Consider the Cox et al. (1985) (CIR) process (left(r_{t} ight)_{t in mathbb{R}_{+}})solution of [d r_{t}=-a r_{t} d

Consider the Cox et al. (1985) (CIR) process \(\left(r_{t}\right)_{t \in \mathbb{R}_{+}}\)solution of

\[d r_{t}=-a r_{t} d t+\sigma \sqrt{r_{t}} d B_{t},\]

where \(a, \sigma>0\) are constants \(\left(B_{t}\right)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion started at 0 .

a) Write down the bond pricing PDE for the function \(F(t, x)\) given by

\[F(t, x):=\mathbb{E}^{*}\left[\exp \left(-\int_{t}^{T} r_{s} d s\right) \mid r_{t}=x\right], \quad 0 \leqslant t \leqslant T\]

Use Itô calculus and the fact that the discounted bond price is a martingale.

b) Show that the PDE found in Question (a) admits a solution of the form \(F(t, x)=\) \(\mathrm{e}^{A(T-t)+x C(T-t)}\) where the functions \(A(s)\) and \(C(s)\) satisfy ordinary differential equations to be also written down together with the values of \(A(0)\) and \(C(0)\).