I'm trying to numerically solve the 1-dimensional stochastic differential equation for brownian motion (Fokker-Planck) with rapidly varying diffusivity $K$,
$$ \textrm{(Ito)}\quad dX_t = \frac{\partial K}{\partial X_t}dt + \sqrt{2K} dW_t.$$
As a toy example, consider $K = 2 + \sin(\lambda X_t)$ with a large $\lambda$. The regular Euler-Maruyama method fails in this case unless the time step is very small and I want to avoid that. I'm not very familiar with stochastic differential equations, but to me this smells like some kind of stiffness. I would be grateful for any suggestions.
In reality, the diffusivity $K$ is obtained by interpolating a high-resolution tabulated quantity. Both the derivative and antiderivative of $K$ are easily obtainable.