White Noise and Wiener Processes

[IAmEric]

Nonius saw through my thinly veiled attempt to bring NCG into the picture Applause



It is true, if someone could tell of any special algebraic relation for



dW' dW'



where W' is white noise, then I could reformulate white noise in terms of NCG, which might be kind of academically interesting.



The real point of this is, as I said, to try to understand the link (if any) between vector autoregression and multifactor stochastic differential equations. I'm just now learning the former and it is reminding me a lot of the latter.



Looking at VAR reminded me of my days back in grad EE and DSP, e.g. z-transforms, digital filters, impulse responses, etc., which made me think of deconvolutions, which made me think... (ad nauseum)

[hammerbacher]

this was precisely how the NYU math in finance "stochastic calculus" course taught by marco avellaneda last semester began--by pounding on the metaphor of white noise as a "derivative" of brownian motion. he's taken the homework files off the website, but i can send them to anyone interested in playing (numerically) with this stuff. it was very good for intuition.



disclaimer: i stopped attending this course after about three lectures, as avellaneda is a horrible lecturer. the psets are still decent, however.

[IAmEric]

hammerbacher,



Sounds cool. I'd be interested in taking a look at that and any notes you might have.



Thanks Beer

[tristanreid]

in regards to vector autoregression, my understanding is this:



AR:



[img]/User%20Files/802/MathML-Equation-3793.gif[/img]



where that last epsilon is your white noise. 



if phi<1, this series is covariance-stationary, so you can take the expected value of both sides.  E(epsilon)=0, so the expected value of the series is:



[img]/User%20Files/802/MathML-Equation-3794.gif[/img], or mu=c/(1-phi).



In other words, it's a mean reverting series.



if phi==1, y_t is a Wiener process (a random walk), it has a 'unit root'.  you can fix this by first differencing the series.  since the derivative is white noise, you are reduced to the above case.



if phi>1, this time series is explosive, which is just no good. 



VAR:



instead of phi, you have a matrix of coefficients, so you take the eigenvalues to see what effect each eigenvector is having.  You check the effect in the same way as above: any lambda>1 is explosive and makes the series worthless.  If all the lambdas are < 1 there is a static equilibrium to the system (kind of like mean-reversion).



The difference with VAR is that some lambdas could be random walks and others stationary.  That's where cointegration comes in (from that other thread)



-t.

[IAmEric]

Making some progress...




[quote]
Here is kind of a neat thing:



g(t)*W(t) = int_0^t g'(tau)*W(tau) dtau + int_0^t g(tau)*W'(tau) dtau



where W(t) is a Wiener process and W'(t) is Gaussian white noise.



I got that from this paper.
[/quote]



In the above expression, the function g(t) must be smooth and have finite support. If we let g(t) = 1 up to some time T and then smoothly taper to zero beyond, then the above reduces to



W(t) = int_0^t W'(tau) dtau



for t < T. This expresses what others have said about Brownian motion being the integral of white noise.



If we approximate this integral as a Riemann sum, we get



W(t) ~= W(t-delt) + delta W'(t).



Comparing this to what tristan said for c = 0 and phi = 1, we have



y(t) = y(t-1) + e(t).



This seems to support the statement that y(t) is Brownian motion for phi = 1. At least if you were to throw in a delt in there and take a limit as delt -> 0. I can buy that Smiley



This is looking like it supports my suspicion that AR could be thought of as a finite difference approximation to a stochastic DE (I think).



Still some mysteries though. If anyone could shed some light, it'd be appreciated.



For example, the integral implies



dW = W' dt.



We know that



dW dW = dt



so that means that W' ~ 1/sqrt(dt) since



dW dW = (W')^2 dt dt = dt.



If we wave the wand of NCG, we convert this to a commutator



[dW, W]

= dW W - W dW

= W' dt W - W W' dt

= W' W dt - W W' dt (W and dt commute)

= dt.



The only way to satisfy this is if



[W, W'] = 1



which looks like a quantization rule to me. I'll try not to get pulled off onto that tangent right now Smiley (Note for anyone interested, the commutative relation [dW, W] = i*hbar*dt leads to the Schrodinger equation and in this case would lead to [W, W'] = i*hbar, which looks even MORE like quantization where white noise and Brownian motion play the role of conjugate variables like position and momentum.)



Back to something remotely practical...



If we want to solve



dy = mu*dt + sigma*dW



approximately we can rewrite it as



dy = mu*dt + sigma*dW = mu*dt + sigma*W'*dt.



The commutative relations suggest W' ~ 1/sqrt(delt) so this becomes



y(t) - y(t-delt) = mu*delt + sigma*W'(t)*delt = mu*delt + sigma*e(t)*sqrt(delt)



where I've set



W'(t) = e(t)/sqrt(delt)



and I really don't know exactly why other than to make it look like a simple Monte Carlo expression (and it seems to somehow relate to the dimension analysis from the commutative relations) Smiley



Rearranging terms gives



y(t) = mu*delt + y(t-delt) + sigma*e(t)*sqrt(delt).



Setting



c = mu*delt



and



epsilon(t) = sigma*e(t)*sqrt(t)



brings us back to something that formally looks like the AR expression.



Ok. As you can see, there are still some holes in my arguments so if anyone can help nail the final missing pieces, I'd appreciate it. That, or show me where I'm totally off the mark.



Just to summarize...



This whole thing is motivated by trying to understand the relation between ARs and SDEs. My hunch is that AR may be interpretted as an approximate numerical (finite difference/Monte Carlo) solution to a simple SDE.



This might be obvious to some, so help me see the light. I'm almost there Smiley



Cheers Beer

Eric

[Cheng]

It is still a bit early to think and maybe I am babbling big nonsense but



y(t) = mu*delt + y(t-delt) + sigma*e(t)*sqrt(delt)



boils down to the Euler scheme for numerical solution of SDEs if e(t) is some N(0,1) distributed rv. Then you get the increment



sigma*sqrt(delt)*N(0,1)



or alternatively



N(0,sigma^2*delt).



Under some technical assumptions this converges to the SDE for delt->0 with order O(delt^(-1/2)). If mu and sigma are C^1 you can get convergence of order O(1/delt) by adding another term involving the first order derivatives (the Milstein scheme). Maybe this helps, if not feel free to trash it.



Regards

[kr]

eric - haven't read all the details, but my intuition is simply that dW should be thought of as a 'generalized function' - i.e. think dirac delta.  That is, it is defined only to the extent that one can compute = int_0^t g(t) dW_t, and the value of this functional is mostly determined by a few key algebraic relations.  In fact, writing is not the right way to go, much better to have W = <1, dW> be thought of as a random variable, and then use the rules for . 



Once that's said, it's clear that "dW'_t" does not operate on the correct space... it would have to be integrated against something else in order to produce a generalized function.  Somewhere under all this is a deep-discount graded algebra involving functions with existing derivatives.

[Cheng]

kr,



do you think along the lines of Sobolev spaces ? This was basically what I had in mind when I started mumbling about generalized derivatives.



Regards

[Nonius]

you are on the right track

[Martingale]

The origional way of Wiener's way to construct the stochastic integral is kind of from white noise as generalized distribution ( throgh some L^2 isometry), but it appoved to be limited, that's why people start to consider this from some other point... I will try to dig some history literature on this