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

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3 The Lévy-Itô decomposition intuitivelyOneofourmainobjectivesfortheremainderofthistextistoestablisharigorousunderstanding of the structure of the paths of Lévy processes. The way we shalldo this is to prove the assertion in Theorem 1.2 that given any characteristicexponent, Ψ, belonging to an infinitely divisible distribution, there exists a Lévyprocesswith the same characteristicexponent. This will be done by establishingthe so-called Lévy–Itô decomposition which describes the structure of a generalLévy process in terms of three independent auxiliary Lévy processes, each withdifferent types of path behaviour.According to Theorem 1.1, any characteristic exponent Ψ belonging to aninfinitely divisible distribution can be written, after some simple reorganisation,in the formΨ(θ) ={iaθ + 1 }2 σ2 θ 2{ ∫}+ Π(R\(−1,1)) (1−e iθx Π(dx))+|x|≥1{ ∫(1−e iθx +iθx)Π(dx)0
The proof of existence of a Lévy process with characteristic exponent givenby (3.1) thus boils down to showing the existence of a Lévy process, X (3) , whosecharacteristic exponent is given by Ψ (3) . Noting that∫(1−e iθx +iθx)Π(dx)0
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 and 12: where β ∈ [−1,1] η ∈ R and
Page 13 and 14: 0.20.10.0−0.10.0 0.2 0.4 0.6 0.8
Page 15: specific distributions, most of the
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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