Levy Process

A Levy process is any stochastic process with independent, stationary increments.

Definition

Formally, we say the stochastic process \({X_t: t \geq 0}\) is a Levy process if it possesses the following properties:

1. X_0 = 0 almost surely (i.e. with probability 1)

2. 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)

\[N_{t_1} - N_{t_0}, N_{t_2} - N_{t_1}, ..., N_{t_n} - N_{t_{n-1}}\]

1. Stationary Increments: \(\forall s < t, X_t - X_s =^D X_{t-s}\)

2. Continuity in probability: For any \(\epsilon > 0\) and \(t \geq 0\), we have

\[\lim_{h \rightarrow 0} P(|X_{t+h} - X_t| > \epsilon) = 0\]

Levy Measure

Examples

The following are all Levy processes

• Poisson process
• Gamma process
• Wiener process