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Hello I need your help. I found the formula for deriving $A(t,T)$ and $B(t,T)$ in Hull White paper is like this

$BB_{tT} - B_{t}B_{T} - B_{T} = 0$ and $ABA_{tT} - BA_{t}A_{T} - AA_{t}B_{T} + \frac{1}{2}\sigma^2(t)A^2B^2B_{T} = 0$

and they are resulting in this $B(t,T) = \frac{B(0,T) - B(0,t)}{\frac{\partial B(0,t)}{\partial t}}$ and $\hat{A}(t,T) = \hat{A}(0,T) - \hat{A}(0,t) - B(t,T)\frac{\partial \hat{A}(0,t)}{\partial t} - \frac{1}{2}\Bigg[B(t,T)\frac{\partial B(0,t)}{\partial t}\Bigg]^{2}\int_{0}^{t} \Bigg[\frac{\sigma(\tau)}{\frac{\partial{B(0,\tau)}}{\partial \tau}}\Bigg]^{2}d\tau$

where $\hat{A}(t,T) = \ln{|A(t,T)|}$. Anyone know how to derive the solution? Please this is for my thesis

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    $\begingroup$ See this and this for an alternative derivation. $\endgroup$ – Gordon Jun 17 at 13:08

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