Currently I'm learning about SDE's. In my document there is a SDE. It is a stochastic BOD model:
$\begin{aligned} \frac{d B_{t}}{d t} &=-K_{1} B_{t}+s_{t}-B_{t} \sigma N_{t} \\ B_{t_{0}} &=B_{0} \end{aligned}$
This is written in terms of a Wiener process as:
$d B_{t}=\left(-K_{1} B_{t}+s_{t}\right) d t-B_{t} \sigma d W_{t}$
$B_{t_{0}}=B_{0}$
From this the Îto formulation is given.
The explaination is:
"since the white noise process in this stochastic model is a mathematical approximation of a noise process with a relatively short correlation scale, this SDE has to be interpreted in the Stratonovitch sense. since the Euler scheme (14) can only be used for İto equations, the model above is first rewritten as an Ito SDE:"
(14):$X_{t}+f\left(X_{t}, t\right) \Delta t+g\left(X_{t}, t\right)\left(W_{t+\Delta t}-W_{t}\right)$
Ito formulation:
\begin{array}{l} d B_{t}=\left(-K_{1} B_{t}+s_{t}+\frac{1}{2} B_{t} \sigma^{2}\right) d t-B_{t} \sigma d W_{t} \\ B_{t_{0}}=B_{0} \end{array}
My question is:
How should I've known to interpretate the SDE as Stratonovitch?
Short answer
If your SDE is derived directly from a physic equation involving the derivative term $\frac{d}{dt}$( or not as often, $\frac{d}{dx}$ ), it is kinda sure that your SDE is in Stratonovitch sense.
Reference
In fact, the choice is casewise, depends largely on the pragmatic aspect of the choice and also the interpretation of the phenomenon through that SDE. More discussion here https://physics.stackexchange.com/questions/245692/it%C3%B4-or-stratonovich-calculus-which-one-is-more-relevant-from-the-point-of-view