# Definition:Martingale

## Definition

### Discrete Time

Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.

Let $\sequence {X_n}_{n \ge 0}$ be an adapted stochastic process.

We say that $\sequence {X_n}_{n \mathop \in \N}$ is a $\sequence {\FF_n}_{n \ge 0}$-martingale if and only if:

$(1): \quad$ $X_n$ is integrable for each $n \in \Z_{\ge 0}$
$(2): \quad \forall n \ge 0: \expect {X_{n + 1} \mid \FF_n} = X_n$.

Equation $(2)$ is understood as follows:

for any version $\expect {X_{n + 1} \mid \FF_n}$ of the conditional expectation of $X_{n + 1}$ given $\FF_n$, we have:
$\expect {X_{n + 1} \mid \FF_n} = X_n$ almost surely.

### Continuous Time

Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a filtered probability space.

Let $\sequence {X_t}_{t \ge 0}$ be an adapted stochastic process.

We say that $\sequence {X_t}_{t \ge 0}$ is a $\sequence {\FF_t}_{t \ge 0}$-martingale if and only if:

$(1) \quad$ $X_t$ is integrable for each $t \in \hointr 0 \infty$
$(2) \quad \forall s, t \in \hointr 0 \infty, \, 0 \le s < t: \expect {X_t \mid \FF_s} = X_s$

Equation $(2)$ is understood as follows:

for any version $\expect {X_t \mid \FF_s}$ of the conditional expectation of $X_t$ given $\FF_s$, we have:
$\expect {X_t \mid \FF_s} = X_s$ almost surely.