submartingale with respect to $({\mathcal{F}}_n{)}_{n\in \mathbb{N}}$ if $M_n\le E(M_{n+1}\mid {\mathcal{F}}_n)$ for all $n\in \mathbb{N}$.

Predictable Random Variables

A sequence of random variables $(A_n{)}_{n\in \mathbb{N}}$ is called predictable with respect to the filtration $({\mathcal{F}}_n{)}_{n\in \mathbb{N}}$ if $A_n$ is ${\mathcal{F}}_{n-1}$-measurable for all $n\in {\mathbb{N}}^{\ast }$.

Martingale Transform

Let $(M_n{)}_{n\in \mathbb{N}}$ be a martingale and $(A_n{)}_{n\in \mathbb{N}}$ a predictable process with respect to the filtration $({\mathcal{F}}_n{)}_{n\in \mathbb{N}}$. Define ${\tilde{M}}_n:={\sum }_{j=1}^n{A}_j(M_j-M_{j-1}),$ for each $j\in \mathbb{N}$. $({\tilde{M}}_n{)}_{n\in \mathbb{N}}$ is called the martingale transform of $(M_n{)}_{n\in \mathbb{N}}$ by $(A_n{)}_{n\in \mathbb{N}}$.

Theorem

Let $(M_n{)}_{n\in \mathbb{N}}$ be a martingale and $(A_n{)}_{n\in \mathbb{N}}$ a predictable process. If $\Vert A_n{\Vert }_{\infty }<\infty$ for each $n\in \mathbb{N}$, then $({\tilde{M}}_n{)}_{n\in \mathbb{N}}$ is a martingale.

$M_n$ be the wealth of the gambler at time $n$ given that they have only bet a unit stake at each round. Suppose that the process $(M_n{)}_{n\in \mathbb{N}}$ is a martingale. This means that the gambler's total wealth is not expected to change. Suppose instead that the gambler decides on the strategy of betting $A_n$ units for the $n$th round of cards. Then their total wealth will be given by the martingale transform of $(M_n{)}_{n\in \mathbb{N}}$ by $(A_n{)}_{n\in \mathbb{N}}$. Will the gambler's strategy improve their fortune? The above theorem states that no matter what strategy is used by the gambler, the total wealth will remain a martingale. This is a rigorous formulation of what is intuitively obvious; you can't beat a fair game.

Consider once again our gambler playing cards. Let