I recently read a paper which I think has an incorrect numerical simulation. I say 'think' because the issue was centered about solving a stochastic ordinary differential equation (SDE) and I have heard from a trusted authority figure that the approach used is wrong. My personal digging and reading around makes me agree despite me being relatively unfamiliar with SDEs. I am aware of the appeal to authority fallacy and my unfamiliarity pushes me to seek more sources and advice to understand if it is wrong or not myself.
The paper solves the SDE of the form
\begin{equation} dz(t) = Az (t)dt + Gdw(t), z(0) = z_0, t\geq 0 \end{equation}
where w(t) wiener process with covariance (incremental) $Q$, $A$ - a constant matrix. The method is solved with an 'implicit' type scheme. \begin{equation} z_{k+1} = z_{k} + Az_{k+1} + Gw_{\Delta t}(k) \end{equation} where $w_{\Delta t}$ has normal distribution with mean zero and variance $Q\Delta t$.
Here is why I think the implicit scheme is wrong.
- Majority of schemes I have seen are low order schemes like Euler-Maruyama in all of literature. I have found $\theta$-Euler-Maruyama methods described in "An Introduction to Computational Stochastic PDEs" by Garbriel J. Lord and others but the convergence was unclear to me.
- Potential computation mistake - In math.stackexchange post in the comments the implicit Eulers of $dz(t) = z(t)dw(t)$ is $z_{k+1} = z_k/(1-dW_{\Delta t})$ which by sampling $dW_{\Delta t}$ can lead to division by zero(however this is not my SDE)
- In the same post as in previous bullet point, it was mentioned that the implicit scheme actually 'By Ito-formalism' is process equivalent to something else that is not related to the original SDE.
- Post 2 talks about RK4 methods not converging to solution of the Ito SDE, but to the one of the Stratonovich SDE.
My limitations in answering the question myself (is the scheme incorrect) is I am not well familiar with the math involved in SDEs and their associated numerics. My close friend has been studying for a month or so and is very good with pure math, but was unable to answer my question(I'm a CS grad) - which tells me that I can get lost in rabbit holes trying to figure things out myself. Does anyone have a good citation/source for me to look at or an explanation to help answer my question.
Many thanks!