Some basic probability and martingle problems

[nguyenxuanson]

Let be standard Brownian, and let then we define . All are martingale for finite time . We see that the filtration of M and Y are independent and it is not possible to construct a predictable process H to represent M in term of Y.

[shinichi9htv]

4) Prove or refute: If you know all the moments of a RV X (you know E(X^n) for all integer n), you'll know the distribution of X.

[nguyenxuanson]

4/ no, Marc Yor course i think? Construct example with log normal process a density which produce the same moments....

[shinichi9htv]

5/ What is a sufficient condition of X to make 4) true?

[shinichi9htv]

5) That's Carleman's crition. Intuitively, sum of integer moments converges to the characteristic function of X under this condition. A weaker one is lim sup E(X^n)^{1/n}/n < \infty

[nguyenxuanson]

If we know the characteristic function, then we can determine the measure. So intuitively the taylor expansion of characteristic function around 0 if converge then knowing the integer moment we can determine uniquely the measure.

I search the Carleman criterion, but it seems apply for the case where state space of random variable is the whole line, what happens if it is half lines...I guess it is something to do with Laguerre polynomials as they form a complete base for function define on half line (as Hermite for whole line)

Son

[shinichi9htv]

It's not that complicated bro. This crition applies to all the cases. A good reference is Feller's book "Introduction to prob, vol II"