A family of martingales generated by a process with independent increments
An explicit procedure to construct a family of martingales generated by a process with independent increments is presented. The main tools are the polynomials that give the relationship between the moments and cumulants, and a set of martingales related to the jumps of the process called Teugels mar...
Gespeichert in:
| Datum: | 2008 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Інститут математики НАН України
2008
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/4559 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | A family of martingales generated by a process with independent increments / J.L. Sole, F. Utzet // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 139–144. — Бібліогр.: 9 назв.— англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859658753925709824 |
|---|---|
| author | Sole, J.L. Utzet, F. |
| author_facet | Sole, J.L. Utzet, F. |
| citation_txt | A family of martingales generated by a process with independent increments / J.L. Sole, F. Utzet // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 139–144. — Бібліогр.: 9 назв.— англ. |
| collection | DSpace DC |
| description | An explicit procedure to construct a family of martingales generated by a process with independent increments is presented. The main tools are the polynomials that give the relationship between the moments and cumulants, and a set of martingales related to the jumps of the process called Teugels martingales.
|
| first_indexed | 2025-11-30T09:12:56Z |
| format | Article |
| fulltext |
Theory of Stochastic Processes
Vol. 14 (30), no. 2, 2008, pp. 139–144
UDC 519.21
JOSEP LLUÍS SOLÉ AND FREDERIC UTZET
A FAMILY OF MARTINGALES GENERATED BY
A PROCESS WITH INDEPENDENT INCREMENTS
An explicit procedure to construct a family of martingales generated by a process
with independent increments is presented. The main tools are the polynomials that
give the relationship between the moments and cumulants, and a set of martingales
related to the jumps of the process called Teugels martingales.
1. Introduction
In this work, we present an explicit procedure to generate a family of martingales from
a process X = {Xt, t ≥ 0} with independent increments and continuous in probability.
We extend our results exposed in [8], where we dealt with Lévy processes (independent
and stationary increments); in that case, the martingales obtained were of the form
Mt = P (Xt, t), where P (x, t) is a polynomial in x and t, and then they are time–space
harmonic polynomials relative to X . Here, the martingales constructed are polynomials
on Xt but, in general, not in t. Part of the paper is devoted to define the Teugels
martingales of a process with independent increments; such martingales, introduced by
Nualart and Schoutens [5] for Lévy processes, are a building block of the stochastic
calculus with that type of processes.
2. Independent increment processes and their Teugels martingales
Let X = {Xt, t ≥ 0} be a process with independent increments, X0 = 0, continuous
in probability and cadlag; such processes are also called additive processes, and we will
indistinctly use both names. Moreover, assume that Xt is centered and has moments of
all orders. It is well known that the law of Xt is infinitely divisible for all t ≥ 0. Let σ2
t
be the variance of the Gaussian part of Xt, and let νt be its Lévy measure; for all these
notions, we refer to Sato [6] or Skorohod [7].
Denote, by ν̃, the (unique) measure on B((0,∞) × R0) defined by
ν̃((0, t] ×B) = νt(B), B ∈ B(R0),
where R0 = R − {0}. By the standard approximation argument, we have that, for a
measurable function f : R0 → R and for every t > 0,∫∫
(0,t]×R0
|f(x)| ν̃(ds, dx) <∞ ⇐⇒
∫
R0
|f(x)| νt(dx) <∞,
and, in this case, ∫∫
(0,t]×R0
f(x) ν̃(ds, dx) =
∫
R0
f(x) νt(dx).
2000 AMS Mathematics Subject Classification. Primary 60G51, 60G44.
Key words and phrases. Process with independent increments, Cumulants, Teugels martingales.
This research was supported by grant BFM2006-06247 of the Ministerio de Educación y Ciencia and
FEDER.
