Past exam of the stochastic differential equations course of the University of Oxford / 2025 / 2 / a / iii / Solution

The Itô formula for $B_t^2$ yields ${\int }_0^t{B}_{u}d{B}_u=(B_t^2-t)/2$. Substituting into the expression for $Z_t$ gives the pathwise identityConsequentlyTaking expected values provesThe random variable on the preceding line is bounded by one for $\alpha \ge 0$, so its integrability is immediate.