Upcrossing Strategy

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a Probability Space . Let $(X_{n}){n \geq 1}$ be a discrete-time Adapted Process wrt Discrete-Time Filtration $\mathcal{F}{*} := (\mathcal{F}{n} \subset \mathcal{B}){n \in \mathbb{N}}$. Let $$H_{n} := \begin{cases} 1 & \text{if } \exists k \geq 1 \text{ s.t. } N_{2k-1} < n \leq N_{2k} \\ 0 & \text{otherwise} \end{cases}$$ for $N_{j}$ defined in Complete Upcrossings (statement 2) for $j \geq 1$. Then $(H_{n}){n \geq 1}$ is a Predictable Sequence, and thus a Strategy for $(X{n})_{n \geq 1}$. We call it the Upcrossing Strategy.

Note that this is a Predictable Sequence since $$H_{n} = \sum\limits_{k=1}^{\infty} 1_{N_{2k-1} \leq n-1} 1_{n-1 < N_{2k}}$$ and $N_{j}$ are all Stopping Times for $j \geq 1$. Therefore $H$ is a Strategy for $X$. $\blacksquare$