Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 27 / 5 / a / Solution

Strict positivity lets us define the continuous local martingaleThe integrand is locally bounded because a positive continuous path has positive minimum on every compact time interval. The Itô formula for $\log M$ givesThis is the stochastic exponential representation of a positive continuous local martingale.

If $⟨X{⟩}_{\infty }$ were finite on an event of positive probability, the finite-bracket convergence lemma proved in Question 2(a) would make $X_t$ converge to a finite limit there. The exponential would then have a strictly positive limit, contradicting the assumed $M_t\to 0$. ThereforeThis is the divergent logarithmic clock for a positive local martingale tending to zero.