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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

The increment variance isso independence givesLinear interpolation makes the supremum of the absolute centered process occur at an integer time. The Doob L2 maximal inequality therefore gives

The L2 martingale convergence theorem gives $M_t\to M_{\infty }$ in $L^2$, soConversely, apply the Doob L2 maximal inequality on $[0,T]$:Monotone convergence as $T\to \infty$ givesThus the two norms are equivalent.

Set $g_{ϵ}(x)=x/\sqrt{{ϵ}^2+x^2}$ and $g(x)=\mathrm{sgn}(x)$, with $g(0)=0$. For every $s>0$, the normal distribution of $B_s$ has no atom at zero, so $g_{ϵ}(B_s)\to g(B_s)$ almost surely. Since $|g_{ϵ}-g|\le 2$, the dominated convergence theorem and Tonelli theorem giveThe Itô isometry followed by the Doob L2 maximal inequality now yieldsThus the stochastic integrals converge ucp to ${\int }_0^t\mathrm{sgn}(B_s)d{B}_s$.

By the stated Cauchy property and completeness of ${\mathcal{M}}^2$, there is a square-integrable continuous martingale $M$ such thatSet $A_t=X_t^2-M_t$. This process is continuous and adapted, andby the Doob L2 maximal inequality. The process $A$ is the quadratic variation $[X]$.

Use the continuous semimartingale decomposition $X=X_0+M+A$, where $M$ is a continuous local martingale and $A$ is a continuous adapted finite-variation process. Pointwise limits preserve predictability, so $H$ is predictable; it is bounded by the common bound for the $H_n$.

Localize so that $[M{]}_{\infty }$ and the total variation $V(A{)}_{\infty }$ are bounded. The Doob L2 maximal inequality and the Itô isometry giveby the dominated convergence theorem. For the finite-variation part,almost surely, again by dominated convergence, now for each sample path. Hence the two integrals converge uniformly in probability after every localization. Part (b) removes the localization and provesu.c.p.

Choose $f_n(0)=0$ and $f_n^{\prime }(x)=\varphi (nx)$, where $\varphi$ is continuously differentiable, nondecreasing, equal to $-1$ on $(-\infty ,0]$ and $1$ on $[1,\infty )$. Then $f_n^{\prime \prime }\ge 0$ and $0\le |x|-f_n(x)\le 2/n$. For $X=M+A$ with continuous adapted finite-variation $A$, the Itô isometry and Doob L2 maximal inequality give local uniform convergence in probability of the martingale integrals. Dominated convergence against the total-variation measure gives pathwise uniform convergence of the drift integrals. The nondecreasing corrections $\frac{1}{2}\int f_n^{\prime \prime }(X)d[M]$ therefore converge locally uniformly in probability to $|X|-\int {\mathrm{sgn}}_{-}(X)dX$. A subsequence on one probability-one event shows that the limit is continuous and nondecreasing. Localization removes the deterministic bounds used in the argument.

For $t>0$, let $M_t={\sup }_{s\le t}{B}_s$ and $A_t={\sup }_{s\le t}|B_s|$. The Brownian reflection principle gives $M_t\stackrel{d}{=}|B_t|$, hence $\Vert M_t{\Vert }_2=\sqrt{t}$. Also $A_t\ge M_t$, with strict inequality on the positive-probability event $\{B_t<-\lambda ,M_t<\lambda \}$: reflection gives its probability as $\mathbb{P}(B_t>\lambda )-\mathbb{P}(B_t>3\lambda )>0$. The Doob L2 maximal inequality bounds $\Vert A_t{\Vert }_2\le 2\sqrt{t}$, so $\sqrt{t}<\Vert A_t{\Vert }_2\le 2\sqrt{t}$. At $t=0$ all quantities vanish and the lower inequality is equality.

// Target: probability-theory.bigb