martingale and filtration

As I understand, martingale is a stochastic process (i.e., a sequence of random variables) such that the conditional expected value of an observation at some time $t$, given all the observations up to some earlier time $s$, is equal to the observation at that earlier time $s$.

A sequence $Y_1, Y_2, Y_3 ...$ is said to be a martingale with respect to another sequence $X_1, X_2, X_3 ...$ if for all $n$:

$E(Y_{n+1}|X_1,...,X_n) = Y_n$

Now I don't understand how it is defined in terms of filtration. Does filtration discretize the time space of a stochastic process so that we can analyze the process as a martingale? A simple explanation or an example on what is filtration and how it relates to martingale theory would be very helpful. I can then read more detailed content.

Solution 1:

A Filtration is a growing sequence of sigma algebras $$\mathcal{F_1}\subseteq \mathcal{F_2}\ldots \subseteq \mathcal{F_n}.$$

Now when talking of martingales we need to talk of conditional expectations, and in particular conditional expectations w.r.t $\sigma$ algebra's. So whenever we write $$ E[Y_n|X_1,X_2,\ldots,X_n]$$

we can alternatively write it as $$E[Y_{n+1}| \mathcal{F_{n}}],$$ where $\mathcal{F}_{n}$ is a sigma algebra that makes random variables $$X_1,\ldots,X_n$$ measurable. Finally a flitration $\mathcal{F_1},\ldots \mathcal{F_n}$ is simply an increasing sequence of simga algebras. That is we are conditioning on growing amounts of information.