Exercise 5.3 Consider the Vasicek model for the short-term interest rate r.

(a) Interpret the so-called mean-reverting property.

(b) Show that limt→∞ E{r(t)} = β.

(c) Show that V ar{r(t)} = σ2 (1−e

−2αt)

2α .

(d) Prove that limt→∞ V ar{r(t)} = σ2 2α.

(e) Show that r(t) converges in law as t → ∞ and find the limiting distribution.