Add multiplicative noise?

[sdwfrost]

In some cases e.g. where states are enforced to be positive, it makes sense for the noise to be multiplicative rather than additive. One option is to model log-transformed states, but it would be less unwieldy to be able to add e.g. gamma distributed noise as in https://www.sciencedirect.com/science/article/pii/S0304414911001761.

[ChrisRackauckas]

Multiplicative noise is a form of diagonal noise, so it's not specifically specialized on in the solvers since things like SOSRI are a form of specialization on them. That said, this is a longer running thread. https://arxiv.org/abs/1804.04344 in section 4 shows how methods for additive noise can in theory be used, and that would be a nice optimization, but it's really really difficult to do this in a general sense inside of the solver. So instead we're looking at SciML/ModelingToolkit.jl#140 . So I'm closing this as an issue that probably should be solved by ModelingToolkit, though there's a chance I revive the numerical transformation approach in the future.

[sdwfrost]

Thanks for the pointer to your paper; I'll take a look at the transform approach. Do you think that in the meantime an Euler scheme with multiplicative noise is OK, along the lines of this?

[ChrisRackauckas]

Yes, and the diagonal noise interface is very natural for multiplicative noise, i.e.:

https://docs.sciml.ai/latest/tutorials/sde_example/#Example-2:-Systems-of-SDEs-with-Diagonal-Noise-1

Here's changing that from additive to multiplicative noise:

function lorenz(du,u,p,t)
  du[1] = 10.0(u[2]-u[1])
  du[2] = u[1]*(28.0-u[3]) - u[2]
  du[3] = u[1]*u[2] - (8/3)*u[3]
end

function σ_lorenz(du,u,p,t)
  du[1] = 3.0u[1]
  du[2] = 3.0u[2]
  du[3] = 3.0u[3]
end

prob_sde_lorenz = SDEProblem(lorenz,σ_lorenz,[1.0,0.0,0.0],(0.0,10.0))
sol = solve(prob_sde_lorenz)
plot(sol,vars=(1,2,3))

FWIW, most of these routines were originally made for the case of multiplicative noise!

[sdwfrost]

Doesn't your example still have noise that is added to the ODE system? What I mean is a system where du = du*dW where dW is Gamma white noise (see rgammawn in https://www.rdocumentation.org/packages/pomp/versions/1.19/topics/eulermultinom)

[ChrisRackauckas]

the system is x' = 10(y-x) + 3x dW, y' = x(28-z) - y + 3y dW, z' = xy - 8/3 z + 3z dW, is that the kind of thing you're looking for? That's what I'd normally think of as multiplicative noise.