White noise for 'SDEs with diagonal noise'

Specific Domains Modelling & Simulations
1

dear all,
I have a SDEs with a stochastic part as a white noise k*N(0,1).
Time step is ‘dt’
Following explanations here I don’t get why the white noise is N(0,dt) and not N(0,1/dt), which seems consitent with my results as when using in the sde code k*dt my outputs make more sense than using k*1/dt … where am I wrong ??
Thx !
Frédéric

2

What do you mean? It’s just using the standard definition.

3

I am trying to represent seismic excitation as input for my SDEs.
To simplify, this excitation can be represented as a white noise with an intensity of So (m2/s3), the only parameter I know
For a given time step, the SDE concerned with this white noise is something like
Mx.. + Cx. + Kx = sqrt(So/dt)*N(0,1)
Right part is then, according to explanations for SDEs with Diagonal Noise something like
du[1] = sqrt(S0/dt)*N(0,dt)/dt
results are then unconsitent with experiments, while
du[1] = sqrt(S0/dt)*N(0,dt)*dt
are !
Is there something I have missed :thinking: ??
Thx for the help !
Frédéric

4

This doesn’t make sense. Mx.. + Cx. + Kx = sqrt(So/dt)*N(0,1) that’s not a well-defined equation. What is the behavior as dt->0? That diverges to infinite variance in infinitesimal steps!

1 Like
5

dt is fixed in my case, (exactly 10^-2s) :slight_smile:

6

For starters, is this the variance or the standard deviation of the white noise in a step?

7

it’s the constant part (intensity) of the spectral density function representing the seismic excitation
for exemple described in this paper
(I am trying the easiest model, not yet Clough-Penzien )

… so the variance inside a step :slight_smile:

8

But continuous equations don’t have steps, so you can’t define the equation by what happens in a “step”. This is wild.

1 Like
9

yes of course, I was just writing the equation that way for someone to understand the sqrt(So/dt) part as it is the homogenous with an acceleration
the continuous equation in that case is
Mx.. + Cx. + Kx = W, with W Gaussian White Noise Process

10

Then W ~ N(0,dt) or sqrt(dt)*N(0,1). It seems like what you’re really saying is that your equation is:

Mx.. + Cx. + Kx = sqrt(S0/1e-4)*W

?

11

ok I get it now :sweat_smile:
as W~N(0,dt) → sqrt(dt)*N(0,1) it is clear now why
sqrt(S0/dt)*W → sqrt(S0/dt)*sqrt(dt)*N(0,1) is working well, and NOT sqrt(S0/dt)*N(0,1)/sqrt(dt)
very sorry for this basic question and thx a lot for the help !