arXiv:0804.2585v1 [math.PR] 16 Apr 2008
The Annals of Probability 2008, Vol. 36, No. 2, 765–795 DOI: 10.1214/07-AOP343 c Institute of Mathematical Statistics, 2008
ON THE ORTHOGONAL POLYNOMIALS ASSOCIATED ´ WITH A LEVY PROCESS1 By Josep Llu´ıs Sol´ e and Frederic Utzet Universitat Aut` onoma de Barcelona Let X = {Xt , t ≥ 0} be a c` adl` ag L´evy process, centered, with moments of all orders. There are two families of orthogonal polynomials associated with X. On one hand, the Kailath–Segall formula gives the relationship between the iterated integrals and the variations of order n of X, and defines a family of polynomials P1 (x1 ), P2 (x1 , x2 ), . . . that are orthogonal with respect to the joint law of the variations of X. On the other hand, we can construct a sequence of orthogonal polynomials pσn (x) with respect to the measure σ 2 δ0 (dx) + x2 ν(dx), where σ 2 is the variance of the Gaussian part of X and ν its L´evy measure. These polynomials are the building blocks of a kind of chaotic representation of the square functionals of the L´evy process proved by Nualart and Schoutens. The main objective of this work is to study the probabilistic properties and the relationship of the two families of polynomials. In particular, the L´evy processes such that the associated polynomials Pn (x1 , . . . , xn ) depend on a fixed number of variables are characterized. Also, we give a sequence of L´evy processes that converge in the Skorohod topology to X, such that all variations and iterated integrals of the sequence converge to the variations and iterated integrals of X.
1. Introduction. Let X = {Xt , t ≥ 0} be a semimartingale with X0 = 0. Define the iterated integrals by the recurrence
(1.1)
(0) Pt
= 1,
(1) Pt
(n) = Xt , . . . , Pt
=
Z
0
t
(n−1)
Ps−
dXs
Received August 2006; revised August 2006. Supported by Grant MTM2006–06427 of the Ministerio de Educaci´ on y Ciencia and FEDER. AMS 2000 subject classifications. Primary 60G51; secondary 42C05. Key words and phrases. L´evy processes, Kailath–Segall formula, orthogonal polynomials, Teugels martingales. 1
This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Probability, 2008, Vol. 36, No. 2, 765–795. This reprint differs from the original in pagination and typographic detail. 1
´ AND F. UTZET J. L. SOLE
2
and consider the sequence of the variations of X, (1)
(1.2) Xt
(2)
= Xt ,
Xt
(n)
= [X, X]t ,
Xt
X
=
(∆Xs )n ,
n ≥ 3,
0 n, the polynomial pm (x) is also identically zero γ-a.e. and not unique.
´ AND F. UTZET J. L. SOLE
22
P
3. When σ > 0 and ν = nk=1 bj δaj , there are also only n + 2 determinate polynomials pσk (x), the last one being pσn+1 (x) = x
(4.6)
n Y
(x − aj ).
j=1
This expression is also deduced from the fact that this polynomial satisfies pσn+1 ≡ 0, γ σ -a.e. The orthogonal polynomials pσn (x), or pn (x) when σ = 0, determine a sequence of strongly orthogonal normal martingales related to the Teugels martingales (see Nualart and Schoutens [13]) and that is why we call them the Teugels polynomials associated with X. In Section 4.3, we provide an explicit expression for those martingales. Remark 4.1. We have changed the notation of Nualart and Schoutens [13] and Schoutens [17] because they write pn (x) for the orthogonal polynomial of degree n − 1, and here it denotes the polynomial of degree n. The next theorem is a modification of the Gauss–Jacobi mechanical quadrature formula (see Szeg¨ o [18], Theorems 3.4.1 and 3.4.2). Theorem 4.2. Assume that the L´evy measure ν has infinite support and let n ≥ 1 be such that pn (0) 6= 0. There are then different nonzero numbers a1 , . . . , an and strictly positive numbers b1 , . . . , bn such that the (L´evy) measure with finite support, νn =
n X
bk δak ,
k=1
satisfies
Z
(4.7)
k
x ν(dx) =
R
Z
R
xk νn (dx),
k = 2, . . . , 2n + 1.