139
140 JOSEP LLUÍS SOLÉ AND FREDERIC UTZET
Note that since, for every t ≥ 0, νt is a Lévy measure, ν̃ is σ–finite. To prove this, we
observe that νt({|x| > 1}) <∞, νt((1/(n+ 1), 1/n]) <∞, and νt([−1/n,−1/(n+ 1))) <
∞, n ≥ 1. So there is a numerable partition of R0 with sets of finite νt measure, ∀t > 0.
Then, we can construct a numerable partition of (0,∞) × R0, each set being with finite
ν̃-measure.
Write
N(C) = #{t : (t,ΔXt) ∈ C}, C ∈ B((0,∞) × R0),
the jump measure of the process, where ΔXt = Xt − Xt−. It is a Poisson random
measure on (0,∞) × R0 with intensity measure ν̃ (Sato [6, Theorem 19.2]). Define the
compensated jump measure
dÑ(t, x) = dN(t, x) − dν̃(t, x).
The process admits the Lévy–Itô representation
Xt = Gt +
∫∫
(0,t]×R0
xdÑ (t, x), (2)
where {Gt, t ≥ 0} is a centered continuous Gaussian process with independent increments
and variance E[G2
t ] = σ2
t .
The relationship between the moments of an infinitely divisible law and the moments
of its Lévy measure is also well known (see Sato [6,Theorem 25.4]). In our case, as the
process has moments of all orders, for all t ≥ 0,∫
{|x|>1}
|x| νt(dx) <∞ and
∫
R0
|x|n νt(dx) <∞, ∀n ≥ 2.
Write
F2(t) = σ2
t +
∫
R0
x2 νt(dx) and Fn(t) =
∫
R0
xn νt(dx), n ≥ 3. (1)
Since
∫
{|x|>1} |x| νt(dx) < ∞ and E[Xt] = 0, the characteristic function of Xt can be
written as
φt(u) = exp
{
− 1
2
σ2
t u
2 +
∫
R0
(
eiux − 1 − iux
)
νt(dx)
}
.
It is deduced that, for n ≥ 2, Fn(t) is the cumulant of order n of Xt (for n = 2,
E[X2
t ] = F2(t)). Also σ2
t is continuous and increasing (Sato [6, Theorem 9.8]).
Proposition 1. The functions Fn(t), n ≥ 2, are continuous and have finite variation
on finite intervals, and, for n even, they are increasing.
Proof.
Consider 0 < u < t < v, and write U = [u, v]. From the continuity in probability of
X ,
lim
s→t, s∈U
Xn
s = Xn
t , in probability.
Moreover, ∀s ∈ U, |Xs| ≤ supr∈U |Xr|, and since X is a martingale, by Doob’s inequality,
E
[
sup
r∈U
|Xr|n
]
≤ C sup
r∈U
E[|Xr|n] ≤ CE[|Xv|n] <∞.
So it follows by dominated convergence that the function t !→ E[Xn
t ] is continuous. Since
the cumulants are polynomials of the moments, the continuity of all functions Fn(t) is
deduced.
MARTINGALES AND INDEPENDENT INCREMENT PROCESSES 141
To prove that Fn(t) has finite variation on finite intervals, consider a partition of [0, t]:
0 < t0 < · · · < tk = t. Then
k∑
j=1
∣∣Fn(tj) − Fn(tj−1)
∣∣ =
k∑
j=1
∣∣ ∫∫
(tj−1,tj ]×R0
xnν̃(ds, dx)
∣∣
≤
k∑
j=1
∫∫
(tj−1,tj ]×R0
|x|n ν̃(ds, dx) =
∫∫
(0,t]×R0
|x|n ν̃(ds, dx) <∞. �
Consider the variations of the process X (see Meyer [4]):
X
(1)
t = Xt,
X
(2)
t = [X,X ]t = σ2
t +
∑
0<s≤t
(
ΔXs
)2
X
(n)
t =
∑
0<s≤t
(
ΔXs
)n
, n ≥ 3.
