= Solution
The \b[<Novikov condition> holds on every horizon], since
$$
\mathbb E\exp\{\tfrac12[M]_T\}\leq e^{C/2}<\infty.
$$
Therefore $Z=\mathcal E(M)$ is a true <martingale> and the <Girsanov theorem> applies through each finite $T$.
The deterministic bound also gives a genuine infinite-horizon density. For any $p>1$, the <stochastic exponential> identity is
$$
Z_t^p=\mathcal E(pM)_t\exp\{\tfrac12p(p-1)[M]_t\}.
$$
A nonnegative <local martingale> is a <supermartingale>, so
$$
\boxed{\sup_t\mathbb E Z_t^p\leq\exp\{\tfrac12p(p-1)C\}.}
$$
This $L^p$ bound gives <uniform integrability>. Also $M$ is a <L2-bounded continuous martingale>, since localization and the <Itô isometry> give $\mathbb E M_t^2=\mathbb E[M]_t\leq C$. The <L2 martingale convergence theorem> gives a finite limit $M_\infty$; hence $Z_\infty=\exp(M_\infty-[M]_\infty/2)>0$ and $\mathbb E Z_\infty=1$. Weighting by $Z_\infty$ defines an equivalent <probability measure> on $\mathcal F_\infty$. This is the <bounded-bracket criterion for a stochastic exponential>.
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