Moreover, let γn (dx) = x2 νn (dx) and γnσ (dx) = σ 2 δ0 (dx) + x2 νn (dx). Then, γ and γn (resp. γ σ and γnσ ) have the same orthogonal polynomials up to order n. By the Gauss–Jacobi formula (Szeg¨ o [18], Theorem 3.4.1), the numbers a1 , . . . , an are the n different nonzero real roots of pn (x), and b1 , . . . , bn are the unique solution of the compatible system m2 .. .
=
a21 b1
m2n+1
=
a2n+1 b1 1
+
a22 b2
+ a2n+1 b2 2
a2n bn .. .
+
··· +
+
· · · + a2n+1 bn , n
´ ORTHOGONAL POLYNOMIALS AND LEVY PROCESSES
23
R
where mk = R xk ν(dx), k ≥ 2. The numbers b1 , . . . , bn , called the Christoffel numbers, are all strictly positive (Szeg¨ o [18], Theorem 3.4.2). From (4.7), it follows that the finite measures γ and γn have the same moments of order 0, 1, . . . , 2n − 1, denoted by m2 , . . . , m2n+1 . Hence, from the expressions (4.2), (4.3) and (4.4), we deduce that they have the same Teugels polynomials up to order n. Corollary 4.3. Let ν be a L´evy measure such that pn (0) 6= 0. There are then λn and λn−1 such that pσn+1 (x) = xpn (x) − λn pσn (x) − λn−1 pσn−1 (x). Proof. The polynomials pσn+1 (x) and xpn (x) are monic and have degree n + 1. The polynomial xpn (x) − pσn+1 (x) can then be written as xpn (x) − pσn+1 (x) =
n X
λj pσj (x),
j=0
where λj = Kj−1 = Kj−1 and
Z
R
(xpn (x) − pσn+1 (x))pσj (x)γ σ (dx)
Z
R
xpn (x)pσj (x)γ σ (dx) Z
(pσj (x))2 γ σ (dx).
Z
xpn (x)pσj (x)γnσ (dx).
Kj =
R
Consider the discrete measures γn and γnσ of the preceding theorem. Then: 1. the measures γ σ and γnσ have the same moments up to order 2n − 1 and, for j = 0, . . . , n − 2, λj = Kj−1 2. denoting by pej (x) [resp. γnσ ), we have and [(4.6)]
pej (x) = pj (x) and
R σ pej (x)]
the Teugels polynomials of γn (resp.
peσj (x) = pσj (x),
peσn+1 (x) = xpen (x) = xpn (x) = 0,
j = 1, . . . , n, γnσ -a.e.,
so it follows that, for j = 0, . . . , n − 2, λj =
Kj−1
= Kj−1
Z
R
Z
R
xpn (x)pσj (x)γnσ (dx) peσn+1 (x)peσj (x)γnσ (dx) = 0.
´ AND F. UTZET J. L. SOLE
24
4.1. An approximating sequence of simple L´evy processes. An interesting consequence of Theorem 4.2 is that it provides a way to construct a sequence of simple L´evy processes that converges in the Skorohod topology to X, satisfying the conditions of Avram [2] in order that all variations and iterated integrals of the sequence converge to the variations and iterated integral of the limit. From the separation of zeros theorem of pn (see [4]), if pn (0) = 0, then pn+1 (0) 6= 0. There is then a sequence m1 < m2 < · · · ր ∞ such that pmk (0) 6= 0, ∀k. Let Xk = {Xk (t), t ∈ R} be a centered Levy process with diffusion coefficient σ and L´evy measure νmk given in Theorem 4.2. That is, the law of Xk is Xk (t) =Law σWt +
mk X
aj (Nj (t) − bj t).
j=1 (n)
(n)
Denote by Pk and Xk the iterated integral and the variation of order n of Xk , respectively, and by P (n) and X (n) the iterated integral and variation of X, respectively. Theorem 4.4. Let X be a L´evy process that satisfies the condition (4.1) and such that ν has infinite support. With the above notation, for every n, (n)
lim Pk k
= P (n)
(n)
and
lim Xk k
= X (n)
(both convergences in the Skorohod sense). Proof. By Avram [2], it suffices to prove that lim(Xk , [Xk , Xk ]) = (X, [X, X])
in the Skorohod sense.
