Consider the two coupled Ito SDEs
$dX_t=-\lambda X_t\cdot dt+\sigma\cdot dB_t$
$dY_t=-\sin Y_t\cdot dt+s\cdot X_t\cdot \cos Y_t dt$
I assume that $X_t$ is an Ornstein-Uhlenbeck process, and that $Y_t$ is the angle of an overdamped physical pendulum, which is subject to the random horizontal force $X_t$.
I think it is a good idea to simulate this system in some timespan, lets say $0\leq t\leq T$, for some fixed $\lambda, \sigma$ and s, and after that plot a single realisation, but how to do this in practice? I have tried with the Euler method, but this did not go well. Furthermore, I use the time step $10^{-3}$. Does anybody have an idea?
I think because the noise $X_t$ fluctates fast compared to the state $Y_t$, I choose to approximate $X_t$ with white noise, but here I need to state the variance spectrum of $X_t$?. I think you can argue here that at low frequencies, $X_t$ may be considered white noise with intensity 1, but I think I must plot the $B_t$ and $\int_{0}^t X_s\cdot ds$, for the same realization as found by the simulation of the system above.
Does anybody have an idea?