= Solution
{c}
Set $\theta_t=(\mu-r)S_t/\sigma$. The <Girsanov theorem> uses the density
$$
Z_T=\exp\left(-\int_0^T\theta_u\,dW_u-\frac12\int_0^T\theta_u^2\,du\right),\qquad
W_t^{\mathbb Q}=W_t+\int_0^t\theta_u\,du.
$$
For these nonexplosive <linear stochastic differential equations>, the <linear drift change of measure> ensures that $Z$ is a true <martingale> on each finite horizon. This matters: a stochastic exponential being a local <martingale> alone would not prove that its terminal expectation is one. Its strict positivity gives an equivalent <risk-neutral measure> $\mathbb Q$, under which
$$
\boxed{dS_t=rS_t\,dt+\sigma\,dW_t^{\mathbb Q}.}
$$
The discounted asset is then a <martingale>. The <Feynman-Kac formula>, equivalently <risk-neutral valuation> of the replicable claim, gives
$$
\boxed{V(t,s)=e^{-r\tau}\mathbb E_{\mathbb Q}[F(S_T)\mid S_t=s].}
$$
Here the payoff must meet the usual integrability and replication assumptions; the bounded digital payoff below does.
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