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I have an stochastic process $r_t$ (short rate model) with dynamics

$dr(t)=a(t,r(t)) \, dt + b(t,r(t)) \, dW(t)$,

where $W$ is a standard brownian motion.

I want to show: If there is a function $F(t,r)$ which satifies (term structure equation)

$F_t + aF_r + \frac{1}{2} b^2F_{rr} - rF =0$

and

$a(t,r)=a(t) + \alpha (t)r$,

$b^2 =b(t) + \beta (t) r$

then $F$ is of the Form

$F(t,r(t)) = \text{exp}[A(t) + B(t)r(t)]$ (affine term structure)

with

$A_t = \frac{1}{2} b(t)B^2-a(t)B$

$B_t = \frac{1}{2} \beta(t)B^2 -\alpha(t)B-1$ .

This statement is well known in short rate theory, but i can't find a proof.

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