= Solution
Take $X=Y=B$, a standard <Brownian motion> with $B_0=0$. The <Itô formula> and $\langle B\rangle_t=t$ give
$$
\boxed{\int_0^tB_s\circ dB_s=\frac12(B_t^2-t)+\frac12t=\frac12B_t^2.}
$$
If this nonnegative <stochastic process>, which starts from zero, were a <local martingale>, question 4(i) would make it a <supermartingale> and force its <expectation> to be at most zero. But $\mathbb E(B_t^2/2)=t/2>0$ for $t>0$. Hence \b[a <Stratonovich integral> is in general not a <local martingale>.]
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