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

Fix approximating sequences $(H^k)$ for $H$ and $(G^k)$ for $G$ in the sense of Existence and Uniqueness of the Mean-Square Extension of the Elementary Stochastic Integral, write $I=\int_0^T H_t\,dM_t$ and $J=\int_0^T G_t\,dM_t$ for the It^{o} integrals, $I^k,J^k$ for the corresponding elementary integrals over $(0,T]$, and $\lVert\cdot\rVert_2$ for the mean-square norm of Square-Integrable Random Variables and the Mean-Square Inner Product. Throughout we use the triangle and Cauchy-Schwarz inequalities, and the elementary pointwise bounds $(x+y)^2\le2x^2+2y^2$ and, for $\delta>0$, $(x+y)^{2}\le(1+\delta)x^{2}+(1+\delta^{-1})y^{2}$ (from $2xy\le\delta x^{2}+\delta^{-1}y^{2}$).

Step 1 (Linearity). Each $aH^k+bG^k$ is a simple adapted process (claim 1 of Linearity, Mean Zero, and Isometry of the Elementary Stochastic Integral). The sequence $(aH^k+bG^k)$ satisfies condition (b) of Existence and Uniqueness of the Mean-Square Extension of the Elementary Stochastic Integral for $aH+bG$ at every $t$, since $\lVert aH^k_t+bG^k_t-(aH_t+bG_t)\rVert_2\le|a|\lVert H^k_t-H_t\rVert_2+|b|\lVert G^k_t-G_t\rVert_2\to0$; and condition (a), since pointwise in $t$

and the Lebesgue integral is monotone and linear (Linearity and Monotonicity of the Lebesgue Integral). Hence $aH+bG$ is It^{o} integrable, and by claim 1 of Linearity, Mean Zero, and Isometry of the Elementary Stochastic Integral its elementary approximants satisfy $\int_0^T(aH^k_t+bG^k_t)\,dM_t=aI^k+bJ^k$, which converges in mean square both to $\int_0^T(aH_t+bG_t)\,dM_t$ (by Existence and Uniqueness of the Mean-Square Extension of the Elementary Stochastic Integral) and to $aI+bJ$; two mean-square limits of one sequence are at mean-square distance $0$, hence almost surely equal by the null-equivalence clause of Square-Integrable Random Variables and the Mean-Square Inner Product.

Step 2 (Moments and polarization). $\mathbb{E}[I]=0$ and $\mathbb{E}[I^{2}]=\lVert H\rVert_M^{2}$ are claim 3 of Existence and Uniqueness of the Mean-Square Extension of the Elementary Stochastic Integral. By Step 1, $H\pm G$ are It^{o} integrable with approximating sequences $(H^k\pm G^k)$, so by claim 3 of Existence and Uniqueness of the Mean-Square Extension of the Elementary Stochastic Integral the limits $\lVert H\pm G\rVert_M^{2}=\lim_k\int\mathbf{1}_{(0,T]}\mathbb{E}[(H^k\pm G^k)^{2}]\rho\,d\lambda$ exist, do not depend on the chosen approximating sequences, and equal $\mathbb{E}[(I\pm J)^{2}]$ (using Step 1 to identify $\int(H\pm G)\,dM$ with $I\pm J$ almost surely; almost surely equal square-integrable random variables have equal second moments, their mean-square distance being $0$). Pointwise in $t$, expanding squares gives $\mathbb{E}[H^k_tG^k_t]=\tfrac14\bigl(\mathbb{E}[(H^k_t+G^k_t)^{2}]-\mathbb{E}[(H^k_t-G^k_t)^{2}]\bigr)$, and all three functions are measurable step functions with $\rho$-weighted integrals over $(0,T]$ (claim 3 of Linearity, Mean Zero, and Isometry of the Elementary Stochastic Integral); by linearity of the integral,

in particular the limit exists and is independent of the chosen sequences. On the other side, $\mathbb{E}[IJ]=\tfrac14(\mathbb{E}[(I+J)^{2}]-\mathbb{E}[(I-J)^{2}])=\tfrac14(\lVert H+G\rVert_M^{2}-\lVert H-G\rVert_M^{2})$, proving claim 2.

Step 3 (Explicit isometry). For each $k$ define $\varphi_k(t)=\liminf_{j}\mathbb{E}[(H^k_t-H^j_t)^{2}]$ for $t\in(0,T]$, extended by $0$; each $\mathbb{E}[(H^k_\cdot-H^j_\cdot)^{2}]$ is a measurable step function and the lower limit of measurable functions is measurable, as recorded in Fatou's Lemma. For every $t$, condition (b) and the triangle inequality give $\lVert H^k_t-H^j_t\rVert_2\to\lVert H^k_t-H_t\rVert_2$ as $j\to\infty$, so $\varphi_k(t)=\mathbb{E}[(H^k_t-H_t)^{2}]$. By Fatou's lemma and condition (a), given $\varepsilon>0$ there is $K$ with, for $k\ge K$,

hence $\int\mathbf{1}_{(0,T]}\varphi_k\rho\,d\lambda\to0$ as $k\to\infty$. Now let $e(t)=\mathbb{E}[H_t^{2}]$ (extended by $0$) be measurable, and write $a_k=\int\mathbf{1}_{(0,T]}\mathbb{E}[(H^k)^{2}]\rho\,d\lambda$, so $a_k\to\lVert H\rVert_M^{2}$. For every $t$ and $\delta>0$, from $\lVert H_t\rVert_2\le\lVert H^k_t\rVert_2+\lVert H_t-H^k_t\rVert_2$ and the second elementary bound above,

