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APPENDIX B
Mean-Reverting Processes
The process equation for the Hull-White, BDT/BK models in Chapter 8, and the Ho-Lee model in Chapter 10 can be expressed as a generalized Ornstein-Uhlenbeck process:
where for the Hull-White/Ho-Lee models x(t, ω) = r(t, ω), while for the BDT/BK models x(t,ω) = ln r (t,ω). Multiplying both sides by the integrating factor
we get
Since the increments of Brownian motion, dB(t, ω), are independent and Normal, x(s, ω) is conditionally Normal (conditioned on x(t, ω)):
Moreover,x(s, ω)ds is also conditionally normal:1
where
where we have used the following property of Ito differentials:

# NORMAL DYNAMICS

For the Vasicek/Hull-White/Ho-Lee models, r (t, ω) = ...

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