I'll use an example of a three-step binomial tree to illustrate my problem (this is exactly equivalent to tossing a coin three times or a one dimensional random walk).

- At each step, a value can randomly move up (u) or down (d).
- Thus, the state space S = {uuu, uud, udu, udd, duu, dud, ddu, ddd} - all possible outcomes of three steps.
- F could then be a collection of subsets of S. I think that F in a discrete state space is taken to be the power set of S: the set of all subsets of F, but I'm not sure.

How can the filtration {F_t} be understood as ths "history" of the process? My idea of what this means is outlined below, but even as I type it I don't think it makes sense.

Is F_2 a sigma algebra over a different state space, say S_2 = {uu, ud, du, dd}? In this case the state space after three steps would have to be reconstructed to include the elements of S_2 (and by extension) S_1 in order to allow F_2 to be a sub-sigma algebra of F (defined over S).

Thus, rewrite S = {u, d, uu, ud, du, dd, uuu, uud, udu, udd, duu, dud, ddu, ddd}

and construct F = pow(S).

Say the first step is up, and the second set is down. So do we construct F_2 as the smallest sigma-algebra over S which contains subset {ud}? This doesn't seem to make sense as such a collection would be (S, null, {ud}, S/{ud}), where S/{ud} is the complement.

I think that I'm rambling now, so I'll stop. Can anybody explain this to me, or tell me if I'm heading in completely the wrong direction?

Any help would be appreciated.

Many thanks

Barry