Assume Xt = et ( X0) + t 0 e(ts) dBs for a Brownian motion B and constants and . Show that

Assume Xt = θ − e−κt

(θ −X0) +σ

t 0

e−κ(t−s)

dBs for a Brownian motion B and constants θ and κ. Show that dX = κ(θ − X)dt +σ dB.

Note: The process X is called an Ornstein-Uhlenbeck process. Assuming

κ > 0, θ is called the long-run or unconditional mean, and κ is the rate of mean reversion. This is the interest rate process in the Vasicek model (Section 18.3).