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How was the Ornstein–Uhlenbeck process originally constructed? The O-U process was introduced as a solution to a Langevin equation dXt = −Xtdt + dBt d X t = − X t d t + d B t by Ornstein Uhlenbeck (1930), and the solution of that equation was made rigourous by Doob in 1942, at a time when Ito's work was little known (if at all) in the West
Exponential of Ornstein-Uhlenbeck process squared Given an Ornstein-Uhlenbeck process dr(t) = −κr(t) + gdWt d r (t) = − κ r (t) + g d W t, the expectation value of the exponential of the OU is a well-known result
stochastic processes - Covariance of Ornstein - Uhlenbeck Process . . . I'm considering the Ornstein - Uhlenbeck process X(t) =x∞ +e−at(x0 −x∞) + b∫t 0 e−a(t−s)dW(s) X (t) = x ∞ + e − a t (x 0 − x ∞) + b ∫ 0 t e − a (t − s) d W (s) where a, b> 0 a, b> 0 are given constants I used the Itô Isometry to compute the variance but I did not figure out how to use it to compute the covariance What is the most general context in which I can