1 Lévy Processes and Infinite Divisibility - Department of ...

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2.52.01.51.00.50.00.0 0.2 0.4 0.6 0.8 1.0Figure 1: A sample path of a Poisson process; Ψ(θ) = λ(1−e iθ ) where λ is thejump rate.0.80.60.40.20.0−0.2−0.40.0 0.2 0.4 0.6 0.8 1.0Figure 2: A sample path of a compound Poisson process; Ψ(θ) = λ ∫ R (1 −e iθx )F(dx) where λ is the jump rate and F is the common distribution of thejumps.12
0.20.10.0−0.10.0 0.2 0.4 0.6 0.8 1.0Figure 3: A sample path of a Brownian motion; Ψ(θ) = θ 2 /2.0.40.30.20.10.0−0.10.0 0.2 0.4 0.6 0.8 1.0Figure 4: A sample path of the independent sum of a Brownian motion and acompound Poisson process; Ψ(θ) = θ 2 /2+ ∫ R (1−eiθx )F(dx).13
Page 3 and 4: The term “Lévy process” honour
Page 5 and 6: Theorem 1.2 (Lévy-Khintchine formu
Page 7 and 8: Using the fact that N has stationar
Page 9 and 10: AccordingtoTheorem1.2thereexistsaL
Page 11: where β ∈ [−1,1] η ∈ R and
Page 15 and 16: specific distributions, most of the
Page 17 and 18: The proof of existence of a Lévy p
Page 19 and 20: as n ↑ ∞ where ‖·‖ := 〈
Page 21 and 22: assume that the processes N (n) are
Page 23 and 24: 4.1 Proof of the Lévy-Itô decompo
Page 25 and 26: With C (k,−) defined in the obvio
Page 27 and 28: ExercisesExercise 1 Using Definitio
Page 30 and 31: Exercise 8 Recall that D[0,1] is th
Page 32 and 33: and Γ (2) is a Gamma subordinator,
Page 34: [17] Lévy, P. (1924) Théorie des

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