= Solution
Work under the money-market <risk-neutral measure>, with $r_0\geq0$. The <zero-coupon bond> price is $P(0,T)=\mathbb E[\exp(-\int_0^T r_sds)]$. For this <CIR model>, write $p(\tau,r)=A(\tau)e^{-B(\tau)r}$. The <Feynman-Kac formula> gives
$$
\partial_\tau p=(a-br)\partial_rp+\tfrac12\sigma^2r\partial_{rr}p-rp,\qquad p(0,r)=1.
$$
Matching the constant and linear coefficients of $r$ yields
$$
B'=1-bB-\tfrac12\sigma^2B^2,\qquad A'/A=-aB,\qquad B(0)=0,\quad A(0)=1.
$$
For <CIR bond pricing>, solving this <Riccati equation> and integrating the scalar equation for $A$, put $\gamma=\sqrt{b^2+2\sigma^2}$ and $D(\tau)=(\gamma+b)(e^{\gamma\tau}-1)+2\gamma$. Then
$$
B(\tau)=\frac{2(e^{\gamma\tau}-1)}{D(\tau)},\qquad A(\tau)=\left[\frac{2\gamma e^{(b+\gamma)\tau/2}}{D(\tau)}\right]^{2a/\sigma^2}.
$$
Substitution verifies both equations and their initial conditions. The discounted candidate solves the <martingale> pricing equation; its boundedness on finite horizons, or <Feynman-Kac formula>, identifies it with the <conditional expectation>. \b[The answer is $\boxed{P(0,T)=A(T)e^{-B(T)r_0}}$.] More generally the same coefficients give $P(t,T)=A(T-t)e^{-B(T-t)r_t}$.
The <CIR model> mean reverts to $a/b$ and has volatility $\sigma\sqrt r$, so it remains nonnegative and its fluctuation size depends on the rate level. For positive initial rate, zero is inaccessible if $2a\geq\sigma^2$; below this threshold zero is reachable but the process remains nonnegative. Its stationary law is a <gamma distribution>, of shape $2a/\sigma^2$ and rate $2b/\sigma^2$. Explicit affine bond prices and nonnegative rates are useful features. Limitations include exclusion of negative rates, restricted volatility behavior and a single stochastic factor; the time-homogeneous parameter family cannot fit an arbitrary initial term structure exactly.
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