k
Since all of the process involved are L´evy process, by Jacod and Shiryaev [6], Corollary VII.3.6, it is sufficient to prove that lim(Xk (1), [Xk , Xk ]1 ) = (X(1), [X, X]1 )
in distribution,
k
and by the Cramer–Wold device, this is equivalent to proving that for every u, v ∈ R, lim(uXk (1) + v[Xk , Xk ]1 ) = uX(1) + v[X, X]1
in distribution.
k
From (3.4), the characteristic function of uX(1) + v[X, X]1 is ψ(z) = exp
− 12 u2 z 2 σ 2
+t
Z
R
2
+ izv σ + 2
Z
R
2
x ν(dx)
(eiz(ux+vx ) − 1 − iz(ux + vx2 ))ν(dx) .
´ ORTHOGONAL POLYNOMIALS AND LEVY PROCESSES
25
From the fact that the characteristic function of X1 is analytic, it follows that ψ(z) also is. So, it suffices to show that all cumulants of uXk (1) + v[Xk , Xk ]1 converge to the corresponding cumulants of uX(1) + v[X, X]1 and this is clear from the construction of νmk . 4.2. The relationship between Kailath–Segall polynomials and Teugels polynomials. 4.2.1. Preliminary results. This subsection is purely algebraic; later, we will give a probabilistic interpretation of the results. First, it is convenient to introduce a new notation. Given a polynomial of order n, P (x) = c0 + c1 x + · · · + cn xn , we denote by L(P )(x1 , . . . , xn+1 ) the polynomial of degree 1 in x1 , . . . , xn+1 associated with the coefficients of P : L(P )(x1 , . . . , xn+1 ) = c0 x1 + · · · + cn xn+1 .
(4.8)
Of course, we can recover P (x) from L(P )(x1 , . . . , xn+1 ): P (x) = L(P )(1, x, . . . , xn ). Second, we need to consider some finite-dimensional vector spaces. Let a = (a1 , . . . , an ), where a1 , . . . , an are different nonzero numbers. Write (
a Sn+1 = (x1 , . . . , xn+1 ) ∈ Rn+1 : x1 =
n X
aj yj ,
j=1
x2 =
n X
a2j yj , . . . , xn+1 =
j=1
n X
yj , an+1 j
j=1
for some (y1 , . . . , yn ) ∈ R
n
)
.
a is subspace of dimension n of Rn+1 , and there is the projection Sn+1 a Rn+1 −→ Sn+1 ,
(x1 , . . . , xn+1 ) → (x1 , . . . , xn , un+1 ), where un+1 is computed as follows. By the Vandermonde determinant property, we can find (y1 , . . . , yn ) ∈ Rn such that (4.9)
x1 =
n X
j=1
aj yj ,
x2 =
n X
j=1
a2j yj , . . . , xn =
n X
j=1
anj yj .
´ AND F. UTZET J. L. SOLE
26
We then write un+1 =
n X
an+1 yj . j
j=1
Lemma 4.5.
With the above notation, un+1 = −L(P )(x1 , . . . , xn , 0),
where P (x) =
n Y
(x − aj ).
j=1
Proof. From the expression of P (x) given in (4.5), 1 .. . L(P )(x1 , . . . , xn+1 ) = Θ−1 n 1 x
1
Hence,
L(P )(x1 , . . . , xn , un+1 ) = L(P )
n X
an1 .. .
a1
···
an x2
··· ann · · · xn+1
aj yj , . . . ,
j=1
n X
.
an+1 yj j
j=1
!
= 0.
The polynomial P (x) is monic, so un+1 = −L(P )(x1 , . . . , xn , 0).
Define the polynomial of degree n + 1, a Jn+1 (x1 , . . . , xn ) = Pn+1 (x1 , . . . , xn , un+1 ) =
n XY 1 j=1
hj !
h
aj j [yj ]hj ,
where the summation is over all nonnegative integers h1 , . . . , hn such that j=1 hj = n + 1, [x]n is the falling factorial and y1 , . . . , yn are given in (4.9). This strange expression is the multiple convolution of Charlier polynomials aC· (yj , 0), j = 1, . . . , n. Note that when working with the polynomials, the variable t does not play the role of time and can be used freely according to our needs.
Pn
Proposition 4.6.
