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

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P)$ be a filtered probability space, let $(M,\rho)$ be an It^{o} integrator of intensity type with respect to $(\mathcal{F}_t)_{t\ge0}$, let $T>0$ be real, and let $\lambda$ denote Lebesgue measure. Let $H=(H_t)_{t\in(0,T]}$ be a family of square-integrable random variables, and write $\lVert\cdot\rVert_{2}$ for the mean-square norm of that definition.

Call a sequence $(H^{k})_{k\in\mathbb{N}}$ of simple adapted processes on $(0,T]$ an approximating sequence for $H$ if:

(a) for every real $\varepsilon>0$ there is $K$ such that all $j,k\ge K$ satisfy

(the integrand is a measurable step function by claim 3 of Linearity, Mean Zero, and Isometry of the Elementary Stochastic Integral, applied to the simple adapted process $H^{j}-H^{k}$);

(b) for every $t\in(0,T]$, $\lVert H^{k}_{t}-H_{t}\rVert_{2}\to0$ as $k\to\infty$.

Then the following hold.

1. (Existence) If $(H^{k})$ is an approximating sequence for $H$, then there is a square-integrable, $\mathcal{F}_T$-measurable random variable $I$ with

where $\int_0^T H^k_t\,dM_t$ is the elementary stochastic integral.

2. (Uniqueness) If $(G^{k})$ is another approximating sequence for the same family $H$, with mean-square limit $I'$ of its elementary integrals, then $P(I=I')=1$.

3. (Norm and moments) If $(H^{k})$ is an approximating sequence for $H$, then the limit

exists, has the same value for every approximating sequence for $H$ (so the notation $\lVert H\rVert_{M}$, in which the horizon $T$ and intensity $\rho$ are left implicit, is unambiguous), and