I would like to now how to solve the PDE of the affine structure under Vasicek.I am delineating the steps:
First let's posit the OU process under a Risk Neutral Measure such as : \begin{align*} \mathrm{d}r_t=\mu(t,r_t)\mathrm{d}t+\sigma(t,r_t)\mathrm{d}W_t \end{align*}
Then comes the bond PDE:
\begin{align*} P_t + \mu(t,r) P_r + \frac{1}{2}\sigma(t,r)^2P_{rr} -rP=0, \end{align*} We Write the penny zero coupon bond's formula and mixed it with the Original PDE,using a latent $r_t$ variable :
\begin{align*} P(t,T)=e^{A(t,T)-r_tB(t,T)} \end{align*}
\begin{align*} P_t(t,T) &=\big(A_t(t,T)-r_tB_t(t,T)\big)\cdot P(t,T), \\ P_r(t,T) &= -B(t,T)\cdot P(t,T), \\ P_{rr}(t,T) &= B(t,T)^2\cdot P(t,T). \end{align*}
\begin{align*} A_t(t,T) - \mu(t,r) B(t,T) + \frac{1}{2}\sigma(t,r)^2B(t,T)^2 +(-B_t(t,T)-1)r &=0. \end{align*}
In the Vasicek case, $\mu(t,r_t)=\kappa(\theta-r_t)$ and $\sigma(t,r_t)=\sigma$.Afterward the calculations are straightforward:
\begin{align*} A_t(t,T) - \kappa \theta B(t,T) + \kappa r B(t,T) + \frac{1}{2}\sigma^2B(t,T)^2 +(-B_t(t,T)-1)r &=0 \\ \implies A_t(t,T) - \kappa \theta B(t,T) + \frac{1}{2}\sigma^2B(t,T)^2-(1+B_t(t,T)-\kappa B(t,T))r &=0. \end{align*}
And we end up with two equations such :
\begin{align*} \begin{cases} A_t(t,T) - \kappa \theta B(t,T) + \frac{1}{2}\sigma^2B(t,T)^2 &= 0, \\ 1+B_t(t,T)-\kappa B(t,T) &= 0,\\ u.c : A(T,T)=B(T,T)=0 \end{cases} \end{align*} However I don't understand the development we must do so as to find $B_t(t,T) = e^{-k(T-t)}$ and hence $B(t,T) = \frac{-1+e^{-k(T-t)}}{k}$.
Thank you for your time