For every (x1 , . . . , xn+1 ) ∈ Rn+1 ,
a Pn+1 (x1 , . . . , xn+1 ) − Jn+1 (x1 , . . . , xn ) =
(−1)n L(P )(x1 , . . . , xn+1 ), n+1
´ ORTHOGONAL POLYNOMIALS AND LEVY PROCESSES
27
where n Y
P (x) =
(x − aj ).
j=1
Equivalently, a Pn+1 (1, x, . . . , xn ) − Jn+1 (1, x, . . . , xn−1 ) =
(−1)n P (x). n+1
Proof. Simply note that Pn+1 is linear in xn+1 , with coefficient (−1)n /(n+ 1) [see (1.3)], and apply Lemma 4.5. Note that this proposition is true if we replace Pn+1 by another polynomial linear in the variable xn+1 , but we will see that with Pn+1 , it has an interesting probabilistic interpretation. In the same way, take σ > 0 and write σ,a Sn+2
(
= (x1 , . . . , xn+2 ) ∈ Rn+2 : x1 = σy0 +
n X
aj yj ,
j=1
x2 =
n X
a2j yj , . . . , xn+2 =
j=1
n X
an+1 yj , j
j=1
)
for some (y0 , y1 , . . . , yn ) ∈ Rn+1 . Consider the projection σ,a Rn+2 −→ Sn+2 ,
(x1 , . . . , xn+2 ) → (x1 , . . . , xn+1 , uσn+2 ), where uσn+2 =
n X
an+2 yj , j
j=1
Pn
P
yj . With y1 , . . . , yn being the solution of x2 = j=1 a2j yj , . . . , xn+1 = nj=1 an+1 j the same proofQas Lemma 4.5, we have that L(P )(x2 , . . . , xn+1 , uσn+2 ) = 0, where P (x) = nj=1 (x − aj ). Also, note that L(xP )(x1 , . . . , xn+2 ) = L(P )(x2 , . . . , xn+2 ).
Define the polynomial σ,a Jn+2 (x1 , . . . , xn+1 ) = Pn+2 (x1 , . . . , xn+1 , uσn+2 ), a with the addition of a Hermite which has an expression similar to Jn+1 polynomial H· (y0 , 0). We then have the following.
´ AND F. UTZET J. L. SOLE
28
Proposition 4.7.
For every (x1 , . . . , xn+2 ) ∈ Rn+2 ,
σ,a Pn+2 (x1 , . . . , xn+2 ) − Jn+2 (x1 , . . . , xn+1 ) =
(−1)n+1 L(xP )(x1 , . . . , xn+2 ), n+2
where P (x) =
n Y
(x − aj ).
j=1
Equivalently, σ,a Pn+2 (1, x, . . . , xn+1 ) − Jn+2 (1, x, . . . , xn ) =
(−1)n+1 xP (x). n+2
4.2.2. Teugels polynomials. The propositions of the previous subsection can be transferred when we have a L´evy measure ν and the corresponding Teugels polynomials pn (x) and pσn (x). We use Corollary 4.3 to identify these polynomials. Corollary 4.8. Fix n ≥ 1 such that pn (0) 6= 0 and let a1 , . . . , an be the roots of pn (x). Then, a Pn+1 (x1 , . . . , xn+1 ) − Jn+1 (x1 , . . . , xn ) =
(−1)n L(pn )(x1 , . . . , xn+1 ) n+1
and σ,a Pn+2 (x1 , . . . , xn+2 ) − Jn+2 (x1 , . . . , xn+1 )
=
(−1)n+1 (L(pσn+1 )(x1 , . . . , xn+2 ) n+2 + λn L(pσn )(x1 , . . . , xn+1 ) + λn−1 L(pσn−1 )(x1 , . . . , xn )).
Equivalently, a Pn+1 (1, x, . . . , xn ) − Jn+1 (1, x, . . . , xn−1 ) =
(−1)n pn (x) n+1
and σ,a Pn+2 (1, x, . . . , xn+1 ) − Jn+2 (1, x, . . . , xn )
=
(−1)n+1 σ (pn+1 (x) + λn pσn (x) + λn−1 pσn−1 (x)). n+2
´ ORTHOGONAL POLYNOMIALS AND LEVY PROCESSES
29
4.3. Main result. The Teugels martingales {Y (n) , n ≥ 1} (see Nualart (n) (n) (n) and Schoutens [13]) are defined by Yt = Xt − E[Xt ], n ≥ 1. Specifically, (1) Yt
(2) Yt
= Xt ,
and (n) = Xt
(n) Yt
(2) = Xt
−t
Z
2
−t σ +
Z
2
x ν(dx)
R
xn ν(dx),
n ≥ 3.
