Write down the bond pricing PDE for the function [F(t, x)=mathbb{E}^{*}left[mathrm{e}^{-int_{t}^{T} r_{s} d s} mid r_{t}=x ight]]

Write down the bond pricing PDE for the function

\[F(t, x)=\mathbb{E}^{*}\left[\mathrm{e}^{-\int_{t}^{T} r_{s} d s} \mid r_{t}=x\right]\]

and show that in case \(\alpha=0\) the corresponding bond price \(P(t, T)\) equals

\[P(t, T)=\mathrm{e}^{-r_{t} B(T-t)}, \quad 0 \leqslant t \leqslant T\]

where

\[B(x):=\frac{2\left(\mathrm{e}^{\gamma x}-1\right)}{2 \gamma+(\beta+\gamma)\left(\mathrm{e}^{\gamma x}-1\right)}, \quad x \in \mathbb{R}\]

with \(\gamma=\sqrt{\beta^{2}+2 \sigma^{2}}\).