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We know that white noise $w_{t}$ is given by the time derivative of Brownian motion $\beta_{t}$, ie that:

$$ w_{t} = \frac{d \beta_{t}}{dt} $$

Now I want to define a new process, called blue noise $b_{t}$, and define it as:

$$ b_{t} = \frac{d w_{t}}{dt} = \frac{d^{2} \beta_{t}}{dt^{2}} $$

Does this make sense? How can I deduce what properties it has? Ie -- its mean and variance?

I take the name 'blue noise' as inspired from signal processing theory. We know that the derivative of some signals $f(t)$ can be found via:

$$ \frac{d f(t)}{dt} = \mathcal{L}^{-1}(s F(s)) $$

where $\mathcal{L}(\cdot)$ is the Laplace transform, and $F(s)$ is the Laplace transform of $f(t)$.

Can I use this to define the properties of this stochastic process?

This problem is arising from trying to determine the differential equation corresponding to the following system driven by white noise input:

$$ H(s) = \frac{s + \gamma}{s^{2} + 2 \alpha s + \gamma^{2}} $$

If $W(s)$ is the Laplace transform of the white noise input, then we have:

$$ Y(s) = W(s)H(s) $$ $$ s^{2}Y(s) + 2 \alpha s Y(s) + \gamma^{2}Y(s) = sW(s) + \gamma W(s) $$ $$ y^{''}_{t} + 2 \alpha y^{'}_{t} + \gamma^{2} y_{t} = w^{'}_{t} + \gamma w_{t} $$

where $w^{'}_{t}$ is this blue noise process I am talking about.

I am trying to make sense of this thing. Can I analyze this somehow in the Ito sense? I want to be able to put it into an SDE and do some analysis -- ie solve it or find the moments of the resulting equation etc.

I'm just a lowly engineer so please keep that in mind haha

Thanks for your help and comments!

the_src_dude
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  • I am not that savvy on signal processing conventions, but as far as probability theory and analysis are concerned, the idea of white noise as the time derivative of Brownian motion is an informal heuristic because of the fact that Brownian motion is nowhere-differentiable with respect to time, almost surely. – Nap D. Lover Nov 26 '19 at 16:38
  • @NapD.Lover Okay, but has this informal ideal been extended to the second derivative? – the_src_dude Nov 26 '19 at 16:47
  • It seems this question is very similar: https://math.stackexchange.com/a/2423515/291100 – Nap D. Lover Nov 26 '19 at 20:22
  • and compare to a general discussion about white noise as the (distributional) derivative of BM: https://math.stackexchange.com/a/1572893/291100 just for reference. – Nap D. Lover Nov 26 '19 at 20:23
  • White noise in continuous time and BM context, can be made mathematically rigorous, have a look here : https://math.stackexchange.com/questions/88183/sobolev-meets-wiener ; concerning your blue noise I don't see how this framework would help to define your objective though. – TheBridge Nov 27 '19 at 10:21
  • @TheBridge I think what I should do is define the blue noise process as its own SDE. Then I can substitute it in and solve. Unfortunately, I can't find such a reference on the net, so I might have to do this myself. It seems this problem is much more complex than I thought! – the_src_dude Nov 27 '19 at 12:05
  • @BenCW Do you have any idea how to define this? I saw your post that is linked here, so maybe you have some insight? – the_src_dude Nov 27 '19 at 12:36

1 Answers1

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Brownian motion $B_{t}$ is nowhere differentiable, so why not just use $dB_{t}$ directly are your noise as is often done in spdes.

In terms of using

$$ \frac{d f(t)}{dt} = \mathcal{L}^{-1}(s F(s)) $$

you could use that Brownian motion $B_{t}$ and the measure $dB_{t}$ have finite Laplace transforms

$$E[(\int_0^\infty e^{-\lambda s}B_s ds)^{2}]=\frac{1}{2\lambda^{3}}$$

and

$$E[(\int_0^\infty e^{-\lambda s}dB_s)^{2}]=\frac{1}{2\lambda}.$$

Thomas Kojar
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