= Solution
For $V(t,r)$ sufficiently smooth, <Itô formula> under the physical measure gives
$$
dV=\left(V_t+bV_r+\frac12\sigma^2V_{rr}\right)dt+\sigma V_r\,dW.
$$
Hence $m=V_t+bV_r+\sigma^2V_{rr}/2$ and $s=\sigma V_r$. Substitute these into the common risk-price condition $m-rV=\theta s$ from part (b), obtaining the <Cox-Ross short-rate pricing equation>
$$
\boxed{V_t+(b-\sigma\theta)V_r+\frac12\sigma^2V_{rr}-rV=0.}
$$
For a terminal payoff $H(r_T)$ the final condition is $V(T,r)=H(r)$. Writing prices as functions of $(t,r)$ requires the relevant <Markov> state to be sufficient for the claim; a path-dependent payoff generally requires additional state variables. The negative sign in $b-\sigma\theta$ is the adjustment from physical to risk-neutral rate drift.
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