Ito Integrable Process and the Ito 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 $(\mathcal{F}_t)_{t\ge0}$, and let $T>0$ be real.

A family $H=(H_t)_{t\in(0,T]}$ of square-integrable random variables is It^{o} integrable on $(0,T]$ with respect to $(M,\rho)$ if there exists an approximating sequence for $H$ in the sense of Existence and Uniqueness of the Mean-Square Extension of the Elementary Stochastic Integral, that is, a sequence $(H^{k})$ of simple adapted processes on $(0,T]$ satisfying conditions (a) and (b) there.

The It^{o} integral of $H$ with respect to $M$ over $(0,T]$, written

is any square-integrable $\mathcal{F}_T$-measurable random variable $I$ as in claim 1 of Existence and Uniqueness of the Mean-Square Extension of the Elementary Stochastic Integral; by claims 1 and 2 there, such an $I$ exists and is determined up to almost sure equality, independently of the choice of approximating sequence, so every assertion about $\int_0^T H_t\,dM_t$ is to be read as an almost sure assertion about any such version. (The notation is consistent with Elementary Stochastic Integral of a Simple Adapted Process: for a simple adapted process, the constant sequence $H^k=H$ is an approximating sequence, so the It^{o} integral agrees almost surely with the elementary stochastic integral, and the same symbol may be used for both.)

Integrals over subintervals. For $t\in(0,T]$, the restriction $(H_u)_{u\in(0,t]}$ is It^{o} integrable on $(0,t]$: if $(H^{k})$ is an approximating sequence for $H$, then the restrictions $(H^{k}_u)_{u\in(0,t]}$ are simple adapted processes on $(0,t]$ (as recorded in The Elementary Stochastic Integral Process is a Square-Integrable Martingale) and satisfy condition (b) at every $u\in(0,t]$; condition (a) holds because $\mathbf{1}_{(0,t]}\le\mathbf{1}_{(0,T]}$ pointwise and the Lebesgue integral is monotone (Linearity and Monotonicity of the Lebesgue Integral). Define

as the It^{o} integral of the restriction, chosen $\mathcal{F}_t$-measurable as in claim 1 of Existence and Uniqueness of the Mean-Square Extension of the Elementary Stochastic Integral applied on $(0,t]$, and set $\int_0^0 H_u\,dM_u=0$.