# Mathematical Expectation Derivation for Ornstein-Uhlenbeck formula

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 $$y_0 = y(0)$$.