This is a follow-up on my previous post about stochastic differential equations.
In the answer from @LuztL, and in the literature, I read commonly that the time-step of the noise should be somewhat larger than that of the numerical solver. To quote LutzL specifically:
Obviously, this assumes that $\Delta t$ is somewhat larger than the step size of the numerical integrator.
I accepted this as a premise without thinking too much about it, but actually in practice, downsampling the noise can have a significant impact on its Fourier spectrum for instance. In fact, for typical runge-kutta methods, which make several intermediary substeps typically of length $dt/2$ or $dt/k$, it would make more intuitive sense that the noise should be over-sampled to a common denominator of these substeps.
My questions are:
- Is it true that the general recommendation is to down-sample the noise? (or was I just unlucky to find it in the litterature and in the answer of LutzL)
- If so, quid of the spectral properties of this noise?
- Would anyone be kind enough to point me to further literature (accessible paper, book chapter, blog post) explaining this specific point in more details?
To make even the slightest sense of the equation $$ \dot X = -X + f(α(X)+η) $$ where $η$ is some random noise term, first assume that $η$ is piecewise constant over intervals of length $Δt$ so that over these intervals an exact solution exists by standard ODE theory and the pieces can be joined together to form one continuous function.
To realize this scheme numerically, of course the standard ODE solver has to use step sizes smaller $Δt$ to stay inside these intervals. Depending on $f$, one step might be enough, but in general several steps are needed where the step size then is "much" smaller than $Δt$.
This all works well for a fixed $Δt$. Now the question is on how to design the distribution of $η(t)$ so that subsampling or/with Brownian bridge constructions result in solutions that are at least similar to each other and at best converge in some sense.
One idea is to set $Z=α(X)+η$ which would strongly suggest $η=W$ so that $$dZ=α(dX)+dW=-α(X)dt+α(f(Z))+dW$$ as mathematically solid interpretation.
Another interpretation would use that over a time step $Δt$ the perturbation $η$ is of magnitude $\sqrt{Δt}$ and thus also infinitesimal so that $$ dX=\bigl(-X+f(α(X))\bigr)dt + f'(α(X))dW $$ is also a properly formed SDE.