SDE noise term equal to zero

Hi everyone,
I am dealing with a Stochastic Differential Equation which, in order to solve, I have divided in two:

I was wondering if it is correct to set the noise term equal to zero in the first equation, this is:

function Helmholtz(du,u,p,t)

du[1] = u[2]
du[2] = -d*u[2]-α*u[1]+β*u[1].^2

function σ_Helmholtz(du,u,p,t)
  du[1] = 0.0
  du[2] = 1.0

I know that matlab has issues with setting the delay=0 when using DDE solvers, I wonder if something similar occurs when setting the noise term=0 with a SDE solver. BTW, I am using the SRIW1() solver.

Thank you in advance for any help.

Yes, this is fine. You can treat it as diagonal noise with only one noise term. Making it scalar noise could be good as well, though maybe a hyperoptimization. Using a solver for additive noise processes would be a good idea here.

Thank you for the quick answer. It is interesting what you said about the solver. I’ve tried SOSRA, SRA3 and SRIW1. SRIW1 is by far the fastest of the three…

Was it more accurate?

What I can say is that I was using SRIW1 with reltol=1e-3,abstol=1e-6. SOSRA is too slow for those tolerance values. I tried SOSRA with the default tolerance values (which I assume are less restrictive) and it is very very fast then. I did 10.000 realizations of the process and the average results are quite similar to the ones using SRIW1 with reltol=1e-3,abstol=1e-6.
I am still working on an analitical solution for the SDE, so I can’t say which one is more accurate yet.

That’s extremely low for strong SDE error. The defaults on SDEs are noted to be 1e-2,1e-2.

Ok, I was reading your paper “Stability-Optimized High Order Methods and Stiffness Detection for Pathwise Stiff Stochastic Differential Equations” trying to understand this a little bit better.
I guess I’m safe using SOSRA with reltol=abstol=1e-2 (which is really fast). And if I ever need a lower tolerance I should use SRA3.

Yes. SOSRA can hit higher tolerances more safely because it has a much larger stability region, but SRA3 has a higher weak convergence rate which can be useful as tolerances decrease.

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