Quadratic variation of an Ito integral

Brownian estimate: for a deterministic partition $\pi$ of $[0,T]$ with mesh $\delta$ and $N_j$ as in [F2], independence of the Brownian increments gives $E{N}_J^2={\sum }_{k}2(s_{k+1}-s_k{)}^2\le 2\delta T$, and Doob's $L^2$ inequality gives $E{\max }_j{N}_j^2\le 8\delta T$; since $[B{]}_t^{\pi ,\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}-t=N_{k(t)}+s_{k(t)}-t$ where $k(t)=\max \{j:0\le j\le J,s_j\le t\}$ and $\mathrm{\mid }{s}_{k(t)}-t\mathrm{\mid }\le \delta$, one has $E{\sup }_{t\le T}\mid [B{]}_t^{\pi ,\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}-t{\mid }^2\le 16\delta T+2{\delta }^2$, which tends to $0$; [F2]'s elementary probability bound turns this into uniform convergence in probability.

For elementary $H$ with partition $0=t_0<\cdots <t_m=T$ and a sub-interval $(u,v]$, the increment $M_v-M_u={\int }_0^T{1}_{(u,v]}HdB$ is the elementary sum of the elementary integrand $1_{(u,v]}H$, whose coefficients are ${\xi }_k$ on $(u\vee t_k,v\wedge t_{k+1}]$, measurable at the left endpoints $u\vee t_k\ge t_k$; consequently $M_v-M_u={\sum }_k{\xi }_k(B_{v\wedge t_{k+1}}-B_{u\vee t_k})$ over the at most two blocks meeting $(u,v]$ when $v-u<{\min }_k(t_{k+1}-t_k)$, and equals ${\xi }_k(B_v-B_u)$ when $(u,v]$ is contained in a single block $(t_k,t_{k+1}]$.

Elementary refinement: fix a bounded elementary $H$ with blocks $(t_k,t_{k+1}]$, $0\le k<m$, and a deterministic bound $K$ for its coefficients on a common probability-one event. For sufficiently large $n$, ${\delta }_n<{\min }_k(t_{k+1}-t_k)$, so each interval of ${\pi }_n$ crosses at most one elementary boundary. Refine by all these boundaries, and write $R_k$ for the induced partition of $[t_k,t_{k+1}]$. Let $S_k^{(n)}(t)$ be the Brownian step sum on $R_k$, extended by zero before $t_k$ and its terminal value after $t_{k+1}$. The refined integral sum is ${\sum }_k{\xi }_k^2{S}_k^{(n)}(t)$. Away from crossing intervals it equals $Q_n(t)$. On a crossing interval, the original increment has absolute value at most $2K{\omega }_B({\delta }_n)$, while each of the at most two refined increments has absolute value at most $K{\omega }_B({\delta }_n)$, where ${\omega }_B(\delta )={\sup }_{\mathrm{\mid }u-v\mathrm{\mid }\le \delta ,u,v\in [0,T]}\mathrm{\mid }{B}_u-B_v\mathrm{\mid }$ on the continuous representative. This also bounds the discrepancy when the refined sum has included the boundary but the original interval is not yet complete. Adding the original squared contribution and the two subtracted refined squared contributions bounds the absolute discrepancy uniformly in $t$ by $6m{K}^2{\omega }_B({\delta }_n{)}^2$, which tends to zero almost surely by [F3].

Apply step 1.1 on each deterministic interval $[t_k,t_{k+1}]$ to its translated Brownian increments and the induced mesh $R_k$, whose mesh is at most ${\delta }_n$. Thus ${\sup }_{t\in [t_k,t_{k+1}]}\mathrm{\mid }{S}_k^{(n)}(t)-(t-t_k)\mathrm{\mid }\to 0$ in probability. Since $A_t={\sum }_k{\xi }_k^2(t\wedge t_{k+1}-t\wedge t_k)$, the refined-sum error is bounded by $K^2$ times the finite sum of these block errors. A finite union bound and step 2.1, with [F6] for its almost-sure vanishing error, give ${\sup }_t\mathrm{\mid }{Q}_n(t)-A_t\mathrm{\mid }\to 0$ in probability for each bounded elementary $H$. No independence of ${\xi }_k$ from these error suprema is needed, because the deterministic bound $K$ is used.

Finite-energy case: use [F9] and [F7] to choose $H^l\to H$ in $L^2(\mathrm{d}t\otimes P)$ with $H^l$ bounded elementary and set $N^l:=H-H^l$; by Cauchy--Schwarz in each partial sum, ${\sup }_t\mathrm{\mid }{Q}_n(H)(t)-Q_n(H^l)(t)\mathrm{\mid }\le Q_n(N^l)(T)+2Q_n(N^l)(T{)}^{1\mathrm{/}2}{Q}_n(H^l)(T{)}^{1\mathrm{/}2}$, and [F1] gives $E{Q}_n(N^l)(T)=E{\int }_0^T(N^l{)}^{2}ds\to 0$ and $E{Q}_n(H^l)(T)=E{\int }_0^T(H^l{)}^{2}ds\le C$ uniformly in $n$, so $E{\sup }_t\mathrm{\mid }{Q}_n(H)-Q_n(H^l)\mathrm{\mid }\to 0$ as $l\to \mathrm{\infty }$ uniformly in $n$; combined with step 3.1 for $H^l$ and $E{\sup }_t\mathrm{\mid }{A}_{H^l}(t)-A_H(t)\mathrm{\mid }\le \mathrm{\parallel }{H}^l-H{\mathrm{\parallel }}_2\mathrm{\parallel }{H}^l+H{\mathrm{\parallel }}_2\to 0$, the finite-energy case follows by a two-parameter argument: for error threshold $\epsilon$, split the total error into the quadratic-sum approximation, the elementary convergence error and the energy approximation, each at threshold $\epsilon \mathrm{/}3$. Their probabilities are bounded by $3\mathrm{/}\epsilon$ times the two expected approximation errors, plus the elementary error probability. The latter vanishes as $n\to \mathrm{\infty }$ at fixed $l$, and the former vanish as $l\to \mathrm{\infty }$, uniformly in $n$.

Localized case: on the event $\{{\tau }_m\ge T\}$ the processes $M$ and $M^{{\tau }_m}$ agree on $[0,T]$ and $A_t^{(m)}=A_t$ for $t\le T$ by [F4], so the squared-increment sums of $M$ coincide with those of the finite-energy integral $M^{{\tau }_m}$ there; consequently, for every $\epsilon >0$, $P({\sup }_t\mathrm{\mid }{Q}_n(M)-A\mathrm{\mid }>\epsilon )\le P({\tau }_m<T)+P({\sup }_t\mathrm{\mid }{Q}_n(M^{{\tau }_m})-A^{(m)}\mathrm{\mid }>\epsilon )$, and the second term tends to $0$ by step 4.1 while the first tends to $0$ as $m\to \mathrm{\infty }$ because ${\tau }_m\uparrow \mathrm{\infty }$ almost surely.

Steps 3.1 and 5.1 establish uniform convergence in probability for the step convention; the partial-increment convention differs from the step value by at most the squared maximal oscillation of the continuous path of $M$ over the partition intervals, which tends to $0$ by [F3] and continuity of $M$, so both conventions have the same limit. The dyadic partitions are included; no almost-sure dyadic conclusion is asserted for general $H$. AC covers the declared interfaces and the countable approximating choices in [F7].