By Kyprianou [2, Theorem 2.7], for n ≥ 3 (the case n = 2 is similar), the characteristic
function of X(n) is
exp
{∫∫
(0,t]×R0
(
eiuxn − 1
)
ν̃(ds, dx)
}
= exp
{∫
R0
(
eiux − 1
)
ν
(n)
t (dx)
}
,
where ν(n)
t is the measure image of νt by the function x !→ xn which is a Lévy measure.
So X(n) has independent increments. Also by Kyprianou [2, Theorem 2.7], for n ≥ 2,
E[X(n)
t ] = Fn(t) and E
[(
X
(n)
t
)2] = F2n(t) +
(
Fn(t)
)2
.
Therefore, combining the independence of the increments and the continuity of Fn(t), it
is deduced that X(n) is continuous in probability.
By Proposition 1, Fn(t) has finite variation on finite intervals. Hence, the process
X
(n)
t = Fn(t) +
(
X
(n)
t − Fn(t)
)
is a semimartingale.
The Teugels martingales introduced by Nualart and Schoutens [5] for Lévy processes
can be extended to additive processes. In the same way as in [5], these martingales are
obtained centering the processes X(n):
Y
(1)
t = Xt,
Y
(n)
t = X(n) − Fn(t), n ≥ 2,
They are square integrable martingales with optional quadratic covariation
[Y (n), Y (m)]t = X(n+m),
and, since F2n(t) is increasing, the predictable quadratic variation of Y (n) is
〈Y (n)〉t = F2n(t).
142 JOSEP LLUÍS SOLÉ AND FREDERIC UTZET
3. The polynomials of cumulants
The formal expression
exp
{ ∞∑
n=1
κn
un
n!
}
=
∞∑
n=0
μn
un
n!
. (3)
relates the sequences of numbers {κn, n ≥ 1} and {μn, n ≥ 0}. When we consider a
random variable Z with moment generating function in some open interval containing
0, then both series converge in a neighborhood of 0, and (3) is the relationship between
the moment generating function, ψ(u) = E[euZ ], and the cumulant generating function,
logψ(u). Moreover, μn (respectively, κn) is the moment (respectively, the cumulant)
of order n of Z, and the well-known relations between moments and cumulants can be
deduced from (3). The first three ones are
μ1 = κ1,
μ2 = κ2
1 + κ2,
μ3 = κ3
1 + 3κ1κ2 + κ3, . . .
If the random variable Z has only finite moments up to order n, the corresponding
relationship is true up to this order.
There is a general explicit expression of the moments in terms of cumulants in Kendall
and Stuart [1], or formulas involving the partitions of a set, see McCullagh [3]. In general,
μn is a polynomial of κ1, . . . , κn, called Kendall polynomial. Denote, by Γn(x1, . . . , xn),
n ≥ 1, this polynomial, that is, we have
μn = Γn(κ1, . . . , κn).
Also write Γ0 = 1. These polynomials enjoy very interesting properties, as the recurrence
formula that follows from Stanley [9, Proposition 5.1.7]:
Γn+1(x1, . . . , xn+1) =
n∑
j=0
(
n
j
)
Γj(x1, . . . , xj)xn+1−j . (4)
We also have
∂Γn(x1, . . . , xn)
∂xj
=
(
n
j
)
Γn−j(x1, . . . , xn−j), j = 1, . . . , n. (5)
Computing the Taylor expansion of Γn(x1 + y, x2, . . . , xn) at y = 0, we get the following
expression we will need later:
Γn(x1 + y, x2, . . . , xn) =
n∑
j=0
(
n
j
)
Γn−j(x1, . . . , xn−j) yj . (6)
Interchanging the roles of x1 and y and evaluating the function at 0, we obtain
Γn(x1, x2, . . . , xn) =
n∑
j=0
(
n
j
)
Γn−j(0, x2, . . . , xn−j)xj
1. (7)
4. A family of martingales relative to the additive process
The main result of the paper is the following Theorem:
Theorem 1. Let X be a centered additive process with finite moments of all orders.
