Hi

I am currently studying a course in financial economics for a professional actuarial qualification and I'm having trouble with some of the probability theory.

I'm having trouble understanding probability spaces and filtrations. Can anyone help? I figure that this level of mathematical theory won't appear on the exam but I'd feel more comfortable if I understood it.

From what I've read, a probability space is a triple (S, F, P)

- S is the space of all possible outcomes
- F is a collection of subsets of S, a sigma-algebra (i.e. closed under complement in S, and under countable (possibly infinite) unions and hence under intersection)
- P is a measure of the elements of F such that P:F -> [0,1] on the reals

For a discrete case, each s in S can be thought of as an event, a single outcome of running through an experiment or observing a share price move. Each element in F is a subset of S, a collection of events (possibly satisfying some condition, like every outcome in which the price increases by a certain amount). The probability measure P assigns a value between 0 and 1 to each element in F.

S and the null-set are elements of any sigma-algebra over S and have probabilities P(S) = 1 and P(null-set) = 0. Intuitively, the probability of anything at all happening is 1 and the probability of nothing happening is 0

I'm comfortable with everything above (though maybe I just think I understand it). My trouble is with filtrations.

A filtration {F_t}t>=0 is a collection of ordered sub-sigma algebras such that F_s is a subset of (or equal to) F_t if s <= t

Question:

- Does this mean that each F_t is also a subset of F? Hence that each F_t is also a sigma-algebra on S?

If t is thought of as the time, then each F_t is the history of the process up to t... This I don't get at all.

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