Exercise: Ornstein-Uhlenbeck via Itô

Prerequisites: Itô's Lemma, Stochastic Differential Equations, Normal Distribution

Problem

The Ornstein-Uhlenbeck (OU) process is the simplest mean-reverting SDE in quant finance — it models short rates (Vasicek), log-volatility (some stochastic-vol setups), and spread dynamics (pairs-trading residuals). Let

with $\kappa, \sigma > 0$.

1. Apply Itô's lemma to $V(t, X) = e^{\kappa t} X$ and show that the drift vanishes. Deduce that $d(e^{\kappa t}X_t) = \sigma e^{\kappa t}\,dW_t$.

2. Integrate the result in part 1 from $0$ to $t$ and solve for $X_t$. Write the solution in the form $X_t = x_0 e^{-\kappa t} + (\text{stochastic integral})$ and identify the integrand explicitly.

3. Use the Itô isometry to compute $\operatorname{Var}(X_t)$. Show that $\operatorname{Var}(X_t) \to \sigma^2/(2\kappa)$ as $t \to \infty$.

4. Explain in one or two sentences why the trick "multiply by $e^{\kappa t}$" is the stochastic analogue of the integrating-factor trick from ordinary linear first-order ODEs, and why the absence of an Itô correction for this particular $V$ is not a coincidence.

Hint