Then the process
M
(n)
t = Γn
(
Xt,−F2(t), . . . ,−Fn(t)
)
MARTINGALES AND INDEPENDENT INCREMENT PROCESSES 143
is a martingale.
Proof.
Let n ≥ 2. We apply the multidimensional Itô formula to the semimartingales
Xt, F2(t), . . . , Fn(t). By Proposition 1, the functions F2(t), . . . , Fn(t) and σ2
t are contin-
uous and of finite variation. From (5) and the fact that [X,X ]ct = σ2
t and [Fj , Fj ]ct = 0,
we have
M
(n)
t = n
∫ t
0
M
(n−1)
s− dXs −
n∑
j=2
(
n
j
)∫ t
0
M (n−j)
s dFj(s)
+
1
2
n(n− 1)
∫ t
0
M (n−2)
s d(σ2
s)
+
∑
0<s≤t
(
Γn
(
Xs− + ΔXs,−F2(s), . . . ,−Fn(s)
)
− Γn
(
Xs−,−F2(s), . . . ,−Fn(s)
)
− nΔXs Γn−1
(
Xs−,−F2(s), . . . ,−Fn(s)
))
.
Applying (6),
Γn
(
Xs− + ΔXs,−F2(s), . . . ,−Fn(s)
)
=
n∑
j=0
(
n
j
)
M
(n−j)
s−
(
ΔXs
)j
.
Then, the jumps part given in the expression of M (n)
t is∑
0<s≤t
n∑
j=2
(
n
j
)
M
(n−j)
s−
(
ΔXs
)j =
n∑
j=2
(
n
j
)∫ t
0
M
(n−j)
s− dX(j)
s −
(
n
2
)∫ t
0
M (n−2)
s d(σ2
s)
=
n∑
j=2
(
n
j
)∫ t
0
M
(n−j)
s− d
(
Y (j)
s + Fj(s)
)
−
(
n
2
)∫ t
0
M (n−2)
s d(σ2
s).
Therefore,
M
(n)
t =
n∑
j=1
(
n
j
)∫ t
0
M
(n−j)
s− dY (j)
s . (8)
Moreover,
(
M
(k)
t
)2 is a polynomial inXt, F2(t), . . . , Fk(t). Taking expectations and using
the relations between moments and cumulants, as well as the fact that the cumulants of
Xt are Fn(t), n ≥ 2, we obtain that
E
[(
M
(k)
t
)2] = P (F2(t), . . . , F2k(t)),
for a suitable polynomial P . Then, for every t ≥ 0, we have
E
[ ∫ t
0
(
M
(k)
s−
)2
d〈Y (j)〉s
]
=
∫ t
0
E
[(
M
(k)
s−
)2
]
dF2j(s)
=
∫ t
0
P (F2(s), . . . , F2k(s)) dF2j(s) <∞,
So all the stochastic integrals on the right-hand side of (8) are martingales. �
Remark 1. It is worth to note that the preceding Theorem implies that the function
gn(x, t) = Γn
(
x,−F2(t), . . . ,−Fn(t)
)
is a time–space harmonic function with respect to Xt. By (7),
gn(x, t) =
n∑
j=0
Γn−j(0,−F2(t), . . . , Fn(t))xj .
144 JOSEP LLUÍS SOLÉ AND FREDERIC UTZET
In general, gn(x, t) is a polynomial in x. If Fn(t), n ≥ 2, are polynomials in t, then
gn(x, t) is a time–space harmonic polynomial; this happens for all Lévy processes with
moments of all orders and for some additive process; see the example below.
Example. Let Λ(t) : R+ −→ R+ be a continuous increasing function, and let J be a
Poisson random measure on R+ with intensity measure μ(A) =
∫
A
Λ(dt), A ∈ B(R+).
