Past exam of the stochastic differential equations course of the University of Oxford / 2025 / 2 / b / i

The expression obtained in part (a)(iii) gives a direct deterministic bound. For every fixed $T<\infty$ and $0\le s\le T$,Thus the bounded local martingale criterion makes $Z$ a true martingale on every finite time interval. More explicitly, if ${\sigma }_n$ is a localizing sequence, the identities $\mathbb{E}[Z_{t\wedge {\sigma }_n}\mid {\mathcal{F}}_s]=Z_{s\wedge {\sigma }_n}$ pass to the limit by dominated convergence, using the displayed bound for $t\le T$. ThereforeThis establishes exactly the finite-horizon property needed for the subsequent change of measure.