Finite-lifetime extension of the Dambis-Dubins-Schwarz theorem (${\tilde{B}}_u=N_u+{\int }_0^u{1}_{\{s>L\}}d{\beta }_s$)

Strict increase of a bracket does not force an infinite terminal value. Up to $L=⟨M{⟩}_{\infty }$, the inverse-clock martingale continues at a finite lifetime by its terminal limit, with bracket $u\wedge L$. On an independent product extension, add ${\int }_0^u{1}_{\{s>L\}}d{\beta }_s$. 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 $M_t$.