If we denote the Ornstein-Uhlenbeck formula like below,

$$ dy(t) = (\lambda y(t-1) + \mu) dt + d\epsilon, $$

where, $y(t)$ is a continuous time-series function, $d\epsilon$ is Gaussian noise, and both $\lambda > 0$ and $\mu > 0$ are constant. Then how can we prove that the mathematical expectation of $y(t)$ can be derived as below,

$$ E(y(t)) = y_0 e^{\lambda t} − \mu/\lambda (1 − e^{\lambda t}) $$

where the domain of $y(t)$ is $[0, \infty)$ and $y_0 = y(0)$.



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