Definition (Supermartingale)

Let $(X_{n})_{n\in\mathbb{N}}$ be a Stochastic Process on $(\Omega,\mathcal{F},P)$ and let $(\mathcal{F}_{n})_{n\in\mathbb{N}}$ be a filtration on $(\Omega,\mathcal{F},P)$. $(X_{n})_{n\in\mathbb{N}}$ is called a $\mathcal{F}_{n}$-Martingale if

1. Integrable: $$X_{n}\in\mathscr{L}^{1(\Omega,\mathcal{F},P),}\ \forall n\in\mathbb{N}$$
2. Adapted: $$(X_{n})_{n\in\mathbb{N}}\text{ is }(\mathcal{F}_{n})_{n\in\mathbb{N}}\text{-adapted}$$
3. Ville’s Criterion: $$\forall m\le n:X_{m}\ge E[X_{n}|\mathcal{F}_{m}]\text{ a.s.}$$or $$X_{n}=E[X_{n+1}|\mathcal{F}_{n}]\text{ a.s.}$$

Definition (Submartingale)

Let $(X_{n})_{n\in\mathbb{N}}$ be a Stochastic Process on $(\Omega,\mathcal{F},P)$ and let $(\mathcal{F}_{n})_{n\in\mathbb{N}}$ be a filtration on $(\Omega,\mathcal{F},P)$. $(X_{n})_{n\in\mathbb{N}}$ is called a $\mathcal{F}_{n}$-Martingale if

1. Integrable: $$X_{n}\in\mathscr{L}^{1(\Omega,\mathcal{F},P),}\ \forall n\in\mathbb{N}$$
2. Adapted: $$(X_{n})_{n\in\mathbb{N}}\text{ is }(\mathcal{F}_{n})_{n\in\mathbb{N}}\text{-adapted}$$
3. Ville’s Criterion: $$\forall m\le n:X_{m}\le E[X_{n}|\mathcal{F}_{m}]\text{ a.s.}$$or $$X_{n}\le E[X_{n+1}|\mathcal{F}_{n}]\text{ a.s.}$$