= Solution
Assume (iii). Fix $t\in\{1,\ldots,T\}$, a coordinate $i$, and an event $A\in\mathcal F_{t-1}$. Choose the bounded predictable risky strategy
$$
\theta_s=\mathbf1_Ae_i\mathbf1_{\{s=t\}},
$$
and choose the numéraire holding to make the strategy self-financing with $V_0=0$. Its terminal discounted value is
$$
V_T=\mathbf1_A(X_t^i-X_{t-1}^i).
$$
Condition (iii) yields
$$
\mathbb E_{\mathbb Q}[
\mathbf1_A(X_t^i-X_{t-1}^i)]=0
$$
for every $A\in\mathcal F_{t-1}$. The defining property of <conditional expectation> implies
$$
\mathbb E_{\mathbb Q}[X_t^i\mid\mathcal F_{t-1}]
=X_{t-1}^i.
$$
Every coordinate is therefore a $\mathbb Q$-martingale, so \b[(iii) $\Rightarrow$ (i)]. This proves the <martingale characterization by bounded self-financing strategies>.
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