Given an SDE: $$ d\mathrm{\boldsymbol{X}}_t = \mu(\mathrm{\boldsymbol{X}}_t)dt + \sigma(\mathrm{\boldsymbol{X}}_t)d\mathrm{\boldsymbol{B}}_t $$
Under which conditions we can model this SDE by a State-Space Model (SSM)?
$$ \mathrm{\boldsymbol{X}}_{t+1} = F(\mathrm{\boldsymbol{X}}_t) + \mathrm{\boldsymbol{\omega}}_{t} $$
Additionally, if $\mathrm{\boldsymbol{\omega}}_{t}$ is zero-mean, could I use a deterministic Runge-Kutta method to predict the mean of $\mathrm{\boldsymbol{X}}_{t+1}$:
$$ \mathbb{E}[\mathrm{\boldsymbol{X}}_{t+1}] = RK(\mathrm{\boldsymbol{X}}_{t}) $$
Thank you very much!