= Finite-lifetime extension of the Dambis-Dubins-Schwarz theorem
{title2=$\widetilde B_u=N_u+\int_0^u\mathbf1_{\{s>L\}}\,d\beta_s$}
Strict increase of a bracket does not force an infinite terminal value. Up to $L=\langle M\rangle_\infty$, the inverse-clock martingale continues at a finite lifetime by its terminal limit, with bracket $u\wedge L$. On an independent product extension, add $\int_0^u\mathbf1_{\{s>L\}}\,d\beta_s$. Its bracket fills $(u-L)^+$ and its cross variation with the original part vanishes. The result is Brownian for all clock times and still represents $M_t$.
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