I'm trying to computationally solve the following SDE:
$\frac{dx(t)}{dt} = DeterministicTerm - \epsilon \, sin(x(t)-\sigma \eta(t)) $.
Here, $\epsilon$ and $\sigma$ are two parameters, and $\eta(t)$ is a white noise term.
From what I understood, the Euler-Maruyama method can only be used for SDEs of the form:
$dX_t=a(X_t, t)dt+b(X_t, t)dW_t$.
Other methods, such as the Stochastic Runge-Kutta methods and the Milstein method, may share the same "problem".
So, do you know any other stochastic integration method that could be appropriate to solve the previous equation? I studied Stochastic Processes from a theoretical point of view during my Master, but the course wasn't focused on numerical methods. Hence, I don't really now where to search for stochastic integration methods for nonlinear white noise terms.