Optional Stopping for Bounded Right-Continuous Square-Integrable Martingales on a Compact Time Interval

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $T>0$ and $K\ge0$ be real numbers, 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 $M=(M_t)_{t\in[0,T]}$ be a square-integrable martingale with respect to $(\mathcal{F}_t)_{t\in[0,T]}$, with time index restricted to $[0,T]$, such that $|M_t(\omega)|\le K$ for every $t\in[0,T]$ and every $\omega\in\Omega$. Assume there is an event $\Omega_0\in\mathcal{F}$ with $P(\Omega_0)=1$ such that for every $\omega\in\Omega_0$ the path $t\mapsto M_t(\omega)$ is right-continuous at every $t\in[0,T)$ in the sense of the supremum lemma for bounded right-continuous processes: for every $t\in[0,T)$ and every $\varepsilon>0$ there is $\eta>0$ with $|M_s(\omega)-M_t(\omega)|\le\varepsilon$ whenever $t\le s\le\min(t+\eta,T)$.

Let $\sigma$ and $\tau$ be stopping times of $(\mathcal{F}_t)_{t\in[0,T]}$ with $\sigma(\omega)\le\tau(\omega)$ for every $\omega\in\Omega$, let $\mathcal{F}_\sigma$ be the $\sigma$-algebra of events prior to $\sigma$, and let $M_\sigma$, $M_\tau$ and $M^\tau=(M_{\min(t,\tau)})_{t\in[0,T]}$ be the sampled functions and the stopped family. Write $\mathbf{1}_{A}$ for the function equal to $1$ on $A$ and $0$ off $A$, and $\mathbb{E}$ for the expectation. Then:

(a) (Measurability.) For every stopping time $\rho$ of $(\mathcal{F}_t)_{t\in[0,T]}$ (in particular for $\rho=\sigma$, $\rho=\tau$, and $\rho=\min(t,\tau)$ with $t\in[0,T]$), the sampled function $M_\rho\mathbf{1}_{\Omega_0}$ is a random variable on $(\Omega,\mathcal{F},P)$ with $|M_\rho\mathbf{1}_{\Omega_0}|\le K$ everywhere. If every path of $M$ is right-continuous at every $t\in[0,T)$, then $M_\rho$ is measurable with respect to $\mathcal{F}_\rho$, the $\sigma$-algebra of events prior to $\rho$.

(b) (Optional stopping.) For every $D\in\mathcal{F}_\sigma$,

in particular, applying this with $\sigma$ replaced by the constant stopping time $0$ and $D=\Omega$, $\mathbb{E}[M_\tau\mathbf{1}_{\Omega_0}]=\mathbb{E}[M_0]$. If every path of $M$ is right-continuous at every $t\in[0,T)$, then $M_\sigma$ is a conditional expectation of $M_\tau$ given $\mathcal{F}_\sigma$.

(c) (The stopped process.) For all $0\le s\le t\le T$ and every $D\in\mathcal{F}_s$,

If every path of $M$ is right-continuous at every $t\in[0,T)$, then $M^\tau$ is a square-integrable martingale with respect to $(\mathcal{F}_t)_{t\in[0,T]}$ with time index restricted to $[0,T]$ (that is, $\mathbb{E}[M^\tau_t\mathbf{1}_D]=\mathbb{E}[M^\tau_s\mathbf{1}_D]$ for all $0\le s\le t\le T$ and $D\in\mathcal{F}_s$, each $M^\tau_t$ being $\mathcal{F}_t$-measurable and square-integrable), with $|M^\tau_t(\omega)|\le K$ for every $t$ and $\omega$, and every path of $M^\tau$ is right-continuous at every $t\in[0,T)$.