Geometric Brownian motion
wanders off to infinity (or zero); many real quantities do not. An interest rate, a volatility,
the spread between two related assets — these revert to a level. The
Ornstein–Uhlenbeck (OU) process is the simplest
SDE
with that behaviour:
dX_t = \theta(\mu - X_t)\,dt + \sigma\,dW_t.
Read the drift \theta(\mu - X_t) as a restoring
force: when X_t is above the level
\mu the bracket is negative and the drift pulls it down; when below,
the bracket is positive and the drift pushes it up. The speed of reversion
\theta > 0 sets how hard the pull is, while the constant diffusion
\sigma keeps kicking it off course. The tug-of-war between the pull
toward \mu and the noise is what gives OU its hallmark: it
fluctuates, but it does not run away.
Solving OU by the integrating factor
The drift contains an -\theta X_t term, so the equation is linear in
X_t but not constant-coefficient. The classical cure for a linear
equation is an integrating factor: multiply by
e^{\theta t} to make the left side a perfect differential. The only
new subtlety over the deterministic case is keeping track of the stochastic term — but
e^{\theta t} is a smooth, deterministic function of time, so it has
finite variation and contributes no covariation term.
Step 1 — apply the Itô product rule to
e^{\theta t} X_t. The
product rule
is d(UV) = U\,dV + V\,dU + d[U, V]. Here
U = e^{\theta t} is smooth in t with
dU = \theta e^{\theta t}\,dt, and V = X_t.
Because U has finite variation, the covariation
d[U, V] = 0, leaving
d\big(e^{\theta t} X_t\big) = e^{\theta t}\,dX_t + X_t\,\theta e^{\theta t}\,dt.
Step 2 — substitute the OU dynamics
dX_t = \theta(\mu - X_t)\,dt + \sigma\,dW_t into the first term:
d\big(e^{\theta t} X_t\big) = e^{\theta t}\big[\theta(\mu - X_t)\,dt + \sigma\,dW_t\big] + \theta e^{\theta t} X_t\,dt.
Step 3 — watch the X_t terms cancel.
Expand the bracket: e^{\theta t}\theta\mu\,dt - e^{\theta t}\theta X_t\,dt + e^{\theta t}\sigma\,dW_t,
then add the last +\theta e^{\theta t} X_t\,dt. The two
\theta e^{\theta t} X_t\,dt terms have opposite signs and annihilate —
which is exactly why e^{\theta t} was the right factor:
d\big(e^{\theta t} X_t\big) = \theta\mu\, e^{\theta t}\,dt + \sigma e^{\theta t}\,dW_t.
Step 4 — integrate from 0 to
t. The left side telescopes. On the right, the first integral is
elementary (\int_0^t \theta\mu\,e^{\theta s}\,ds = \mu(e^{\theta t} - 1)),
and the second is an Itô integral we leave as is:
e^{\theta t} X_t - X_0 = \mu\big(e^{\theta t} - 1\big) + \sigma\int_0^t e^{\theta s}\,dW_s.
Step 5 — multiply back by e^{-\theta t} to isolate
X_t. Distribute the factor across every term:
X_t = e^{-\theta t} X_0 + \mu\big(1 - e^{-\theta t}\big) + \sigma\int_0^t e^{-\theta(t - s)}\,dW_s.
Step 6 — tidy. Group the two deterministic terms around
\mu to expose the reversion cleanly:
X_t = \mu + (X_0 - \mu)\,e^{-\theta t} + \sigma\int_0^t e^{-\theta(t - s)}\,dW_s.
Read it off: the process is the level \mu, plus a deterministic
memory of the start (X_0 - \mu)e^{-\theta t} that
decays exponentially, plus a weighted average of past noise in which older
shocks (small s) are exponentially discounted by
e^{-\theta(t - s)}. The process forgets both its start and its old
kicks at rate \theta.
Gaussian, mean-reverting, and stationary
Everything we want now reads off the solution. The stochastic integral
\int_0^t e^{-\theta(t-s)}\,dW_s is an Itô integral of a
deterministic function against Brownian motion, which is a
Gaussian random variable with mean zero. So X_t is a
constant plus a Gaussian — hence Gaussian itself.
The mean. The stochastic integral has mean zero, so taking expectations of the
solution kills the last term:
\mathbb{E}[X_t] = \mu + (X_0 - \mu)\,e^{-\theta t} \;\xrightarrow[t\to\infty]{}\; \mu.
The mean glides exponentially from X_0 toward the level
\mu — this is mean reversion made precise.
The variance comes from the
Itô isometry,
which turns the variance of a stochastic integral into an ordinary integral of the squared
integrand:
\operatorname{Var}(X_t) = \sigma^2\,\mathbb{E}\!\left[\left(\int_0^t e^{-\theta(t-s)}\,dW_s\right)^{\!2}\right] = \sigma^2 \int_0^t e^{-2\theta(t - s)}\,ds.
Substitute u = t - s (so du = -ds, and the
limits flip to 0 to t) and integrate the
exponential:
\operatorname{Var}(X_t) = \sigma^2 \int_0^t e^{-2\theta u}\,du = \sigma^2\cdot\frac{1 - e^{-2\theta t}}{2\theta} \;\xrightarrow[t\to\infty]{}\; \frac{\sigma^2}{2\theta}.
As t \to \infty the memory of the start fades and the variance
settles to the stationary value \sigma^2/(2\theta):
a balance between the noise \sigma^2 pumping spread in and the
reversion 2\theta draining it out. Stronger pull or weaker noise gives
a tighter stationary band.
The SDE dX_t = \theta(\mu - X_t)\,dt + \sigma\,dW_t with
\theta > 0 has the unique solution
X_t = \mu + (X_0 - \mu)\,e^{-\theta t} + \sigma\int_0^t e^{-\theta(t - s)}\,dW_s,
with the following properties:
- X_t is Gaussian (a constant plus an Itô integral of a deterministic function).
-
Mean reverting:
\mathbb{E}[X_t] = \mu + (X_0 - \mu)e^{-\theta t} \to \mu as
t \to \infty.
-
Variance
\operatorname{Var}(X_t) = \dfrac{\sigma^2}{2\theta}\big(1 - e^{-2\theta t}\big),
with stationary value \dfrac{\sigma^2}{2\theta}.
Mean reversion is the economic content. A short-term interest rate cannot drift to infinity or
plunge arbitrarily — central banks and markets pull it back toward a long-run level. The
Vasicek model (1977) is exactly OU applied to the short rate
r_t:
dr_t = \theta(\mu - r_t)\,dt + \sigma\,dW_t,
with \mu the long-run rate, \theta the
reversion speed, and \sigma the rate's volatility. Its Gaussian,
closed-form law makes bond prices analytically tractable — the payoff of choosing an
analytically friendly process. Its one notorious flaw is the flip side of being Gaussian: the
rate can go negative (once a deal-breaker, now occasionally realistic).
The stationary variance from the isometry. The long-run spread
\sigma^2/(2\theta) is worth re-deriving as a sanity check. In
stationarity, take the variance of dX = \theta(\mu - X)\,dt + \sigma\,dW
and demand it not change: the noise injects variance at rate \sigma^2
per unit time while reversion removes it at rate 2\theta\operatorname{Var}(X)
(the factor two from squaring). Setting injection equal to removal,
\sigma^2 = 2\theta\operatorname{Var}(X), gives
\operatorname{Var}(X) = \sigma^2/(2\theta) — the same value the Itô
isometry produced above, now seen as a steady-state balance.