Submartingale

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a Probability Space. Let $X: \Omega \to \mathbb{R}^{T}$ be an Adapted Process wrt Filtration $\mathcal{F}{*} := (\mathcal{B}{t})_{t \in T}$ on $\Omega$. $X$ is a Supermartingale if it satisfies the following properties:

1. $X_{t} \in L^{1}(\Omega)$ $\forall t \in T$
2. $\mathbb{E}(X_{t} | \mathcal{B}{r}) \geq X{r}$ Almost Surely for any $r \leq t$.

1. An Adapted Process is a Supermartingale iff its negative is a Submartingale
2. Expectations of a Submartingale are Non-Decreasing

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a Probability Space. Let $X: \Omega \to \mathbb{R}^{\mathbb{N}}$ be an Adapted Process wrt Filtration $\mathcal{F}{*} := (\mathcal{B}{n})_{n \in \mathbb{N}}$ on $\Omega$. $X$ is a Submartingale if it satisfies the following properties:

1. $X_{n} \in L^{1}(\Omega)$ $\forall n \in \mathbb{N}$,
2. $\mathbb{E}(X_{n+1} | \mathcal{B}{n}) \geq X{n}$ Almost Surely for any $n \in \mathbb{N}$.

1. Like in Martingale, Definition (2) and (1) are equivalent. The proof is very similar, except we also require Conditional Expectation is Non-Decreasing.