: Pricing exotic options and modeling "volatility smiles" where market returns have heavier tails than a normal distribution.
: The statistical properties of an increment depend only on the length of the time interval, not when it occurred. Levy processes and stochastic calculus
Traditional calculus fails when dealing with the non-differentiable paths of random processes. Stochastic calculus provides the tools to integrate and differentiate these paths, which is critical for: : Pricing exotic options and modeling "volatility smiles"
: Modeling systems where noise is driven by Lévy processes rather than just Gaussian noise. Stochastic calculus provides the tools to integrate and
: Estimating risk and claim sizes in aggregate loss processes.
: Modeling turbulence, laser cooling, and bursty arrival patterns in communication networks.
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: