Let X_t = ∫_0^t e^{-(t-s)} dW_s. Give the mean, the variance, and Cov(X_s, X_t) for s ≤ t, decide whether X is a martingale, and identify the process.

Let X_t = ∫_0^t e^{-(t-s)} dW_s. Give the mean, the variance, and Cov(X_s, X_t) for s ≤ t, decide whether X is a martingale, and identify the process.

Approach: Pull the factor e^{-t} outside the integral to expose a martingale, then use the Ito isometry for the variance and the tower property for the covariance.

X is a centred Gaussian process with Var(X_t) = (1 - e^{-2t})/2 and Cov(X_s, X_t) = e^{-(t-s)}(1 - e^{-2s})/2 for s ≤ t, it is mean reverting rather than a martingale, and it is the Ornstein-Uhlenbeck process solving dX_t = -X_t dt + dW_t with X_0 = 0. Write X_t = e^{-t} M_t with M_t = ∫_0^t e^{s} dW_s, a true martingale with E[M_t^2] = ∫_0^t e^{2s} ds = (e^{2t} - 1)/2 by the Ito isometry, so Var(X_t) = e^{-2t}(e^{2t}-1)/2 = (1 - e^{-2t})/2. Since it is a Wiener integral with a deterministic kernel, the law is Gaussian at every time. For s ≤ t the covariance is Cov(X_s, X_t) = e^{-s} e^{-t} E[M_s M_t] = e^{-s-t} E[M_s^2] = e^{-(t-s)}(1 - e^{-2s})/2. Applying Ito to e^{-t} M_t produces the drift -X_t dt, which is what rules out the martingale property.

Follow-up: What is the stationary autocorrelation function of this process, and at what lag does it fall to 0.5?

Key concepts: Wiener integral, Ornstein-Uhlenbeck, covariance, mean reversion.