a Levy process
/ lev-EE /
A Lévy process is the continuous-time analogue of a random walk: a process whose increments over disjoint intervals are independent and whose increment law depends only on the length of the interval, not its position. It is the master class that contains both Brownian motion (purely continuous) and the Poisson process (purely jumping) as special cases, and everything in between — drift, diffusion, and jumps of all sizes living together on one timeline.
Formally, a real-valued process (X_t)_{t>=0} with X_0 = 0 is a Lévy process if (i) it has independent increments: for 0 <= t_0 < t_1 < ... < t_n the increments X_{t_1} - X_{t_0}, ..., X_{t_n} - X_{t_{n-1}} are independent; (ii) it has stationary increments: the law of X_{t+s} - X_s depends only on t; and (iii) it is stochastically continuous (for every epsilon, P(|X_{t+h} - X_t| > epsilon) -> 0 as h -> 0). Stochastic continuity allows fixed-time jumps to be ruled out and guarantees a version with cadlag paths (right-continuous with left limits). The law of X_t is infinitely divisible with characteristic function e^(t psi(theta)), and the Lévy-Itô decomposition writes X_t = b t + sigma B_t + (compensated small jumps) + (large jumps), exhibiting the three ingredients of the Lévy-Khintchine triplet as actual path components.
Lévy processes are the building blocks of jump models in finance (variance-gamma, normal-inverse-Gaussian), of anomalous diffusion in physics, and of branching and queueing theory. The essential warning: stochastic continuity is NOT path continuity — a Lévy process other than Brownian motion with drift has jumps, even infinitely many small ones in any interval. Also, the increments are independent but the process is strongly Markov, so do not treat X_t as if its increments were iid copies of X_1 placed at integer times; they live on a continuum.
The Poisson process N_t is a Lévy process whose increments over [s, t] are Poisson(lambda(t-s)); Brownian motion B_t is a Lévy process with N(0, t-s) increments. A jump-diffusion X_t = mu t + sigma B_t + sum of Poisson-timed jumps combines both and is again Lévy: its increments over any interval are independent and depend only on the interval's length.
Brownian motion, Poisson, and their jump-diffusion fusion are all Lévy.
Stochastic continuity forbids jumps at deterministic times but not at random times; almost every Lévy path other than (drift + Brownian) jumps. Right-continuity (the d in cadlag) is a genuine modelling choice — it makes X_t equal to the value AFTER any jump at time t.