The Stochastic Integral

Each summand $h_i(M_{t_{i+1}\wedge t}-M_{t_i\wedge t})$ is adapted and integrable, and the conditional expectation of each increment given $\F_s$ equals its current value, because $h_i$ is $\F_{t_i}$-measurable and $M$ has martingale increments. A finite sum of martingales is a martingale. For square integrability at each $t$, apply the isometry Theorem 2, whose proof uses only the increments of $M$ and not this proposition, to the truncated integrand $H\ind_{(0,t]}$, which satisfies $(H\ind_{(0,t]}\cdot M)_T=(H\cdot M)_t$, so $\E[(H\cdot M)_t^2]=\E[\int_0^t H_s^2\,d\qv_s]\le\norm{H}_M^2<\infty$.

Expand the square of Equation (2) at $t=T$ into diagonal and off-diagonal terms. For $i<j$ the cross term $h_i h_j(M_{t_{i+1}}-M_{t_i})(M_{t_{j+1}}-M_{t_j})$ has zero expectation, since conditioning on $\F_{t_j}$ leaves the $\F_{t_j}$-measurable factors times $\E[M_{t_{j+1}}-M_{t_j}\mid\F_{t_j}]=0$. The diagonal terms give

where the last equality uses that $M_t^2-\qv_t$ is a martingale, so $\E[(M_{t_{i+1}}-M_{t_i})^2\mid\F_{t_i}]=\E[\qv_{t_{i+1}}-\qv_{t_i}\mid\F_{t_i}]$. The final sum is $\E[\int_0^T H_s^2\,d\qv_s]$, which is Equation (3).

Let $\mu_M(d\omega,ds)=d\P\,d\qv_s$ on $(\Omega\times[0,T],\mathcal P)$, a finite measure since $\E[\qv_T]<\infty$, so $\mathcal H=L^2(\mu_M)$. The indicators $\ind_{A\times(s,t]}$ with $A\in\F_s$ are simple predictable, and their linear span is an algebra generating $\mathcal P$. By the functional monotone class theorem this span is dense in $L^2(\mu_M)$, so bounded predictable $H$ are $\norm{\cdot}_M$-approximable by simple predictable processes, and a general $H\in\mathcal H$ follows by truncating to $H\ind_{|H|\le n}$. Given $H\in\mathcal H$ choose simple $H^n\to H$ in $\norm{\cdot}_M$. The isometry Theorem 2 makes $(H^n\cdot M)$ Cauchy in $L^2$, and by Doob's inequality $\E[\sup_{t\le T}((H^n-H^m)\cdot M)_t^2]\le 4\,\norm{H^n-H^m}_M^2\to 0$, giving a.s. uniform Cauchy convergence, so the limit $H\cdot M$ exists, is independent of the approximating sequence, and is a square-integrable martingale by closure of the square-integrable martingales under $L^2$ limits. For the quadratic variation, the same conditioning as in Theorem 2 on $\F_s$ rather than full expectation gives, for simple $H$ and $s<t$,

Since $H\cdot M$ is a martingale, $\E[(H\cdot M)_t^2\mid\F_s]-(H\cdot M)_s^2=\E[((H\cdot M)_t-(H\cdot M)_s)^2\mid\F_s]$, so $(H\cdot M)_t^2-\int_0^t H_s^2\,d\qv_s$ is a martingale, whence the predictable increasing process $\int_0^t H_s^2\,d\qv_s$ is by definition $\langle H\cdot M\rangle_t$. The identity extends to general $H\in\mathcal H$ by the $L^2$ convergence above.