Lévy processes

Lévy processes are a type of mathematical model used to describe random movements or changes over time, characterized by continuous fluctuations and sudden jumps. Think of stock prices that usually change gradually but sometimes experience sharp spikes or drops. These processes have independent, stationary increments—meaning the change in one period doesn't affect future changes, and the statistical properties stay consistent over time. Lévy processes are widely used in finance, physics, and engineering to model complex systems with both smooth variability and abrupt shifts, combining predictability with randomness in a mathematically rigorous way.