Help understanding filtered probability spaces

[tensorproduct]

Hi

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 (W, F, P) using W, because my keboard doesn't have an Omega key.

W is the space of all possible outcomes, F is a collection of subsets of W, and P is a measure such that P:W -> [0,1] on the reals.

Each w in W 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 W, 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 F.

A couple of side notes:

• F is a sigma-algebra, meaning that the collection of subsets is closed under complement and countably infinite unions (and countably infinite intersections by de Morgan's theorem).
• P(W) = 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

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

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.

• At each step, a value can randomly move up (u) or down (d).
• Thus, the state space W = {uuu, uud, udu, udd, duu, dud, ddu, ddd}
• F could then be a collection of subsets of W.
• How can F be constructed to correspond to a particular path? I can't see anyway to do this and to maintain the definition of a sigma-algebra above. i.e. closed under complements: if subset {uud} is an element of F then so is subset {uuu, udu, udd, duu, dud, ddu, ddd}. Do we understand then that the probability of uud, is the same as the probability of not-uud? Clearly that's not right.
• How can the filtration {F_t} be understood as ths "history" of the process? If we know that after two steps, both are up then is F_2 the collection of subsets containing both uud and uuu? In that case, F_1 would be the collection of subsets containing all states starting with u. To me, it seams that this would imply that F_2 is a subset of F_1 rather than the other way around.

Any help would be appreciated.

Many thanks
Barry

 

[John Potter]

Barry,

Could the state space in your example include {u, d, uu, ud, dd}? This way we can have Ft as the history up to time t, for example Ft could be {u, ud, udd}. The share price moves up then down twice. Does this help?

As you say, understanding Ft from a sigma-algebra point of view isn't going to help you at all in the CT8 exam. As long as you think of Ft as the history up to time t, you will be fine,

Good Luck!
John

 

[kylie jane]

In CT8 do they use filtration and sigma-algebra interchangeably, so effectively the filtration F equal the sigma-algebra F

 

[tensorproduct]

In CT8 do they use filtration and sigma-algebra interchangeably, so effectively the filtration F equal the sigma-algebra F

Not really. A sigma-algebra F is defined over the underlying state-space S, and a filtration {F_t} for t>=0 is defined as an ordered collection of sub-sigma-algebras of F.

So each F_t is a sigma-algebra in itself (closed under complement, countable infinite union and countably infinite intersection) and a subset of F. Also F_s is a subset or equal to F_t where s <= t.

All of this is pretty much standard from what I've read in other sources.

As I understand it now, for a three step binomial tree with state-space

S = {uuu, uud, udu, udd, duu, dud, ddu, ddd}

F can be defined as the powerset of S. That is, the set of all subsets of S, which would take far too long too enumerate here.

Now, we can say that

F_0 = {the empty set, S}

and

F_1 = {the empty set, S, {uuu, uud, udu, udd}, {duu, dud, ddu, ddd}}

It is trivial to show that these both satisfy the conditions to be algebras.

Under the probability measure P, defined against F, P(the empty set) = 0 and P(S) = 1.

So, as I understand it, to say that we are given the filtration up to a time t is to say that we know in which subset of F_t the underlying process lies at t.

In this example, if the first step on the binomial tree is up, then the process is in the {uuu, uud, udu, duu} element of F_1, which has finite measure under P.

I'm not 100% sure that all of the above is correct, but in the interests of passing the exam I've decided to just accept (for now) the general idea of a filtration as the history of the process.