Suppose I have a SDE of the form:
$$dx_i = x_i\left(b_i-\sum_{j=1}^n a_{ij}x_j\right) \,dt + \sigma_i x_i \, d\eta(t)$$ where $\eta$ solves the Ornstein-Uhlenbeck process:
$$d\eta(t) = \lambda \eta(t)\, dt + \sigma \, dW(t)$$
Two possible quantities I'm interested in are: $x_i(t)^p$ or $\sum_{i=1}^n x_i(t)^p$
Is there anyway to determine $Cov(x_i(t)^p, \eta(t))$ or $Cov(\sum_{i=1}^n x_i(t)^p, \eta(t))$? Or, is it at all possible to determine the sign of the covariance ($>0$ or $<0$?)