R
P
(2)
(Nualart and Schoutens [13] write Xt = s≤t (∆Xs )2 , instead of [X, X], as we have done; however, both definitions give the same Y (2) .) By an orthogonalization procedure, they obtain a family {H (n) , n ≥ 1} of normal martingales, pairwise strongly orthogonal, that, under the hypothesis (4.1), generate all of L2 (Ω) by sums of iterated integrals. In order to strongly orthogonalize {Y (n) , n ≥ 1}, if σ > 0, they show that you can look for the sequence of orthogonal polynomials pσn (x) and take (n+1)
(1)
(n+1)
= L(pσn )(Yt , . . . , Yt
Ht
and the same expression with pn replacing
pσn
),
if σ = 0.
Theorem 4.9. Let X be a centered L´evy process with moments of all orders and fix n ≥ 1 such that the Teugels polynomial of order n, pn (x), does not have a zero root. Let a1 , . . . , an be the roots of pn (x). If σ = 0, then (1)
(n+1)
Pn+1 (Xt , . . . , Xt
(4.10)
=
)
(−1)n (n+1) (−1)n (1) (n) a Ht + Jn+1 (Xt , . . . , Xt ) + Cn t, n+1 n+1
where Cn = (1)
Z
(n)
R
x(pn (x) − pn (0))ν(dx) n
(1)
(1)
(−1) a (X and Jn+1 t , . . . , Xt ) + n+1 Cn t is orthogonal to P1 (Xt ), . . . , Pn (X1 , (n)
. . . , Xt ). If σ > 0, then (1)
(n+2)
Pn+2 (Xt , . . . , Xt
)=
(−1)n+1 (n+2) (n+1) (n) (Ht + λn Ht + λn−1 Ht ) n+2 (1)
(n+1)
σ,a + Jn+2 (Xt , . . . , Xt
)+
where λn and λn−1 are given in Corollary 4.3 and Dn+1 = σ 2 pn (0) +
Z
R
x2 pn (x)ν(dx).
(−1)n+1 Dn+1 t, n+2
´ AND F. UTZET J. L. SOLE
30 (1)
(n+1)
σ,a Moreover, Jn+2 (Xt , . . . , Xt (1)
(n−1)
. . . , Pn−1 (X1 , . . . , Xt
n+1
(1)
) + (−1) n+2 Dn+1 t is orthogonal to P1 (Xt ),
).
Proof. Since the proof for σ > 0 is very similar to the case σ = 0, we consider only the latter one. Formula 4.10 follows from Corollary 4.8 and (n+1)
Ht
(1)
(n+1)
(1)
(n+1)
= L(pn )(Yt , . . . , Yt
= L(pn )(Xt , . . . , Xt
) )−t
Z
R
x(pn (x) − pn (0))ν(dx). (−1)n n+1 Cn t a definition of Jn+1 (n)
(1)
a (X To prove the orthogonality between Jn+1 t , . . . , Xt ) +
and
(1) (j) Pj (X1 , . . . , Xt )
and
for j = 1, . . . , n, observe that, by
Lemma 4.5,
(−1)n (n+1) (1) (n) ), Cn t = Pn+1 (X1 , . . . , Xt , Vt n+1
(n)
(1)
a Jn+1 (Xt , . . . , Xt ) +
where (n+1)
Vt
(n)
(1)
= −L(pn )(Xt , . . . , Xt , 0) + Cn t.
