A Levy process is any stochastic process with independent, stationary increments.
Formally, we say the stochastic process \({X_t: t \geq 0}\) is a Levy process if it possesses the following properties:
X_0 = 0 almost surely (i.e. with probability 1)
Independence of Increments: For any \(0 \leq t_1 < t_2 < ... < t_n < \infty\), the following random variables are mutually independent (independent of any intersection of the other events)
Stationary Increments: \(\forall s < t, X_t - X_s =^D X_{t-s}\)
Continuity in probability: For any \(\epsilon > 0\) and \(t \geq 0\), we have
The following are all Levy processes