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For a simple predictable process, define its stochastic integral byCommon refinement of the deterministic partitions makes this definition independent of the representation. Each summand is a martingale increment multiplied by information already available at its left endpoint, so the resulting process is a continuous square-integrable martingale starting at zero. Its terminal value is the same sum with $t$ replaced by $t_n$.

Write ${\Delta }_{k}M=M_{t_{k+1}}-M_{t_k}$. If $j<k$, then $Z_{t_j}{\Delta }_{j}M{Z}_{t_k}$ is ${\mathcal{F}}_{t_k}$-measurable. By the conditional expectation identity for increments of a martingale,The conditional bracket isometry for stopped martingale increments givesFor completeness, this identity follows by applying the optional stopping theorem to $M$ and $M^2-[M]$ and expanding the square. The latter process is a true uniformly integrable martingale here: the Doob L2 maximal inequality bounds $\mathbb{E}{\sup }_t|M_t{|}^2$, stopped square identities and Fatou's lemma give $\mathbb{E}[M{]}_{\infty }<\infty$, and its absolute supremum is bounded by ${\sup }_t|M_t{|}^2+[M{]}_{\infty }$. Consequently all the stopped identities pass to the limit.

Multiplying the conditional identity by $Z_{t_k}^2$, taking expectations, and using the vanished cross terms proves the Itô isometry:

To extend the stochastic integral, use the finite quadratic-variation measure ${\nu }_M(C)=\mathbb{E}\int 1_{C}d[M]$ on the predictable sigma-algebra. The density of simple predictable processes for finite measures gives approximations to every $H\in L^2({\nu }_M)$. The Itô isometry makes the corresponding terminal integrals Cauchy in $L^2$, and the Doob L2 maximal inequality makes the whole processes Cauchy in expected squared supremum. Their limit defines a continuous martingale, and the isometry persists. For a locally bounded predictable process, choose increasing stopping times tending to infinity on which it is bounded, also restrict the time horizon, and perform this construction after stopping. The isometry shows that the stopped definitions agree on overlaps, so they patch into a uniquely defined continuous local martingale $H\cdot M$.

First justify optional sampling for the square-minus-bracket process. The Doob L2 maximal inequality gives $\mathbb{E}{\sup }_{t\ge 0}|M_t{|}^2\le 4\mathbb{E}|M_{\infty }{|}^2<\infty$. Stop the local martingale $M^2-[M]$ using a localizing sequence and bounded bracket levels. The stopped expectation identity, followed by Fatou, bounds $\mathbb{E}[M{]}_{\infty }$ by the expected squared path supremum. Hence $M^2-[M]$ has an integrable absolute supremum, and localization with conditional dominated convergence makes it a uniformly integrable martingale. The optional stopping theorem stated in part (b) is therefore applicable to both $M$ and $M^2-[M]$.

For ordered stopping times $\sigma \le \tau$, expand the squared increment and condition on ${\mathcal{F}}_{\sigma }$. The mean increment is zero, soThis is the conditional bracket isometry for stopped martingale increments. Apply it with $\sigma =T_j$, $\tau =T_{j+1}$ and multiply by the bounded ${\mathcal{F}}_{T_j}$-measurable variable $h_j^2$. The orthogonality proved in part (b) gives the Itô isometry

To identify the whole bracket process, let $N=H\cdot M$ and $C_t={\int }_0^t{H}_s^{2}d[M{]}_s$. For deterministic $s<t$ and $E\in {\mathcal{F}}_s$, apply the proved isometry to the simple integrand $1_E{1}_{(s,t]}H$. It yieldsThus $\mathbb{E}[(N_t-N_s{)}^2\mid {\mathcal{F}}_s]=\mathbb{E}[C_t-C_s\mid {\mathcal{F}}_s]$. Since $N$ is a martingale, the cross term $2N_s(N_t-N_s)$ has conditional mean zero. Consequently $N^2-C$ is a martingale. The process $C$ is continuous, adapted, nondecreasing and starts at zero, so uniqueness in the definition of quadratic variation provesThis deduction uses only the proved elementary isometry, not a previously assumed formula for the quadratic variation of stochastic integrals.