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For a continuous local martingale and a locally bounded predictable process, stop at a localizing time, an integrand bound, a time bound, and level bounds for the centered martingale and its bracket. The stopped martingale is bounded and its bracket and integrand are deterministically bounded. The Itô isometry, applied to $1_F{1}_{(s,t]}H$ for $F\in {\mathcal{F}}_s$, gives the conditional second-moment identity. Expanding the square proves that $(H\cdot M{)}^2-H^2\cdot [M]$ is a martingale after stopping. Uniqueness of an increasing square compensator identifies the bracket, and increasing stopping times patch the identities into the displayed local result.

Subtract $M_0$, which does not affect either the integral or its quadratic variation. Let $N=M-M_0$. Choose a localizing sequence ${\tau }_n$ for $N$. By local boundedness of the previsible process $H$, there are increasing stopping times ${\rho }_n\uparrow \infty$ and finite deterministic constants $K_n$ with $|H_s|\le K_n$ on $0<s\le {\rho }_n$. PutContinuity and finiteness on compact intervals imply ${\sigma }_n\uparrow \infty$ almost surely. The stopped process $N^{{\sigma }_n}$ is a bounded square-integrable martingale, its bracket is bounded by $n$, and its stopped integrand is bounded by $K_n$. The indicator $1_{\{s\le {\sigma }_n\}}$ is predictable, so the stopped stochastic integral identity is valid.

Fix $n$ and abbreviateThe Itô isometry gives $\mathbb{E}{J}_t^2=\mathbb{E}{Q}_t\le K_n^{2}n$, so $J$ is a square-integrable martingale. For $F\in {\mathcal{F}}_s$, apply the same isometry to the predictable integrand $1_F{1}_{(s,t]}H$, stopped at ${\sigma }_n$. This yieldsAlso $\mathbb{E}[J_t-J_s\mid {\mathcal{F}}_s]=0$. Expanding $J_t^2-J_s^2$ therefore proves that $J^2-Q$ is a true martingale. The process $Q$ is continuous adapted increasing and starts at zero. By the characterization of continuous martingale quadratic variation as its increasing square compensator, and uniqueness of an increasing square compensator, we have $[J]=Q$.

Stochastic integration and quadratic variation commute with stopping; for the latter this follows either from squared-increment sums or from the same square-compensator characterization. Hence the desired identity holds up to every ${\sigma }_n$. Since these times increase to infinity, they exhaust every finite time interval almost surely, and the continuous identities patch to giveThis is the localized isometry proof of stochastic-integral quadratic variation. No unstopped second-moment or bracket-integrability assumption has been made.

For two candidates $A,C$, their difference is a continuous adapted finite-variation process. It is also a local martingale, sinceApply part (b) after subtracting its initial value. ConsequentlyThis proves the uniqueness of an increasing square compensator when the initial value is prescribed, in particular under the standard quadratic variation normalization $A_0=C_0=0$.

The printed assertion omits that normalization and is literally false. For standard Brownian motion $Y$, both $A_t=t$ and $C_t=t+1$ are continuous adapted increasing processes; both $Y_t^2-t$ and $Y_t^2-t-1$ are martingales. Thus uniqueness holds after fixing the initial value; otherwise it holds only up to an initial constant.

As with quadratic variation, the increasing process must be normalized by $A_0=0$. Suppose $A,A^{\prime }$ have this initial value and both $M^2-A$ and $M^2-A^{\prime }$ are continuous local martingales. Since $M^2$ and these differences are continuous, both $A$ and $A^{\prime }$ are continuous. Their difference is a finite-variation process, andis a continuous local martingale. The result proved in part (a) makes this difference constant. Its initial value is zero, so $A$ and $A^{\prime }$ are indistinguishable. This is the uniqueness of an increasing square compensator; it uses no existence argument.

If no initial-value normalization is imposed, the literal uniqueness assertion is false: whenever $A$ works, $A+c$ works for every deterministic constant $c$, since subtracting a constant preserves the local martingale property. The exact unnormalized conclusion is $A_t-A_t^{\prime }=A_0-A_0^{\prime }$ for all $t$, up to indistinguishability of stochastic processes.