Calculus - Levy Processes And Stochastic

Lévy processes and stochastic calculus are essential for modeling systems with "jumps"—sudden, discontinuous changes that standard Brownian motion cannot capture. While Brownian motion is continuous and smooth, Lévy processes represent the continuous-time equivalent of a random walk, allowing for both gradual drift and abrupt shocks. Core Concepts A Lévy process is defined by three fundamental properties:

, representing its variation (diffusion), jump measure, and location (drift). Key Examples Levy processes and stochastic calculus

: Generalizes the Poisson process by allowing jumps of random sizes. Lévy processes and stochastic calculus are essential for

: Pricing exotic options and modeling "volatility smiles" where market returns have heavier tails than a normal distribution. Key Examples : Generalizes the Poisson process by

The behavior of any Lévy process is entirely determined by its

: Heavy-tailed processes often used to model extreme market volatility. Why Stochastic Calculus is Necessary