SDE noise term equal to zero

New to Julia
1

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

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)
d=0.1;
α=6.0;
β=1.0;

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

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

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.

2

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.

5

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…

6

Was it more accurate?

7

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.

8

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

9

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.

10

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.

1 Like