# Ito or Stratonovich equation

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?

• It depends a lot on parameters. For small $\lambda$, $X_t$ is nearly Brownian (not white), while for large $\lambda$ it is nearly zero. In any case, Euler-Maruyama does in fact work when implemented correctly and with quite small steps. – Ian Oct 26 '18 at 13:54
• The formula $X_t=σe^{-λt}\tilde B_{(e^{2λt}-1)/(2λ)}$ demonstrates the properties of the previous comment. $\tilde B_t$ is a different version of the Brownian process. – LutzL Oct 26 '18 at 13:59
• So you mean that the system of SDE can be solved numerically with the Euler-Maruyama method? I have no experience with this scheme, would you know how to do this? I assume that $\lambda=100$, which I can see that $X_t$ is virtually zero, that makes sense! – Jonathan Kiersch Oct 26 '18 at 14:16
• Euler-Maruyama is essentially forward Euler, where $dB_t$ is replaced by $\Delta B_t$ (which are iid normal random variables). If I remember correctly (I don't work in this area much anymore), there is a slight modification when the variable coefficient on the diffusion creates additional drift due to the Ito formula. But this is definitely no issue when the diffusion coefficient is constant. In any case, with such large $\lambda$, you will need considerably smaller time steps to make Euler-Maruyama stable. – Ian Oct 26 '18 at 15:24
• Sorry, But I am still struggling with seeing how Euler Maruyama can help me with solving a system of equations, I am only able to solve the problem for one equation: – Jonathan Kiersch Oct 27 '18 at 6:26