ハル・ホワイトの三項木

Deriving the alpha variable where alpha=R-R* is non-trivial. Here, R* is the rate where the long-term mean, theta/a, is 0. The differential equation derived for alpha by subtracting dR* from dR is:
d(alpha) = (theta-a*alpha)dt
Theta is derived from the term structure to be:
theta = dF/dt + (sigma^2/2a)(1-exp(-2at))
Substituting theta yields:
d(alpha) = (dF/dt + (sigma^2/2a)(1-exp(-2at))-a*alpha)dt
Direct integration of this expression is challenging, as there is an alpha term on the right. It is unclear which differential equation approachyields the correct alpha, but differentiating the text's alpha with respect to t does yield the d(alpha) expression (after dividing both sides by dt) above.
alpha(t) = F(0,t)+(sigma^2/(2a^2))(1-exp(-at))^2