Assume Xt = et ( X0) + t 0 e(ts) dBs for a Brownian motion
Question:
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).
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