# Cross-correlation in SDEs

Is it possible to derive a cross-correlation function between a stochastic variable and a state-variable in an SDE, such as for the simple model here, or better between two state variables in a two variable (plus one stochastic variable) model:

sys = ItoProcess[{\[DifferentialD]n[t] == (ibar + rP p[t] - lN n[t] - u1 n[t])
\[DifferentialD]t + sd \[DifferentialD]w[t],
\[DifferentialD]p[t] == (u1 n[t] - (rP + lP) p[t]) \[DifferentialD]t},
{n[t], p[t]}, {{n, p}, {n0, p0}}, t, w \[Distributed] WienerProcess[0, sd]];


I am interested in the long-term limit of correlations between variables, starting from initial conditions that are equal to the means. For instance for the one variable case, I am able to compute expressions for the mean and variance (and autocorrelation, but not shown):

proc = ItoProcess[\[DifferentialD]n[t] == ( ibar - q n[t]) \[DifferentialD]t +
sd \[DifferentialD]w[t], n[t], {n, ibar/q}, t,
w \[Distributed] WienerProcess[0, sd]](*this is white noise*);

Mean[proc[t]]
Variance[proc[t]]


Out[2]= ibar/q

Out[3]= (E^(-2 q t) (-1 + E^(2 q t)) sd^4)/(2 q)

 Limit[E^(-2 q t) (-1 + E^(2 q t)) /. q -> 1, t -> Infinity]


Out[4]= 1

However, large outputs are generated for the two variable case. Is there a way to solve for variances (mean can be computed easily on paper) and the cross correlation between any of the variables (ibar + sd dw[t]/dt, n[t], p[t]) in the limit shown above?

Is a symbolic approach possible? If not, are there approaches to approximate a function, either from numerical simulations, Taylor expansion or another technique? Suggestions and links to any existing resources are appreciated.

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