Strategy

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a Probability Space and let $(\mathcal{F}{n}){n=1}^{\infty}$ be a Discrete-Time Filtration on $\mathcal{B}$. Let $(H_{n}){n=1}^{\infty}$ be a Predictable Sequence on $(\mathcal{F}{n}){n=1}^{\infty}$ and let $(X{n}){n=1}^{\infty}$ be a $\mathcal{F}{n}$-Adapted Process $\Omega, \mathcal{B}$. Then we call $(H_{n}){n=1}^{\infty}$ a Strategy on $(X{n}){n=1}^\infty$with winnings $$(H \cdot X){n} = \sum\limits_{m=1}^{n} H_{m} (X_{m} - X_{m-1})$$ for $n \geq 1$, where $X_{0} = 0$.

1. Nonnegative Bounded Strategy on Supermartingale is a Supermartingale
2. Nonnegative Bounded Strategy on Submartingale is a Submartingale
3. Bounded Strategy on Martingale is a Martingale