Lévy measure
E1020436
A Lévy measure is a mathematical tool used in probability theory to characterize the jump behavior of Lévy processes and more general infinitely divisible distributions.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
measure in probability theory ⓘ |
| appearsIn |
Lévy–Khintchine formula
NERFINISHED
ⓘ
characteristic exponent of Lévy processes ⓘ generator of Lévy processes ⓘ |
| associatedWith | Lévy triplet NERFINISHED ⓘ |
| characterizes |
jump behavior of Lévy processes
ⓘ
jump intensity ⓘ jump size distribution ⓘ |
| componentOf | Lévy triplet NERFINISHED ⓘ |
| constraint |
integrability condition near 0 via (1 ∧ |x|^2)
ⓘ
ν({x: |x|>1}) < ∞ for many Lévy processes ⓘ |
| determines |
distribution of jumps of a Lévy process
ⓘ
infinitely divisible law ⓘ |
| domain | ℝ^d \ {0} ⓘ |
| field |
measure theory
ⓘ
probability theory ⓘ stochastic processes ⓘ |
| generalizes | intensity measure of a Poisson process ⓘ |
| namedAfter | Paul Lévy NERFINISHED ⓘ |
| property |
measure of {0} equals 0
ⓘ
σ-finite measure ⓘ ∫_{ℝ^d \ {0}} (1 ∧ |x|^2) ν(dx) < ∞ ⓘ |
| relatedConcept |
Lévy process
NERFINISHED
ⓘ
Lévy–Itô decomposition NERFINISHED ⓘ characteristic function ⓘ infinitely divisible measure ⓘ |
| relatedTo |
Poisson random measure
ⓘ
compound Poisson process ⓘ jump kernel in integro-differential operators ⓘ jump measure of a process ⓘ |
| role |
encodes frequency of jumps of different sizes
ⓘ
separates small and large jumps in Lévy–Itô decomposition ⓘ |
| symbol |
Π
ⓘ
ν ⓘ |
| usedIn |
CGMY processes
NERFINISHED
ⓘ
Lévy process theory NERFINISHED ⓘ financial mathematics ⓘ infinitely divisible distributions ⓘ jump process modeling ⓘ risk theory ⓘ stable distributions ⓘ stochastic calculus with jumps ⓘ tempered stable processes ⓘ variance gamma processes ⓘ |
| usedToDefine |
jump-diffusion process
ⓘ
pure-jump Lévy process ⓘ subordinator ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.