Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 202 / 2 / a

The bounded continuous local martingale $X$ is a square-integrable martingale. Since $M^{(n)}$ is a discrete predictable transform of $X$, it has mean zero. The identity $A_t^{(n)}=X_t^2-M_t^{(n)}$ therefore givesuniformly in $t$ and $n$.

The Itô isometry for the elementary predictable integrand in $M^{(n)}$ gives

Because $A_t^{(n)}=X_t^2-M_t^{(n)}$, the inequality $(x-y{)}^2\le 2x^2+2y^2$ and part (ii) yield

By the stated Cauchy property and completeness of ${\mathcal{M}}^2$, there is a square-integrable continuous martingale $M$ such thatSet $A_t=X_t^2-M_t$. This process is continuous and adapted, andby the Doob L2 maximal inequality. The process $A$ is the quadratic variation $[X]$.

If $A_t=0$ almost surely, then $X_t^2=M_t$ is a nonnegative martingale starting from zero. A nonnegative random variable of expectation zero vanishes almost surely, so $X_t=0$ almost surely for each $t$. Applying this on the nonnegative rational times and using path continuity shows that $X_t=0$ simultaneously for every $t\ge 0$ almost surely.