Lévy process

Lévy process

A Lévy process is a type of stochastic process that is widely used in the field of probability theory and financial mathematics. Named after the French mathematician Paul Lévy, these processes are characterized by their stationary and independent increments. Lévy processes are a generalization of the Poisson process and include other well-known processes such as Brownian motion and stable processes.

Definition

A Lévy process \((X_t)_{t \geq 0}\) is a stochastic process with the following properties:

• Independent increments: For any \(0 \leq t_1 < t_2 < \cdots < t_n\), the random variables \(X_{t_2} - X_{t_1}, X_{t_3} - X_{t_2}, \ldots, X_{t_n} - X_{t_{n-1}}\) are independent.
• Stationary increments: The distribution of \(X_{t+s} - X_t\) depends only on \(s\), not on \(t\).
• Stochastic continuity: For any \(\epsilon > 0\), \(\lim_{s \to t} P(|X_s - X_t| \geq \epsilon) = 0\).
• Starting point: \(X_0 = 0\) almost surely.

Examples

• Brownian motion: A continuous-time stochastic process with continuous paths and normally distributed increments.
• Poisson process: A process with jumps occurring at a constant average rate, used to model random events happening over time.
• Stable process: A process with stable distributions, which generalizes the normal distribution to allow for heavy tails.

Properties

Lévy processes have several important properties:

• Infinite divisibility: The distribution of a Lévy process at any fixed time \(t\) is infinitely divisible.
• Lévy-Khintchine representation: The characteristic function of a Lévy process can be expressed using the Lévy-Khintchine formula, involving a drift term, a Gaussian term, and a jump term.
• Path properties: Depending on the specific type of Lévy process, the paths can be continuous (as in Brownian motion) or can have jumps (as in the Poisson process).

Applications

Lévy processes are used in various fields such as:

• Financial mathematics: Modeling stock prices and interest rates.
• Queueing theory: Analyzing the behavior of queues and waiting times.
• Physics: Describing random movements of particles.

Related Pages

• Stochastic process
• Brownian motion
• Poisson process
• Stable process
• Probability theory
• Financial mathematics