Strategy Corresponding to a Stopping Time

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a Probability Space and let $(\mathcal{F}{n}){n=0}^{\infty}$ be a Discrete-Time Filtration on $\mathcal{B}$. Let $\nu$ be a Stopping Time on $(\mathcal{F}{n}){n=0}^{\infty}$. Then $(H_{n} := \mathbb{1}{\nu \geq n}){n=1}^{\infty}$ is a Predictable Sequence. Therefore, $(H_{n})$ defines a Strategy for a $(\mathcal{F}{n}){n=0}^{\infty}$-Adapted Process.

This follows because $[\nu \geq n] = [\nu > n-1] \in \mathcal{F}{n-1}$ by equivalent condition (3). Since Indicators are Measureable iff their Set is Measureable, we have that $H{n} \in \mathcal{F}_{n-1}$ and is predictable. $\blacksquare$