Proof of Existence and Uniqueness of the Mean-Square Extension of the Elementary Stochastic Integral

Throughout, $\lVert\cdot\rVert_{2}$ is the mean-square norm of Square-Integrable Random Variables and the Mean-Square Inner Product and all integrals $\int\cdots\rho\,d\lambda$ are over $(0,T]$ as in the statement. For simple adapted processes $F,F'$ on $(0,T]$, claims 1 and 3 of Linearity, Mean Zero, and Isometry of the Elementary Stochastic Integral give

Step 1 (Existence). By (1) and hypothesis (a), the sequence of elementary integrals $\bigl(\int_0^T H^k_t\,dM_t\bigr)_k$ is Cauchy in mean square. Each elementary integral is square-integrable (claim 2 of Linearity, Mean Zero, and Isometry of the Elementary Stochastic Integral) and $\mathcal{F}_T$-measurable: it is a finite sum of products of $\mathcal{F}_T$-measurable random variables (the coefficients $\xi_i$, being $\mathcal{F}_{t_i}$-measurable with $t_i\le T$, and the increments of the adapted process $M$ at times $\le T$), and sums and products of measurable functions are measurable by the arguments recorded in Square-Integrable Random Variables and the Mean-Square Inner Product. By mean-square completeness applied with the sub-$\sigma$-algebra $\mathcal{G}=\mathcal{F}_T$, there is a square-integrable $\mathcal{F}_T$-measurable random variable $I$ with $\lVert\int_0^T H^k_t\,dM_t-I\rVert_2\to0$.

Step 2 (Uniqueness). Let $(G^k)$ be another approximating sequence for $H$ and $I'$ the mean-square limit of its elementary integrals (which exists by Step 1). Fix $j$ and $t\in(0,T]$. By the triangle inequality of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm, for every $k$,

and the middle term tends to $0$ as $k\to\infty$, since $\lVert H^k_t-G^k_t\rVert_2\le\lVert H^k_t-H_t\rVert_2+\lVert H_t-G^k_t\rVert_2\to0$ by hypothesis (b) for both sequences. Using $(x+y)^2\le2x^2+2y^2$, it follows that for every $t\in(0,T]$,

All the functions of $t$ appearing here are measurable step functions (claim 3 of Linearity, Mean Zero, and Isometry of the Elementary Stochastic Integral), and the lower limit of measurable functions is measurable, as recorded in Fatou's Lemma. Multiplying by $\mathbf{1}_{(0,T]}\rho$ and integrating, Fatou's lemma gives

Given $\varepsilon>0$, choose $K$ so that hypothesis (a) holds with bound $\varepsilon$ for both sequences whenever $j,k\ge K$. Then for $j\ge K$ the right side is at most $4\varepsilon$, so by (1),

Now fix $\varepsilon>0$ and let $j\ge K(\varepsilon)$ grow. By the triangle inequality,

and letting $j\to\infty$ gives $\lVert I-I'\rVert_2\le2\sqrt{\varepsilon}$. Since $\varepsilon>0$ was arbitrary, $\lVert I-I'\rVert_2=0$, and $P(I=I')=1$ by the null-equivalence clause of Square-Integrable Random Variables and the Mean-Square Inner Product.

Step 3 (Norm and moments). By the isometry (claim 3 of Linearity, Mean Zero, and Isometry of the Elementary Stochastic Integral),

By the triangle inequality, $\bigl|\,\lVert\int_0^T H^k_t\,dM_t\rVert_2-\lVert I\rVert_2\,\bigr|\le\lVert\int_0^T H^k_t\,dM_t-I\rVert_2\to0$, so $a_k\to\lVert I\rVert_2^{2}=\mathbb{E}[I^{2}]$: the limit $\lVert H\rVert_M^{2}$ exists and equals $\mathbb{E}[I^2]$. If $(G^k)$ is another approximating sequence with limit $I'$, then $P(I=I')=1$ by Step 2, so $\lVert I'\rVert_2=\lVert I\rVert_2$ (their mean-square distance is $0$, and the triangle inequality gives equality of norms); hence the value $\lVert H\rVert_M^{2}$ does not depend on the approximating sequence. Finally, by the Cauchy-Schwarz inequality with the constant random variable $1$ and claim 2 of Linearity, Mean Zero, and Isometry of the Elementary Stochastic Integral,

so $\mathbb{E}[I]=0$. $\square$