A stochastic process with stationary, independent increments and càdlàg (right-continuous with left limits) paths; it generalizes Brownian motion by permitting discontinuous jumps and is characterized by an infinitely divisible increment distribution and a Lévy triplet specifying drift, Gaussian component, and jump measure.

Principle

Stationary independent increments imply that increment distributions depend only on time differences and that nonoverlapping increments are independent; the process decomposes into drift, continuous Gaussian part, and a jump component whose statistical properties are encoded in a Lévy measure.

Demonstration

A compound Poisson process with random jump sizes is a Lévy process: jumps occur at Poisson times and the sum of jump magnitudes produces discontinuous sample paths; adding a Brownian term and deterministic drift yields a richer Lévy process with both continuous and jump behavior.

Misapplication

Modeling phenomena with strong temporal dependence or memory as a Lévy process (which enforces independent increments), or approximating small-time jump behavior by a Gaussian when jumps dominate tail risk.

Consequence

Using a Lévy process allows explicit modeling of heavy tails, skewness, and sudden discontinuities in sample paths; it provides tractable characteristic functions and limit theorems for sums of independent increments and underlies many jump‑driven models in finance and physics.

Reversal

A pure Brownian motion is the continuous-limit reversal of a Lévy process with zero jump measure and only a Gaussian component; conversely pure jump processes arise when the Gaussian part is zero and only discrete jumps remain.

Boundary

Applies to processes with independent increments and càdlàg paths; excludes processes with dependent increments, deterministic chaos, or continuous-time processes with pathologies violating right-continuity with left limits.

Semantic Tension

Tension exists between modeling with Lévy processes for tractability of independent increments and the need to capture temporal correlations, leading to alternatives like fractional processes or subordinated models.

Synthesis

A Lévy process is the canonical Markovian model of stochastic motion that preserves increment stationarity and independence while allowing both continuous diffusion-like evolution and abrupt jumps, parameterized by a triplet that separates drift, Gaussian fluctuation, and jump intensity and distribution.