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On a fixed horizon $T$, an arbitrage is an admissible trading strategy that is self-financing, has $X_0=0$, and satisfies $X_T\ge 0$ 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 ${\tilde{S}}^i=S^i/B$ a martingale.

For a predictable, stochastically integrable risky holding vector $\pi$, the discounted self-financing identity isIndeed, since the riskless account is a finite-variation process, its quadratic covariations vanish, and the Itô formula for $X/B$, combined with $X=\varphi B+\pi \cdot S$ and $dX=\varphi dB+\pi \cdot dS$, 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 satisfyIt follows that $X_T=0$ $Q$-almost surely, hence also physically almost surely by equivalence, a contradiction. Therefore 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.

The limit is in probability, or more strongly in uniform convergence on compacts in probability; a pathwise limit for every continuous local martingale is not implicit. The original PDF uses $\lceil t{2}^n\rceil$, whereas the supplied TeX changes this to $\lfloor t{2}^n\rfloor$; both versions have the same limit by continuity.

Put $\delta =2^{-n}$ and $m=\lfloor t/\delta \rfloor$. For the completed intervals, the algebraic identityseparates a left-point sum from a quadratic covariation sum.

The results from stochastic calculus used are these: predictable left-step approximations to a continuous adapted integrand converge in their Itô integrals in uniform convergence on compacts in probability; and products of increments of continuous semimartingales on deterministic partitions of an interval with vanishing mesh converge to their quadratic covariation in the same sense. The first follows after localization from the Itô isometry and the Doob L2 maximal inequality, since the integrand approximations converge uniformly on compact intervals; the second is the deterministic-partition characterization of quadratic covariation.

Thus the two sums converge to ${\int }_0^t{Y}_{s}d{X}_s$ and $\frac{1}{2}⟨Y,X{⟩}_t$. For the ceiling convention, at most one additional full interval extends past $t$; on $[0,T+1]$ its absolute contribution is at most ${\sup }_{s\le T+1}|Y_s|$ times the modulus of continuity of $X$ at scale $\delta$, which tends to zero almost surely. Therefore