Then the process X = {Xt, t ≥ 0} defined pathwise,
Xt(ω) =
∫ t
0
J(ds, ω) − Λ(t)
, is an additive process; it is a Cox process with deterministic hazard function Λ(t). From
the characteristic function of Xt, we deduce that the Lévy measure is
νt(dx) = Λ(t)δ1(dx),
where δ1 is a Dirac delta measure concentrated in the point 1. Hence,
Fn(t) = Λ(t), n ≥ 2.
Note that the conditions we have assumed on Λ are necessary to obtain an additive
process, but it is not necessary (though not very restrictive) to assume that Λ is absolutely
continuous with respect to the Lebesgue measure.
The function defined in Remark 1 is
gn(x, t) = Γn
(
x,−Λ(t), . . . ,−Λ(t)
)
.
Hence, when Λ(t) is a polynomial, gn(x, t) is a time–space harmonic polynomial.
Denote, by Cn(x, t), the Charlier polynomial with leading coefficient equal to 1. Then
(see [8])
gn(x, t) =
n∑
j=1
λ
(n)
j Cj(x,Λ(t)),
where λ(n)
1 = 1 and
λ
(n)
k+1 =
n−1∑
j=k
(
n
j
)
λ
(j)
k , k = 1, . . . , n− 1.
Bibliography
1. M.Kendall and A.Stuart, The Advanced Theory of Statistics, Vol. 1, 4th edition, MacMillan,
New York, 1977.
2. A.E.Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications,
Springer, Berlin, 2006.
3. P.McCullagh, Tensor Methods in Statistics, Chapman and Hall, London, 1987.
4. P.A.Meyer, Un cours sur les integrales stochastiques, Séminaire de Probabilités X, Springer,
New York, 1976, pp. 245–400. (French)
5. D.Nualart and W.Schoutens, Chaotic and predictable representation for Lévy processes, Sto-
chastic Process. Appl. 90 (2000), 109–122.
6. K.Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press,
Cambridge, 1999.
7. A.V.Skorohod, Random Processes with Independent Increments, Kluwer Academic Publ., Dor-
drecht, Boston, London, 1986.
8. J.L.Solé and F.Utzet, Time-space harmonic polynomials relative to a Lévy process, Bernoulli
(2007).
9. R.P.Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge,
1999.
'���������� �� %����A���C���� (������� �� )�A������� ���"������� ���1����� �� 9����2
����� � #*� 9��������� D9��������E� ������
E-mail : jllsole@mat.uab.cat, utzet@mat.uab.cat
|
| id | nasplib_isofts_kiev_ua-123456789-4559 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-11-30T09:12:56Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Sole, J.L. Utzet, F. 2009-12-03T16:42:16Z 2009-12-03T16:42:16Z 2008 A family of martingales generated by a process with independent increments / J.L. Sole, F. Utzet // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 139–144. — Бібліогр.: 9 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4559 519.21 An explicit procedure to construct a family of martingales generated by a process with independent increments is presented. The main tools are the polynomials that give the relationship between the moments and cumulants, and a set of martingales related to the jumps of the process called Teugels martingales. This research was supported by grant BFM2006-06247 of the Ministerio de Educacion y Ciencia and FEDER. en Інститут математики НАН України A family of martingales generated by a process with independent increments Article published earlier |
| spellingShingle | A family of martingales generated by a process with independent increments Sole, J.L. Utzet, F. |
| title | A family of martingales generated by a process with independent increments |
| title_full | A family of martingales generated by a process with independent increments |
| title_fullStr | A family of martingales generated by a process with independent increments |
| title_full_unstemmed | A family of martingales generated by a process with independent increments |
| title_short | A family of martingales generated by a process with independent increments |
| title_sort | family of martingales generated by a process with independent increments |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4559 |
| work_keys_str_mv | AT solejl afamilyofmartingalesgeneratedbyaprocesswithindependentincrements AT utzetf afamilyofmartingalesgeneratedbyaprocesswithindependentincrements AT solejl familyofmartingalesgeneratedbyaprocesswithindependentincrements AT utzetf familyofmartingalesgeneratedbyaprocesswithindependentincrements |