I have the equations defined as follows: $$ \begin{align} \frac{dX}{dt}&=f(X,U), \\ U(t)&=g(X)+\xi_t, \end{align} $$ where $\xi_t$ is a Gaussian white noise, $f(X,U)$ and $g(X)$ are such functions as one could rewrite the first equation by substituting the second equation in it as follows: $$ \frac{dX}{dt}=\psi(X)+\sigma \xi_t. $$ It's possible to apply the Euler-Maruyama scheme to solve the differential equation only as we can write down its Ito's form: $$ dX=\psi(X)dt+\sigma dW. $$ The numeric scheme for a differential equation would be $$ X(t_{n+1})=X(t_n)+\psi(X(t_n))\Delta t+ \sigma \sqrt{\Delta t} \hat{\xi},\quad \hat{\xi}\in N(0,1). $$ But how to compute the algebraic variable $U(t_{n+1})$ on the same time step? The problem is the random variable $\hat{\xi}$ somehow depends on $\xi_t$.
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I don't understand your question, $dW$ is basically $xi_tdt$. What is $xi_t$ in your case? White noise? – quallenjäger Oct 04 '17 at 10:05
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Yes. I modified the question. – Artem Zefirov Oct 04 '17 at 11:51
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It's obvious that $\xi_t dt = dW$. And if I just replace $dt=t_{n+1}-t_n$ then I get $\hat{\xi}=W(t_{n+1})-W(t_n)=\xi_t (t_{n+1}-t_n)$.So, all I need is just to generate $\xi_t$ as a random normal variable? – Artem Zefirov Oct 04 '17 at 12:20