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

The term “Lévy process” honours the work of the French mathematicianPaul Lévy who, although not alone in his contribution, played an instrumentalroleinbringingtogetheranunderstandingandcharacterisationofprocesseswithstationary independent increments. In earlier literature, Lévy processes can befound under a number of different names. In the 1940s, Lévy himself referred tothem as a sub-class of processus additifs (additive processes), that is processeswith independent increments. For the most part however, research literaturethrough the 1960s and 1970s refers to Lévy processes simply as processes withstationary independent increments. One sees a change in language through the1980sandbythe1990stheuseofthe term“Lévyprocess”hadbecomestandard.From Definition 1.1 alone it is difficult to see just how rich a class of processesthe class of Lévy processes forms. De Finetti [4] introduced the notionof an infinitely divisible distribution and showed that they have an intimaterelationship with Lévy processes. This relationship gives a reasonably good impressionof how varied the class of Lévy processes really is. To this end, let usnow devote a little time to discussing infinitely divisible distributions.Definition 1.2 We say that a real-valued random variable Θ has an infinitelydivisible distribution if for each n = 1,2,... there exist a sequence of i.i.d. randomvariables Θ 1,n ,...,Θ n,n such thatΘ d = Θ 1,n +···+Θ n,nwhere = d is equality in distribution. Alternatively, we could have expressed thisrelation in terms of probability laws. That is to say, the law µ of a real-valuedrandom variable is infinitely divisible if for each n = 1,2,... there exists anotherlaw µ n of a real valued random variable such that µ = µ ∗nn . (Here µ ∗ n denotesthe n-fold convolution of µ n ).In view of the above definition, one way to establish whether a given randomvariable has an infinitely divisible distribution is via its characteristic exponent.Suppose that Θ has characteristic exponent Ψ(u) := −logE(e iuΘ ) for all u ∈ R.Then Θ has an infinitely divisible distribution if for all n ≥ 1 there exists acharacteristic exponent of a probability distribution, say Ψ n , such that Ψ(u) =nΨ n (u) for all u ∈ R.The full extent to which we may characteriseinfinitely divisible distributionsis described by the characteristic exponent Ψ and an expression known as theLévy–Khintchine formula. 2Theorem 1.1 (Lévy–Khintchine formula) A probability law µ of a realvaluedrandom variable is infinitely divisible with characteristic exponent Ψ,∫e iθx µ(dx) = e −Ψ(θ) for θ ∈ R,R2 Note, although it is a trivial fact, it is always worth reminding oneself of that one worksin general with the Fourier transform of a measure as opposed to the Laplace transform onaccount of the fact that the latter may not exist.3