Progressive Measurability: Sections, Right-Continuous Adapted Processes, Arithmetic, and Indefinite Time Integrals

Let $\mathbb{R}$ be the real numbers, let $(\Omega,\mathcal{F},P)$ be a probability space, let $T>0$ be real, and let $(\mathcal{F}_t)_{t\in[0,T]}$ be a filtration on $(\Omega,\mathcal{F},P)$ with time index restricted to $[0,T]$. Progressive measurability of a family $X=(X_t)_{t\in[0,T]}$ of real-valued functions on $\Omega$ is with respect to $(\mathcal{F}_t)_{t\in[0,T]}$, with $\mathcal{B}_{[0,t]}$ and $\otimes$ as in that definition and $\mathcal{B}(\mathbb{R})$ the Borel $\sigma$-algebra on the real line; $X$ is called adapted when $X_t$ is an $\mathcal{F}_t$-measurable random variable for every $t\in[0,T]$ (adaptedness with time index restricted to $[0,T]$). A path of $X$ is a map $s\mapsto X_s(\omega)$ on $[0,T]$ at a fixed $\omega\in\Omega$; a path is called right-continuous if for every $s\in[0,T]$ and every sequence $(s_j)_{j\in\mathbb{N}}$ in $[s,T]$ converging to $s$, the sequence $(X_{s_j}(\omega))_{j\in\mathbb{N}}$ converges to $X_s(\omega)$. Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$, both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line. For $t\in(0,T]$ write $\lambda_{[0,t]}$ for the restricted Lebesgue measure on $[0,t]$ and $\int_{[0,t]}\cdot\,ds$ for the Lebesgue integral with respect to it, and set $\int_{[0,0]}\cdot\,ds=0$.

1. (Sections and joint measurability.) Let $X$ be progressively measurable. Then $X$ is adapted; for every $\omega\in\Omega$ and every $t\in[0,T]$ the path section $s\mapsto X_s(\omega)$, restricted to $[0,t]$, is measurable with respect to $\mathcal{B}_{[0,t]}$ and $\mathcal{B}(\mathbb{R})$; and the function $(s,\omega)\mapsto X_s(\omega)$ on $[0,T]\times\Omega$ is measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{F}_T$, hence also with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{F}$.

2. (Adapted processes with right-continuous paths.) If $X$ is adapted and every path of $X$ is right-continuous, then $X$ is progressively measurable.

3. (Arithmetic and deterministic weights.) Let $X$ and $Y$ be progressively measurable, let $c$ be real, and let $u:[0,T]\to\mathbb{R}$ be continuous on $[0,T]$. Then the families $(cX_t)_{t\in[0,T]}$, $(X_t+Y_t)_{t\in[0,T]}$, $(X_tY_t)_{t\in[0,T]}$, the deterministic family $(u(t))_{t\in[0,T]}$ — regarded as the function $(s,\omega)\mapsto u(s)$ — and the weighted family $(u(t)X_t)_{t\in[0,T]}$ are progressively measurable.

4. (Indefinite time integrals.) Let $X$ be progressively measurable and bounded: for some real $K\ge0$, $|X_s(\omega)|\le K$ for all $s\in[0,T]$ and $\omega\in\Omega$. Then for every $\omega\in\Omega$ and every $t\in[0,T]$ the Lebesgue integral $Y_t(\omega)=\int_{[0,t]}X_s(\omega)\,ds$ is defined; the family $Y=(Y_t)_{t\in[0,T]}$ satisfies $|Y_t(\omega)-Y_r(\omega)|\le K(t-r)$ for all $0\le r\le t\le T$ and every $\omega$; every path of $Y$ is continuous on $[0,T]$; and $Y$ is adapted and progressively measurable.