Levy-Sheer and IID-Sheer polynomials with applications
to stochastic integrals
Wim Schoutens∗
Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Heverlee, Belgium
Received 30 October 1997; received in revised form 15 April 1998
Abstract
In [11] an unusual connection between orthogonal polynomials and martingales has been studied. There, all orthogonal Sheer polynomials, were linked to a unique Levy process, i.e., a continuous time stochastic process with stationary and independent increments. The connection between the polynomials and the Levy process is expressed by a martingale relation.
As an application of these martingales we show that the Charlier polynomials are the counterparts for Itˆo’s integral with respect to a variant of the Poisson process of the customary powers.
A simpler approach is possible when trying to obtain discrete time martingales from a Sheer set. We illustrate this by for example relating Krawtchouk polynomials to partial sums of Bernoulli IID variables. c 1998 Elsevier Science B.V. All rights reserved.
AMS classication:60G42; 60J30; 60H05; 33C45; 11B83
Keywords:Orthogonal polynomials; Levy processes; Martingales; Stochastic integration; IID Random variables; Sheer polynomials
1. Introduction
1.1. Levy processes
Let {X(t); t¿0} be a stochastic process, with X(0) = 0. For 06s¡t the random variable X(t)−
X(s) is called the increment of the process X(t) over the interval [s; t]. A stochastic process X(t) is said to be a process with independent increments if the increments over nonoverlapping intervals (common endpoints are allowed) are stochastically independent. A process X(t) is called homoge-neous or stationary if the distribution of the increment X(t+s)−X(t) depends on s, not on t. A homogeneous process with independent increments is called a Levy process [1].
∗E-mail: [email protected].
Let X(t) be a Levy process. We denote the characteristic function of the distribution of X(t+
s)−X(t) by (; s). It is known that (; s) is innitely divisible and that (s¿0),
(; s) = ((;1))s=: (())s
where () =E{eiX} with X:=X(1).
Next we list the Levy processes which will appear in this paper:
Name Distribution (u;1)
Brownian motionW(t) N(0; t) exp(−u2=2) Poisson processN(t) Poisson(t) exp(eiu−1) Gamma process G(t) Gamma(t;1) (1−iu)−1
Pascal process P(t) Pascal(t; p) (p=(1−(1−p)eiu))
Meixner process H(t) – cosh((cos(u−a=2)ia)=2)2
1.2. Sheer systems
Using the classical Faa di Bruno formula [6] one can easily show that the equation
f(z) exp(xu(z)) =
∞
X
m=0
Qm(x)
zm
m! (1)
generates a family of polynomials {Qm(x); m¿0} when both functions u(z) and f(z) can be
ex-panded in a formal power series and if u(0) = 0; u′
(0)6= 0 andf(0)6= 0. The polynomial Qm(x) so
dened is of exact degree m. Any set of such polynomials is called aSheer-set since the treatment of such polynomials had been started by Sheer in [12, 13].
Dene as the inverse function of u, so that (u(z)) =z. Then also can be expanded formally in a power series with (0) = 0 and ′
(0)6= 0.
We now introduce an additional time parameter n∈N into the polynomials dened in (1) by
replacing the function f(z) by (f(z))n.
Denition 1. A polynomial set{Qm(x; n); m¿0; n∈N}is called aIID-Sheer systemif it is dened
by a generating function of the form
∞
X
m=0
Qm(x; n)
zm
m!= (f(z))
nexp(xu(z)) (2)
where
(i) f(z) and u(z) are analytic in the neighborhood of z= 0, (ii) u(0) = 0, f(0) = 1 and u′
(0)6= 0, (iii) 1=f((i)) is a characteristic function.
The quantityn can be considered to be a discrete positive parameter, as such the function Qm(x; n)
If condition (iii) is satised, then there are IID random variables X1; X2; : : :dened by the function
() =Xi() =
1
f((i)) (3) through the characteristic function. The basic link between the polynomials and the corresponding IID variables is provided by the requirement that for each m¿0 the following martingale equality
holds for 06k6n
E{Qm(Sn; n)|Sk}=Qm(Sk; k); (4)
where
Sn=X1+X2+· · ·+Xn:
We combine this assumption with the form of the generating function. On the left hand side we nd, using Fubini and condition (i),
∞
X
m=0
E{Qm(Sn; n)|Sk}
zm
m! =E
(∞
X
m=0
Qm(Sn; n)
zm
m!
Sk )
=E{(f(z))nexp(u(z)Sn)|Sk}
= (f(z))nexp(u(z)Sk)E{exp(u(z)(Sn−Sk))|Sk}:
On the right we immediately nd
∞
X
m=0
Qm(Sk; k)
zm
m!= (f(z))
kexp(u(z)S k):
Combination of both expressions leads to the relationship
E{exp(u(z)(Sn−Sk))|Sk} = (f(z))k
−n:
If we compare this relationship with the equation determining the IID system
E{exp(i(Sn−Sk))|Sk} = (())n
−k:
then we realize that (4) will be satised i (3) holds.
