= Solution
Put $F_t=\exp(-i\theta X_t-\frac12\theta^2A_t)$. The <Itô formula> gives
$$
dF_t=-i\theta F_t\,dX_t-\theta^2F_t\,dA_t.
$$
The explicit decreasing exponential contributes half the finite-variation term; the other half is the quadratic correction from $e^{-i\theta X_t}$. Now the <Itô product rule> and part (b) yield
$$
d(F_tM_t)=F_t\,dM_t-i\theta F_tM_t\,dX_t-\theta^2F_tM_t\,dA_t-i\theta F_t\,d\langle M,X\rangle_t=F_t\,dM_t-i\theta F_tM_t\,dX_t.
$$
The finite-variation terms cancel because $-i\theta(i\theta)=\theta^2$. This proves the product is a <local martingale>. Moreover $|F_t|=e^{-\theta^2A_t/2}\le1$ and $|M_t|\le1$. The <bounded local martingale criterion> therefore proves \b[the product is a true martingale].
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