Lévy theorem - Ch9 - once again

[Sandor Kelemen]

I asked the following around 5 months ago and I did not get any satisfactory answer. Can anybody help me out now?

Can anybody give me a solid reference to the form of Levy theorem used in Chapter 9 to characterize Brownian motions?

This article
http://individual.utoronto.ca/norma.../Levy_characterization_of_Brownian_motion.pdf
is talking about quadratic variations as a central concept.

Comparing the covariation definition mentioned in
https://en.wikipedia.org/wiki/Quadratic_variation
and in the CMP I see some risk that for stochastic processes there are two different definitions (with different, i.e. not equivalent, underlying meaning).

Therefore I think that there is a chance that Levy's theorem is used inappropriately in the CMP.

Hopefully, there will be somebody who could help me out on this. Thanks in advance!

Thx in advance!

S.

 

[Steve Hales]

Hi

Can anybody give me a solid reference to the form of Levy theorem used in Chapter 9 to characterize Brownian motions?

The short answer is no.

The slightly longer answer is that (as you've clearly already discovered) this isn't really Levy's Theorem. Levy's Theorem is a characterisation of Brownian motion in terms of quadratic variation - a concept which isn't covered in CM2.

The closest thing to the Core Reading's version of 'Levy's Theorem' is Theorem 3.3 in the Introduction to Stochastic Calculus with Applications by Klebaner: "A Gaussian process with zero mean function, and covariance function min(t,s) is a Brownian motion."

As I mentioned before, this has been referred back to the IFoA for clarification.

Hope that helps!

Steve

 

[Sandor Kelemen]

Thank you Steve,

I think this is quite a critical theoretical misuse. The core reading states

a stoch. process with Cov(X_s,X_t) = min(s,t) is a Wiener process

.

1. If W is a Wiener process and c is a constant then for M_t := W_t +c we still have Cov(M_s,M_t) = min(s,t), however, M is clearly not a standard Brownian motion. This is the negligible deficiency as it is usually not hard to check the expected value of the process.
2. The material bug is, however, the Gaussian property mentioned above. Is there a quick/reasonably general way to prove about an f(t,W_t) formed stochastic process (with W_t being Wiener pr.) the gaussian property? I do not think so.

I checked the examples in Ch9 and I think that all can be proved honestly referring to the definition of Wiener process and omitting this inaccurate/deficient Lévy theroem approach. However, it is a bit more time-consuming. Hence, what would be a piece of wise advice for the exam to solve concisely such exercises?

 

[Steve Hales]

I understand - I feel your pain.
If I needed to use this Cov(M(s),M(t)) = min(s,t) result in an exam I would write "Levy's Theorem according to the Core Reading says that M(t) is a Brownian motion."