Numerical evidence for the square root of a Wiener process

02/02/2012

Brownian motion is a very kind mathematical object being very keen to numerical simulations. There are a plenty of them for any platform and software so that one is able to check very rapidly the proper working of a given hypothesis. For these aims, I have found very helpful the demonstration site by Wolfram and specifically this program by Andrzej Kozlowski. Andrzej gives the code to simulate Brownian motion and compute Itō integral to verify Itō lemma. This was a very good chance to check my theorems recently given here by some numerical work. So, I have written a simple code on Matlab that I give here (rename from .doc to .m to use with Matlab).

Here is a sample of output:

As you could note, the agreement is almost perfect. I have had to rescale with a multiplicative factor as the square root appears somewhat magnified after the square but the pattern is there. You can do checks by yourselves. So, all my equations are perfectly defined as is a possible square root of a Wiener process.

Of course, improvements, advices or criticisms are very welcome.

Update: I have simplified the code and added a fixed scale factor to make identical scale. The code is available at Simulation. Here is an example of output:

Marco Frasca (2012). Quantum mechanics is the square root of a stochastic process arXiv arXiv: 1201.5091v2

27/01/2012

Disclaimer: This post is somewhat technical.

Recently, I posted a paper on arXiv (see here) claiming that quantum mechanics is the square root of a Wiener process. In order to get my results I have to consider some exotic Itō integrals that Didier Piau showed not existent (see here and here). In my argument I have a critical definition and this is the process $|dW(t)|$ that I defined using the sum

$S_n=\sum_{i=1}^n|W(t_i)-W(t_{i-1})|$

so that I assumed the limit $\lim_{n\rightarrow\infty}\langle S_n^2\rangle$ exists and is finite. This position appears untenable as Didier showed in the following way. In this case one has ($s,\ t>0$)

$\langle|W(t+s)-W(t)|\rangle=\sqrt{2s/\pi}$

and increments are independent so that $i\ne k$

$\langle|W(t_i)-W(t_{i-1})||W(t_k)-W(t_{k-1})|\rangle=$

$\langle|W(t_i)-W(t_{i-1})|\rangle\langle|W(t_k)-W(t_{k-1})|\rangle=\frac{2}{\pi}\sqrt{t_i-t_{i-1}}\sqrt{t_k-t_{k-1}}.$

Now, if you want to compute the limit in $L^2$ you are in trouble. Just choose $t_i=i/n$ and you will get

$\langle\left(\sum_{i=1}^n|W(t_i)-W(t_{i-1})|\right)^2\rangle$

that is

$\frac{2}{\pi}\frac{1}{n}\sum_{i=1}^n\sum_{j=1}^n.$

If you compute these sums you will get finally a term proportional to $n$ that blows  up in the limit of increasingly large $n$. The integral simply does not exist from a mathematical standpoint.

Of course, a curse for a mathematician is a blessing for a theoretical physicist, mostly when an infinity appears. Indeed, let us consider the sum

$\sum_{i=1}^\infty=1+1+1+1+\ldots$

People who have read Hardy’s book know for sure that this sum is just $-1/2$ (see also discussion here). This series can be regularized and so the limit can be taken to be finite!

$\langle S_n^2\rangle\rightarrow\ {\rm finite\ value}.$

This average is just finite and this is what I would expect for this kind of process. With this idea of regularization, the generalized Itō integral $\int_{t_0}^tG(t')|dW(t')|$ exists and is meaningful. The same idea can be applied to the case $\int_{t_0}^tG(t')(dW(t'))^\alpha$ with $0<\alpha<1$ and my argument is just consistent as I show that for $(dW(t))^\frac{1}{2}$ the absolute value process enters.

As a theoretical physicist I can say: Piau’s paradox is happily evaded!

Marco Frasca (2012). Quantum mechanics is the square root of a stochastic process arXiv arXiv: 1201.5091v1