The idea of the proof is to construct a simple L´evy process Zt such that, for r ≤ n, (1)
(4.11)
(n)
(1)
(r)
(n+1)
)]
(n+1)
)]
E[Pr (X1 , . . . , Xt )Pn+1 (Xt , . . . , Xt , Vt (r)
(1)
(1)
= E[Pr (Z1 , . . . , Zt )Pn+1 (Z1 , . . . , Zt
and by the orthogonality of the iterated integrals of different order, the expectation on the right is zero. (n+1) (n) (1) ) With this objective, consider the L´evy process (Xt , . . . , Xt , Vt that has characteristic function [see (3.3)] Z
ϕX (z) = exp t
R
Pn+1
i(
(e
j=1
zj xj −zn+1 xpn (x))
− 1 − ix(z1 − zn+1 pn (x)))ν(dx) ,
where z = (z1 , . . . , zn+1 ). On the other hand, let b1 , . . . , bn the Christoffel numbers corresponding to ν given in Theorem 4.2. Define (on the same probability space or another) Zt =
n X
aj (Nn (t) − bj t),
j=1
where N1 , . . . , Nn are independent Poisson processes with respective intensi(1) (n+1) ties b1 , . . . , bn . The characteristic function of (Zt , . . . , Zt ) is [see (3.3)] Z
ϕZ (z) = exp t
R
Pn+1
i(
(e
j=1
zj x j )
− 1 − ixz1 )νn (dx) ,
´ ORTHOGONAL POLYNOMIALS AND LEVY PROCESSES
31
P
where νn = nj=1 bj δaj . By Theorem 4.2, ν and νn have the same moments up to order 2n + 1, both ν and νn have the same first n Teugels polynomialsPand pn (x) ≡ 0, νn -a.e. Let j1 , . . . , jn+1 be nonnegative integers such that n+1 k=1 kjk ≤ 2n + 1. Therefore, the (joint) cumulant of order j1 in the first component, order j2 in the second component and so on, of the vectors (n+1) (1) (n+1) (n) (1) ) are the same. So, a polyno) and (Zt , . . . , Zt (Xt , . . . , Xt , Vt (1) (n) (n+1) mial up to degree 2n + 1 of (Xt , . . . , Xt , Vt ) and the same polynomial (1) (n+1) of (Zt , . . . , Zt ) have the same expectation. In particular, for r ≤ n, we have the identity (4.11). Example. A very simple example will help to interpret Theorem 4.9. Consider a L´evy process X with σ = 0 and L´evy measure ν, and let a be the root of its Teugels polynomial of order 1, p1 (x). Assume a 6= 0 and let b be the solution of Z
x2 ν(dx) = ba2 .
R
Let Zt = a(Nt − bt), where Nt is a Poisson process of intensity b. Zt is then a simple L´evy process that has L´evy measure ν1 = bδa . By Gauss–Jacobi Theorem 4.2, ν and ν1 have the same moments of order 2 and 3. So, Xt and Zt have the same cumulants of order 2 and 3, and, since both are centered, they have the same moments of those orders. Then, on one hand, (1) J2a (Xt ) = P2 (Xt , aXt ), so J2a (Xt ) − 12 t
Z
2
x ν(dx) = P2 Xt , aXt + t
R
Z
2
x ν(dx) .
R
On the other hand, (2)
(1)
P2 (Zt , Zt ) = P2 (Zt , aZt + ba2 t). We then have
J2a (Xt ) − 21 t
E P1 (Xt )
Z
R
R
2
x ν(dx)
(1)
(1)
(2)
= E[P1 (Zt )P2 (Zt , Zt )]
because P1 (Xt )(J2a (Xt ) − 12 t R x2 ν(dx)) is a product of a polynomial of degree 1 and a polynomial of order 2, in Xt , which is centered, so the expectation of that product depends only on the moments of order 2 and 3. So, Theorem 4.9 says that we have a decomposition (1)
(2)
P2 (Xt , Xt ) = J2a (Xt ) − 21 t
Z
R
(2)
x2 ν(dx) − 21 Ht ,
such that: 1. J2a (Xt ) − 12 t
R
Rx
2 ν(dx)
(1)
is orthogonal to P1 (Xt );
´ AND F. UTZET J. L. SOLE
32
2. J2a (Xt ) − 12 t
R
Rx
2 ν(dx) = P (X (1) , V (2) ), 2 t t
(1)
(2)
where (Xt , Vt ) is a L´evy pro(1)
(2)
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´ ORTHOGONAL POLYNOMIALS AND LEVY PROCESSES
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` tiques Departament de Matema Facultat de Ci` encies ` noma de Barcelona Universitat Auto 08193 Bellaterra (Barcelona) Spain E-mail:
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