Lévytype

Lévytype refers to a class of stochastic processes that generalize Lévy processes by allowing the local characteristics—diffusion, drift, and jump intensity—to depend on the current state. These processes are typically Markov processes with càdlàg paths and are often called Lévy-type processes in the literature. For each state x in R^d, the process possesses a Lévy triplet (b(x), a(x), ν(x,·)) describing the drift vector, diffusion matrix, and jump measure, respectively, with ν depending on x.

The infinitesimal generator A acts on smooth test functions f by

Af(x) = b(x)·∇f(x) + 1/2 trace(a(x) ∇^2 f(x)) + ∫_{R^d} [f(x+z) - f(x) - ∇f(x)·z 1_{|z|≤1}] ν(x, dz).

Equivalently, the process is a space-inhomogeneous, or Lévy-type, Markov process whose semigroup may be described by a symbol q(x, ξ) that varies with x, reflecting the state-dependent jump and diffusion characteristics.

Lévy-type processes include stable-like processes (where ν(x, dz) ≈ c(x)|z|^{-d-α(x)}dz) and other models with spatially varying jump intensity or diffusion. They reduce to classical Lévy processes when b, a, and ν are constant in x.

Applications appear in mathematics, finance, physics, and biology to model phenomena with spatially varying randomness, such as state-dependent jump risk or diffusion in heterogeneous media. Theoretical studies focus on existence and uniqueness of solutions, regularity of the associated semigroups, and heat kernel estimates.

See also: Lévy process, Feller process, stable-like process, pseudo-differential operator.