Furthermore, if the characteristic function in condition (iii) is an innitely divisible characteristic function, then we can extend the range of the quantity n to the nonnegative real numbers. For convenience we will write in this case t≡n, as such the function Qm(x; t) will also be polynomial
in t.
We can then associate a Levy process {X(t); t¿0} dened by the function () =X() =
1=f((i)) through the characteristic function.
The basic link between the polynomials and the corresponding Levy processes is again provided by the requirement that for each m¿0 the following martingale equality holds for 06s6t [11]
E{Qm(X(t); t)|X(s)}=Qm(X(s); s): (5)
2. Examples
2.1. IID-Sheer systems
Examples for this procedure can be easily found; let us give somewhat unusual applications.
Example 2. The generalized Bernoulli polynomials are determined by the generating function [4]
∞
where n∈N. We identify the ingredients of (2) and (3) for this example
u(z) =z; f(z) = z
ez−1; () =
ei−1
i :
Hence we conclude that the generalized Bernoulli polynomials induce martingales for the sequenceSn
with the Xi uniformly distributed on [0,1]:
E{Bm(n)(Sn)|Sk}=Bm(k)(Sk); 06k6n:
Example 3. The Krawtchouk polynomials are determined by the generating function [7]
n
Note that in order to simplify the notation we write the generating function as power series in w
for which the nth partial sum equals the left-hand side. Therefore we use the notation ≃ instead of the = sign.
We identify as before to nd
u(z) = log
the sequence Sn with the Xi Bernoulli distributed with parameter 0¡p¡1:
E
Example 4. The Euler polynomials of order are determined by the generating function
We identify as before to nd now
with the Xi Binomial(;1=2) distributed
E{E(n)
m (Sn)|Sk}=Em(k)(Sk):
Example 5. Next we will consider the polynomials with generating function
∞
where a∈N. Except for a shift in index, we have a set of polynomials considered by Narumi
[5, p. 258].
We identify as before to nd now
u(z) = log(1 +z); f(z) =
with the Xi distributed as a sum of a independent Uniform (0;1) r.v.
E{N(an)
m (Sn)|Sk}=Nm(ak)(Sk):
2.2. Levy–Meixner systems
In [8] Meixner determined all sets of orthogonal polynomials [3] that also satisfy the generating function relation (1). It turned out that there were ve such families of orthogonal polynomials. In [11] a detailed study was made of these polynomials and their associated Levy processes, called
Levy–Meixner systems. Summarizing the results, we have the following martingales, 0¡q¡1 and 0¡¡:
In the next section we will make use of the two rst martingale transformations of the Brownian motion and the Poisson process, respectively.
3. Stochastic integration
We follow notations and denitions as in [10]. All stochastic integrals are Itˆo integrals. A standard result in stochastic calculus with respect to Brownian motion is the following theorem.
Theorem 6.
Z t
0
˜
Hn(W(s); s)dW(s) =
˜
Hn+1(W(t); t)
n+ 1 ; (6)
where H˜n(x; t) = (t=2)n=2Hn(x= √
2t) are the monic Hermite polynomials with parameter t.
An interpretation of this result is that the monic, i.e., with leading coecient 1, Hermite polyno-mials with parameter are the counterparts for Itˆo’s integral of the customary powers (W(s))n; n¿1.
For the compensated Poisson process we now prove a similar result:
Theorem 7.
Z
(0; t]
˜
Cn(N(s−); s)d(N(s)−s) =
˜
Cn+1(N(t); t)
n+ 1 ; (7)
with C˜n(x; s); the monic Charlier polynomials.
Proof. LetM(s) =N(s)−s; dene i the time of theith jump in the Poisson process {N(t); t¿0}.
For convenience we set 0= 0. Using the generating function of the monic Charlier polynomials,
one can easily see that it is sucient to prove
Z t
0
Y(s−; w) dM(s) =Y(t; w)−1
w ;
where
Y(s; w) = exp(−sw)(1 +w)N(s)=
∞
X
m=0
˜
Cm(N(s); s)
wm
m!:
Next we will give an explicit calculation of the above stochastic integral. First note that
N(i−) = lim
s→i; s¡iN(s) = (i−1); i
¿1:
So we have
Z t
0
Y(s−; w) dM(s)
=
Z t
0
esw(1 +w)N(s−)dN(s)
− Z t
0
=
The interpretation is now that the monic Charlier polynomials are the counterparts for Itˆo’s integral of the customary powers (N(s)−s)n= ( ˜C
1(N(s); s))n, n¿1 and that Y(t;1) is the stochastic exponent
of the compensated Poisson process {M(t); t¿0}.
Acknowledgements
The author is Research Assistant of the Fund for Scientic Research – Flanders (Belgium). The author thanks Agnieska Plucinska for showing him the content of her paper [9] prior to publication and the referees for their useful comments.
References
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