I have a following question:
Given a Gaussian (in general complex) noise with fixed mean and correlations, is it possible to write the stochastic differential equation it fulfills?
So, fixing: $<X(t)>=a(t), <X(t)X^*(s)>=\alpha(t,s), <X(t)X(s)>=\beta(t,s)$, is it for all functions $a(t), \alpha(t,s), \beta(t,s)$ possible to find such $A(t), B(t)$, that $dX(t)=A(X(t))dt + B(X(t))dW$, with $W$ Wiener process?
How $A(t), B(t)$ depend on X(t)?