Properties of the Ito Integral: Linearity, Isometry, Martingale Property, and Mean-Square Continuity

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, let $H=(H_t)_{t\in(0,T]}$ and $G=(G_t)_{t\in(0,T]}$ be It^{o} integrable on $(0,T]$ with respect to $(M,\rho)$, and let $a,b$ be real numbers. Write $\lambda$ for Lebesgue measure and $\lVert\cdot\rVert_{M}$ for the norm of claim 3 of Existence and Uniqueness of the Mean-Square Extension of the Elementary Stochastic Integral. All identities between random variables below are almost sure identities.

1. (Linearity) $aH+bG=(aH_t+bG_t)_{t\in(0,T]}$ is It^{o} integrable on $(0,T]$, and

2. (Moments and polarization) $\mathbb{E}\bigl[\int_0^T H_t\,dM_t\bigr]=0$ and $\mathbb{E}\bigl[(\int_0^T H_t\,dM_t)^{2}\bigr]=\lVert H\rVert_{M}^{2}$; moreover, for any approximating sequences $(H^k)$ for $H$ and $(G^k)$ for $G$, the limit below exists, does not depend on the choice of the two approximating sequences, and satisfies

3. (Explicit isometry) If the function $t\mapsto\mathbb{E}[H_t^{2}]$, extended by $0$ off $(0,T]$, is measurable for the Borel $\sigma$-algebra, then $\mathbf{1}_{(0,T]}\,\mathbb{E}[H_\cdot^{2}]\,\rho$ is integrable and

4. (Martingale property) With $I_t=\int_0^t H_u\,dM_u$ for $t\in[0,T]$ as in Ito Integrable Process and the Ito Integral, the process $(I_{\min(s,T)})_{s\ge0}$ is a square-integrable martingale with respect to $(\mathcal{F}_s)_{s\ge0}$; in particular each $I_t$ is $\mathcal{F}_t$-measurable and $\mathbb{E}[(I_t-I_s)\mathbf{1}_A]=0$ for all $0\le s\le t\le T$ and $A\in\mathcal{F}_s$.

5. (Mean-square continuity) The family $(I_t)_{t\in[0,T]}$ is uniformly mean-square continuous on $[0,T]$: for every $\varepsilon>0$ there is $\delta>0$ such that all $s,t\in[0,T]$ with $|s-t|<\delta$ satisfy $\lVert I_s-I_t\rVert_{2}<\varepsilon$, with $\lVert\cdot\rVert_{2}$ the mean-square norm of Square-Integrable Random Variables and the Mean-Square Inner Product.