Math 2.0

Probability theory and stochastic processes Definition

Lévy process

Builds on Stochastic processIndependence of random variablesProbability distribution

Definition

Let d≥1d \ge 1. A Lévy process in ℝd\R^d is a stochastic process (Xt)t≥0(X_t)_{t \ge 0} with values in ℝd\R^d such that:

  1. X0=0X_0 = 0 almost surely;
  2. for all 0≤t0<t1<⋯<tn0 \le t_0 < t_1 < \cdots < t_n, the increments Xt1−Xt0,…,Xtn−Xtn−1X_{t_1} - X_{t_0}, \ldots, X_{t_n} - X_{t_{n-1}} are independent;
  3. for all 0≤s≤t0 \le s \le t, the increment Xt−XsX_t - X_s has the same distribution as Xt−sX_{t-s};
  4. for every t≥0t \ge 0 and ε>0\varepsilon > 0, lim⁡h→0P(∣Xt+h−Xt∣>ε)=0\lim_{h \to 0} P(\abs{X_{t+h} - X_t} > \varepsilon) = 0.

Remarks

Example: the Poisson process NN of rate λ\lambda satisfies the four clauses; Nt−NsN_t - N_s is Poisson with mean λ(t−s)\lambda(t - s), and P(Nt+h≠Nt)=1−e−λ∣h∣→0P(N_{t+h} \ne N_t) = 1 - e^{-\lambda \abs{h}} \to 0. Non-example: the deterministic process Xt=t2X_t = t^2 satisfies clauses 1, 2 and 4, but Xs+1−Xs=2s+1X_{s+1} - X_s = 2s + 1 depends on ss, so clause 3 fails. Every Lévy process has a modification whose paths are right-continuous with left limits, and one usually works with it. There is a function ψ\psi, the characteristic exponent, with 𝔼[ei⟨u,Xt⟩]=etψ(u)\mathbb{E}[e^{i \langle u, X_t \rangle}] = e^{t \psi(u)} for all u∈ℝdu \in \R^d and t≥0t \ge 0.

Reading path

To understand this page from the ground up, read these 42 pages in order. The foundations come first.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Independence of random variables concept
  4. Integers object
  5. Ordered field definition
  6. Probability distribution concept
  7. Proposition concept
  8. Random experiment concept
  9. Rational numbers object
  10. Real numbers object
  11. Set definition
  12. Ring of integers statement
  13. Disjunction definition
  14. Implication definition
  15. Predicate definition
  16. Ordered pair definition
  17. Union definition
  18. Sample space definition
  19. Existential quantifier definition
  20. Universal quantifier definition
  21. The rationals form a field statement
  22. Cartesian product definition
  23. Subset definition
  24. Uniqueness quantifier definition
  25. The rationals form an ordered field statement
  26. Binary relation definition
  27. Universal set definition
  28. Power set definition
  29. Order on the reals definition
  30. Complement definition
  31. Function definition
  32. Interval definition
  33. Sigma-algebra definition
  34. Preimage of a set definition
  35. Event definition
  36. Borel sigma-algebra definition
  37. Probability axioms concept
  38. Probability measure definition
  39. Probability space definition
  40. Random variable definition
  41. Stochastic process concept
  42. Lévy process definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Lévy_process
About this page
TagCP7Q2FWWNT
URIm2://CP7Q2FWWNT
Kinddefinition
WritingW0 stub
MSC60G51, 60J25
DefinesLévy process
UsesStochastic process, Independence of random variables, Probability distribution
Changed2 November 2026

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