Definition (Martingale — discrete time)

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: $$\forall m\le n:X_{m}=E[X_{n}|\mathcal{F}_{m}]\text{ a.s.}$$or $$X_{n}=E[X_{n+1}|\mathcal{F}_{n}]\text{ a.s.}$$

Definition (Martingale — continuous)

Let $(X_{t})_{t\ge 0}$ be a process on $(\Omega,\mathcal{F},P)$ and let $(\mathcal{F}_{t})_{t\ge 0}$ be a filtration on $(\Omega,\mathcal{F},P)$. $(X_{t})_{t\ge 0}$ is called a $(\mathcal{F}_{t})_{t\ge 0}$-Martingale if

1. Integrable: $$X_{t}\in\mathscr{L}^{1}(\Omega,\mathcal{F},P)\ \forall t\ge 0$$
2. Adapted: $$(X_{t})_{t\ge 0}\text{ is }(\mathcal{F}_{t})_{t\ge 0}\text{-adapted}$$
3. Ville’s Criterion: $$\forall 0\le s\le t:X_{s}=E[X_{t}|\mathcal{F}_{s}]\text{ a.s.}$$or $$X_{t}=E[X_{t+1}|\mathcal{F}_{t}]\text{ a.s.}$$