Consider the following SDE:
$$dy(t) = a(t)dt + b(t) dW$$
In Euler-Maruyama (EM) integration method, we substitute the Wiener process during each time step of during $dt$ with $\sqrt{dt}\mathcal N(0,1)$ and use the usual Euler method for integration. My question is as follows:
Is it correct to do the same with other deterministic solvers such as Runge-Kutta (RK). i.e., feed the solver with a deterministic gradient according to $\frac{dy}{dt} = a(t) + \frac{b(t)}{\sqrt{dt}}\mathcal N(0,1)$?
My intuition says it is flawed since, in RK, derivatives are estimated at fractional timesteps, $dt/4, dt/2$, etc. In a real stochastic process, the Wiener process can diffuse during such fractional timesteps, but the deterministic treatment stated earlier does not. Is my intuition correct?