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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.

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Did you notice the distinction between a "discrete-time martingale" and a "continuous-time martingale"? – Shai Covo Dec 9 '10 at 4:37
An intuitive way to think about a filtration is that each $\sigma$-algebra $\mathcal F_n$ in a filtration carries the information of what happened up to time $n$. – trutheality Dec 9 '10 at 4:46
Yes I did Shai. But need better understanding of filtration in continuous-time martingale – user957 Dec 9 '10 at 7:15
@user957: Can you be more specific? Also, have you noticed the distinction between a "natural filtration" and just a "filtration"? – Shai Covo Dec 9 '10 at 7:38
@user957: I suggest you take a look at, pages 13-14. – Shai Covo Dec 9 '10 at 7:49
up vote 6 down vote accepted

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.

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