# Floquet stochastic process

Let $$X_t$$ be defined by the SDE

$$dX_t = A(t, X_t)dt + dW_t$$

where $$A(t, X_t)$$ is linear in $$X_t$$ and periodic in $$t$$. Assume also that the process is stable. If $$A(\cdot)$$ didn't have $$t$$ dependency, the process would be an $$n$$ dimensional OU process, where the eigenvalues of matrix $$A$$ in $$A(X_t) = A\cdot X_t$$ all have negative real part. In this case it can be seen that it would reach an equilibrium (given by an invariant measure), so $$X_t$$ would exponentially fast approach a stationary solution. But what happens in the case of a $$t$$-periodic $$A(t,X_t)$$? Will the process be cyclostationary/what can we say about it?

• I assumed the process is stable (so all the eigenvalues of the matrix $A$ are negative). Is that what you mean? – nabla Oct 26 '18 at 13:16