Overview
A martingale is a random process where the best guess (in the expected value sense) of the process at future times is the value of the process now.
Informally, given a stochastic process {% X(t) %},
{% \mathbb{E}(X(t)) = X(s) %}
for {% s < t %}
Measure Theoretic Definition
Given a Probability Space ({% \Omega, \mathcal{F}, \mathbb{P} %}), {% X %} is a martingale if
- {% X %} is adapted
- {% \mathbb{E}[X] < \infty %}
- {% \mathbb{E}[X_n | \mathcal{F}_{n-1}] = X_{n-1} %}
If {% X %} is a function of the continuous variable {% t %}
{% \mathbb{E}(X(t)| \mathcal{F}_s) = X(s) %}
for {% s<t %}