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

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 $$X$$ .

### MSC:

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

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