= Solution
On a fixed horizon $T$, an <arbitrage> is an <admissible trading strategy> that is <self-financing>, has $X_0=0$, and satisfies $X_T\geq0$ almost surely with $\mathbb P(X_T>0)>0$. Here admissibility means that wealth in units of the strictly positive riskless <numéraire> has a deterministic lower bound throughout the horizon. An <equivalent martingale measure> $Q$ is equivalent to the physical measure on $\mathcal F_T$ and makes each discounted risky price $\widetilde S^i=S^i/B$ a <martingale>.
For a predictable, stochastically integrable risky holding vector $\pi$, the discounted self-financing identity is
$$
\widetilde X_t:=\frac{X_t}{B_t}
=\frac{X_0}{B_0}+\int_0^t\pi_s\cdot d\widetilde S_s.
$$
Indeed, since the riskless account is a <finite-variation process>, its quadratic covariations vanish, and the <Itô formula> for $X/B$, combined with $X=\phi B+\pi\cdot S$ and $dX=\phi\,dB+\pi\cdot dS$, gives $d(X/B)=\pi\cdot d(S/B)$.
Use two standard facts from <stochastic calculus>: a stochastic integral against a continuous local martingale is a <local martingale>, and a local martingale bounded below by a deterministic constant is a <supermartingale>. The latter follows by adding the constant to make the process nonnegative, stopping at a localizing sequence, and applying the conditional <Fatou lemma>. Thus admissible discounted wealth is a $Q$-supermartingale, and an arbitrage would satisfy
$$
0\leq\mathbb E_Q[\widetilde X_T]\leq\widetilde X_0=0.
$$
It follows that $X_T=0$ $Q$-almost surely, hence also physically almost surely by equivalence, a contradiction. Therefore \b[an equivalent martingale measure rules out arbitrage among admissible self-financing strategies]. The lower-bound restriction is essential to this argument; stochastic integrals need not be true martingales for arbitrary unbounded holdings.
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