How was the Ornstein–Uhlenbeck process originally constructed?

Question

It is natural to wonder about a Brownian motion with a drift toward $0$ whose rate is equal to the current value of the process. Unlike the standard Wiener process, which is null-recurrent, this is positive-recurrent and seldom wanders very far from $0.$

But the way I have seen it presented is this:

$$\tag 1 \text{Let } X(t) = e^{-t}W(e^{2t}),\qquad\qquad\qquad\qquad$$ where $W$ is the standard Wiener process. Now derive its covariance function and its drift.

Would any reasonable person think something like that is of interest other than as a routine exercise for undergraduates unless they knew the (to me) unexpected result?

What sort of thought process could possibly lead someone from wondering about the mean-reverting Brownian motion to coming up with line $(1)$ above as the answer?

Answer 1

The O-U process was introduced as a solution to a Langevin equation $dX_t = - X_t \, dt +dB_t$ 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.

Because the integrand in the solution $X_t=X_0e^{- t}+\int_0^t e^{-(t-s)} \, dB_s$ ($X_0$ standard normal and independent of the Brownian motion $B$) is non-random, it's clear that the process is Gaussian, with mean $0$ and covariance $\Gamma (s,t) = e^{-t}\sinh(s)$ for $0\le s\le t$. For those of us of a certain age, who learned about things like Brownian bridge theorem from Patrick Billingsley's book, it's natural to try to cook up a Gaussian process with a given covariance by looking at something like $g(t)[Z+\tilde W(h(t))]$, with $g$ and $h$ to be chosen to obtain the desired covariance. (Here $\tilde W$ is a standard Brownian motion and $Z$ is an independent standard normal.) The choices $g(t) = e^{-t}$ and $h(t) = e^{2t}-1$ yield the "out-of-left-field" construction of O-U; notice that $W(u) = Z+\tilde W(u-1)$ for $u\ge 1$. (One advantage of $t^{-t}W(e^{2t})$ is that it's defined for all real $t$ and is a stationary version of the O-U process.)

I'm not sure who came up with this recipe for O-U, but I first saw it in a paper by David Williams circa 1980, using the idea to construct the infinite-dimensional O-U, with values in Wiener space rather that Euclidean space.

Answer 2

The answer by John Dawkins already covers most of the bases; I'll add two points:

1. The Langevin equation was considered independently by Ornstein in a 1919 paper 1; Ornstein attributes it to the PhD thesis of Haas-Lorentz, although further digging reveals that the Langevin equation was also considered by Einstein and Smoluchowski as well.

2. In fact, it was Doob in 1942 who first studied Brownian motion as a time-changed OU process (i.e. the inverse operation you are proposing). To see this, look at equation 1.2.1 of 2, set $\gamma = 2$ and let $u(t)$ be an OU process.

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1 Ornstein, L. S. "On the Brownian motion." Proc. Amst. Vol. 21. 1919. Link

2 Doob, Joseph L. "The Brownian movement and stochastic equations." Annals of Mathematics (1942): 351-369. Link