Martingales

Wrt a filtration $( \mathcal{F}_n )^\infinity_{n=0}$ , a martingale is a sequence of random variables $( X_n )^\infinity_{n=0}$ with:

• $X_n \in \mathcal{F}_n$
• $E[X] \lt \infinity$
• $E[X_{n+1} | \mathcal{F}_n] = X_n$ .

Properties:

• $E[X_m | \mathcal{F}_n] = X_n$ if $n \lt m$ ,
• $E[X_n] = E[X_{n-1}] = E[X_0]$

Submartingales have $E[X_{n+1} | \mathcal{F}_n] \ge X_n$ . Supermartingales have $E[X_{n+1} | \mathcal{F}_n] \le X_n$ .

Regular martingales

If $( \mathcal{F}_n )^\infinity_{n=0}$ is a filtration and $X$ is a rv with $E[|X|] \lt \infinity$ then $M_n = E[X | \mathcal{F}_n]$ makes a martingale wrt $( \mathcal{F}_0)$ . Many martingales have this form.

Martingale transforms

A process (collection of rv’s) $(C_n)^\infinity_{n=1}$ is predictable or previsible wrt $(\mathcal{F}_n)^\infinity_{n=1}$ if $C_n \in F_{n-1}$ .

If $C_n$ is predictable and $(M_n)$ a martingale, then let $Y_n = \sum_{k=\phi}^n C_k (M_k = M_{k-1})$ , then $(Y_n)$ is a martingale.

Proof: $E[Y_{n+1} | \mathcal{F}_n] = E[Y_n + C_{n+1} (M_{n+1} = M_n) | \mathcal{F}] = Y_n + C_n+1 E[M_{n+1} - M_n | \mathcal{F}_n] = Y_n$