Integrating the first bound shows $\mathbf{1}_{(0,T]}e\rho$ is integrable (dominated by an integrable function; monotonicity from Linearity and Monotonicity of the Lebesgue Integral) with $\int\mathbf{1}_{(0,T]}e\rho\,d\lambda\le(1+\delta)a_k+(1+\delta^{-1})\int\mathbf{1}_{(0,T]}\varphi_k\rho\,d\lambda$; letting $k\to\infty$ and then $\delta\downarrow0$ gives $\int\mathbf{1}_{(0,T]}e\rho\,d\lambda\le\lVert H\rVert_M^{2}$. Integrating the second bound and passing to the same limits gives $\lVert H\rVert_M^{2}\le\int\mathbf{1}_{(0,T]}e\rho\,d\lambda$. Together with claim 2, this proves claim 3.

Step 4 (Martingale property). By Ito Integrable Process and the Ito Integral, for each $t\in(0,T]$ the restricted sequence approximates $(H_u)_{u\in(0,t]}$ and $I_t$ is an $\mathcal{F}_t$-measurable square-integrable mean-square limit of the elementary integrals $I^k_t=\int_0^t H^k_u\,dM_u$; also $I_0=0=I^k_0$. Let $0\le s\le t\le T$ and $A\in\mathcal{F}_s$. By The Elementary Stochastic Integral Process is a Square-Integrable Martingale, $\mathbb{E}[(I^k_t-I^k_s)\mathbf{1}_A]=0$ for every $k$, and by the Cauchy-Schwarz inequality (with $\lVert\mathbf{1}_A\rVert_2\le1$),

so $\mathbb{E}[(I_t-I_s)\mathbf{1}_A]=0$. For the process $\tilde I_s=I_{\min(s,T)}$, $s\ge0$: it is adapted ($I_{\min(s,T)}$ is $\mathcal{F}_{\min(s,T)}$-measurable, and $\mathcal{F}_{\min(s,T)}\subseteq\mathcal{F}_s$) with square-integrable values, and for $0\le s\le t$ and $A\in\mathcal{F}_s$ the averaged identity $\mathbb{E}[\tilde I_t\mathbf{1}_A]=\mathbb{E}[\tilde I_s\mathbf{1}_A]$ holds: it is trivial when $s\ge T$, and for $s<T$ it is the displayed identity with $t$ replaced by $\min(t,T)$. By the averaged-form equivalence in Square-Integrable Martingale, Submartingale, and Supermartingale, $(\tilde I_s)_{s\ge0}$ is a square-integrable martingale.

Step 5 (Mean-square continuity). Fix $\varepsilon>0$. By condition (a), choose $K$ such that $\int\mathbf{1}_{(0,T]}\mathbb{E}[(H^k-H^{K})^{2}]\rho\,d\lambda\le\varepsilon$ for all $k\ge K$; then let $C>0$ bound the step function $\mathbb{E}[(H^{K}_\cdot)^{2}]$, and finally choose $\delta>0$ by absolute continuity of the Lebesgue integral (interval form, applied to $\rho$ on $(0,T]$, which is integrable by clause (iv) of Ito Integrator of Intensity Type) so that $\int\mathbf{1}_{(s,t]}\rho\,d\lambda<\varepsilon/C$ whenever $0\le s\le t\le T$ with $t-s<\delta$.

Now let $0\le s\le t\le T$ with $t-s<\delta$ and let $k\ge K$. The family $\tilde H^k$ equal to $H^k_u$ for $u\in(s,t]$ and to $0$ for $u\in(0,s]$ is a simple adapted process on $(0,t]$ (insert $s$ into a representation of $H^k$ and replace the coefficients on $(0,s]$ by $0$; the zero coefficients are measurable for every $\sigma$-algebra). Its elementary integral over $(0,t]$ is $I^k_t-I^k_s$: writing the refined sum of $I^k_t$ over a partition of $(0,t]$ containing $s$, the terms over $(0,s]$ sum to $I^k_s$ (Elementary Stochastic Integral of a Simple Adapted Process), and the remaining terms are exactly the refined sum of the elementary integral of $\tilde H^k$. The isometry (claim 3 of Linearity, Mean Zero, and Isometry of the Elementary Stochastic Integral) then gives

using the first elementary bound and monotonicity. Letting $k\to\infty$ in $\lVert I_t-I_s\rVert_2\le\lVert I_t-I^k_t\rVert_2+\lVert I^k_t-I^k_s\rVert_2+\lVert I^k_s-I_s\rVert_2$ yields $\lVert I_t-I_s\rVert_2^{2}\le4\varepsilon$ whenever $|t-s|<\delta$ (the case $t<s$ by symmetry). Since each $I_t$ is square-integrable, $(I_t)_{t\in[0,T]}$ is uniformly mean-square continuous on $[0,T]$, as claimed. $\square$