The Pre-Stopping-Time Indicator and Stopped Time Integrals

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $T>0$ be a real number, let $(\mathcal{F}_t)_{t\in[0,T]}$ be a filtration on $(\Omega,\mathcal{F},P)$ with time index restricted to $[0,T]$, and let $\tau$ be a stopping time of $(\mathcal{F}_t)_{t\in[0,T]}$. Progressive measurability of a family of real-valued functions on $\Omega$ indexed by $[0,T]$ is with respect to $(\mathcal{F}_t)_{t\in[0,T]}$, with $\mathcal{B}_{[0,t]}$ the trace Borel $\sigma$-algebra on $[0,t]$ for $t>0$ and $\mathcal{B}_{[0,0]}=\{\emptyset,\{0\}\}$, both as in that definition; measurability of a path $s\mapsto X_s(\omega)$ on $[0,T]$ is with respect to $\mathcal{B}_{[0,T]}$ and the Borel $\sigma$-algebra of the real line. For $t\in(0,T]$ write $\int_{[0,t]}\cdot\,ds$ for the Lebesgue integral with respect to the restricted Lebesgue measure on $[0,t]$, and set $\int_{[0,0]}\cdot\,ds=0$. 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.

The pre-stopping-time indicator of $\tau$ is the family $I=(I_t)_{t\in[0,T]}$ of real-valued functions on $\Omega$ defined by $I_t(\omega)=1$ if $t<\tau(\omega)$ and $I_t(\omega)=0$ otherwise.

Then the following hold.

1. (Progressive measurability.) The family $I$ is progressively measurable. In particular, $\{\omega\in\Omega:t<\tau(\omega)\}\in\mathcal{F}_t$ for every $t\in[0,T]$ (also immediate from the definition of a stopping time by complementation), and for every $\omega\in\Omega$ the path $s\mapsto I_s(\omega)$, equal to $1$ on $[0,\tau(\omega))$ and to $0$ on $[\tau(\omega),T]$, is measurable on $[0,T]$.

2. (Stopped time integrals.) Let $X=(X_t)_{t\in[0,T]}$ be a family of real-valued functions on $\Omega$, and let $\omega\in\Omega$ and the real number $K\ge0$ be such that the path $s\mapsto X_s(\omega)$ is measurable on $[0,T]$ and satisfies $|X_s(\omega)|\le K$ for every $s\in[0,T]$. Then for every $t\in[0,T]$ the two Lebesgue integrals below exist and

$$\int_{[0,\min(t,\tau(\omega))]}X_s(\omega)\,ds=\int_{[0,t]}I_s(\omega)\,X_s(\omega)\,ds .$$

3. (The stopped integral family.) Let $X=(X_t)_{t\in[0,T]}$ be progressively measurable and suppose there is a real number $K\ge0$ with $|X_s(\omega)|\le K$ for every $s\in[0,T]$ and every $\omega\in\Omega$. Then the family $J=(J_t)_{t\in[0,T]}$ defined by

$$J_t(\omega)=\int_{[0,t]}I_s(\omega)\,X_s(\omega)\,ds\qquad(t\in[0,T],\ \omega\in\Omega)$$

is progressively measurable, satisfies $J_t(\omega)=\int_{[0,\min(t,\tau(\omega))]}X_s(\omega)\,ds$ for every $t\in[0,T]$ and $\omega\in\Omega$ as well as $|J_t(\omega)-J_r(\omega)|\le K\,(t-r)$ for all $0\le r\le t\le T$ and every $\omega\in\Omega$, and every path of $J$ is continuous on $[0,T]$.