On the orthogonal polynomials associated with a Lévy process. (English) Zbl 1149.60028

Solé, Josep Lluís; Utzet, Frederic

Ann. Probab. 36, No. 2, 765-795 (2008).

Let \(X_t , t\geq 0\) be a semimartingale with \(X_0=0\). Consider the variation of \(X\), \[ X^{(1)}_t = X_t , \;X^{(2)}_t = [X,X]_t , \;X^{(n)}_t = \sum_{0<s\leq t} (\Delta X_s)^n , \;n\geq 3 \]
and the iterated integrals of \(X\)
\[ P^{(0)}_t =1, \;P^{(1)}_t = X_t, \;P^{(n)}_t =\int_0^t P^{(n-1)}_{s-} dX_s. \]
The authors deduce that \(P^{(n)}_t\) is a polynomial in \(X^{(1)}_t ,\dots, X^{(n)}_t\) and denote it by \(P_n(x_1,\dots,x_n)\).
In this paper the authors study the probabilistic properties of, and relationship between \(P_n(x_1,\dots,x_n)\) and the Teugels polynomial \(p^{\sigma}_n(x), \;n\geq 1\). Their three main results are interesting in themselves. First they prove that for a general semimartingale \(X\) the Doleans exponential \({\mathcal E}(uX_t)\) is analytic in a certain neighborhood of the origin and that the iterated integrals are the Taylor coefficients.
The second result is related to the Kailath-Segall polynomials that are expressible as polynomials of a fixed set of variables. It is known that the Brownian motion and the compensated Poisson process are the unique Lévy processes such that the Kailath-Segall polynomials can be written as polynomials in \(x\) and \(t\). So, a natural question is how to characterize the Lévy processes with a similar property for a finite number of variables. The key of their proof is that only the application of linear functions to a Lévy process gives rise to another Lévy process.
The third result is that it is possible to give a sequence of simple Lévy processes \(\{X_k\}\) that converges in the Skorohod topology to \(X\) so that, under the appropriate hypothesis, all variations and iterated integrals of \(X_k\) converge to the variations and iterated integrals of \(X\) .

Reviewer: Elisaveta Pancheva (Sofia)

MSC:

60G51	Processes with independent increments; Lévy processes
42C05	Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis

Keywords:

Lévy processes; Kailath-Segall formula; orthogonal polynomials; Teugels martingales

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