in PROBABILITY
REPRESENTATIONS OF URBANIK’S CLASSES AND
MULTIPARAM-ETER ORNSTEIN-UHLENBECK PROCESSES
SVEND-ERIK GRAVERSEN
Department of Mathematical Sciences, Aarhus University, Ny Munkegade, 8000 Aarhus C, Denmark email:
♠❛ts❡❣❅✐♠❢✳❛✉✳❞❦
JAN PEDERSEN
Department of Mathematical Sciences, Aarhus University, Ny Munkegade, 8000 Aarhus C, Denmark email:
❥❛♥❅✐♠❢✳❛✉✳❞❦
SubmittedNovember 26, 2010, accepted in final formMarch 9, 2011 AMS 2000 Subject classification: 60G51, 60G57, 60H05
Keywords: Lévy bases, stochastic integrals, Urbanik’s classes, multiparameter Ornstein-Uhlenbeck processes
Abstract
A class of integrals with respect to homogeneous Lévy bases on ❘k is considered. In the one-dimensional case k =1 this class corresponds to the selfdecomposable distributions. Necessary and sufficient conditions for existence as well as some representations of the integrals are given. Generalizing the one-dimensional case it is shown that the class of integrals corresponds to Ur-banik’s classLk−1(❘). Finally, multiparameter Ornstein-Uhlenbeck processes are defined and
stud-ied.
1
Introduction
The purpose of this note is twofold. First of all, for any integerk≥1 we study the integral Z
❘k +
e−t.M(dt), (1.1)
whereM={M(A):A∈ Bb(❘k)}is a homogeneous Lévy basis on❘kandt
.=t1+· · ·+tkis the sum of the coordinates. Recall that a homogeneous Lévy basis is an example of an independently scattered random measure as defined in [9]; see the next section for further details. The one-dimensional case k =1, where M is induced by a Lévy process, is very well studied; see[10] for a survey. For example, in case of existence when k=1 the integral has a selfdecomposable distribution ([7,14]) and it is thus the marginal distribution of a stationary Ornstein-Uhlenbeck process. Moreover, necessary and sufficient conditions for the existence of (1.1) for k = 1 are also well-known. In the present note we give necessary and sufficient conditions for the existence of (1.1) for arbitrary k and provide several representations of the integral. The main result, Theorem 3.1, shows that for arbitrary k≥1 the law of (1.1) belongs to Urbanik’s class Lk−1(❘)
and conversely that any distribution herein is representable as in (1.1). The proof of the main theorem is in fact very easy. It relies only on a transformation rule for random measures (see Lemma 2.1) and well-known representations of Urbanik’s classes.
Assuming that (1.1) exists we may define a processY={Yt:t∈❘k}as
Yt = Z
s≤t
e−(t.−s.)M(ds),
wheres ≤ t should be understood coordinatewise. The second purpose of the note is to study some of the basic properties of this process. It is easily seen thatY is stationary and can be chosen lamp, where we recall that the latter is the multiparameter analogue of being càdlàg. In the case k =1, Y is often referred to as an Ornstein-Uhlenbeck process and we shall thus call Y a k-parameter Ornstein-Uhlenbeck process. In the casek=1,Y is representable as
Yt=Y0−
Z t
0
Ysds+M((0,t]) fort≥0. We give the analogous formula to this equation in the casek=2.
In the Gaussian case, Hirsch and Song[5]gave an alternative definition ofk-parameter Ornstein-Uhlenbeck processes; however, in Remark 4.2(2) we show that the two definitions give rise to the same processes.
The next section contains a few preliminary results. Section 3 concerns the main result, namely characterizations of (1.1). Finally, in Section 4 we study multiparameter Ornstein-Uhlenbeck processes.
2
Preliminaries
Let Leb denote Lebesgue measure on❘k. Throughout this note all random variables are defined on a probability space(Ω,F,P). The law of a random vector is denoted byL(X)and for a setNand two families{Xt:t∈N}and{Yt:t∈N}of random vectors write{Xt:t∈N}D={Yt:t∈N}if all finite dimensional marginals are identical. Furthermore, we say that{Xt:t∈N}is a modification of{Yt:t∈N}ifXt =Yt a.s. for all t∈N. Let ID=ID(❘)denote the class of infinitely divisible distributions on❘. That is, a distributionµon❘is in ID if and only if
b µ(z):=
Z
❘
eiz xµ(dx) =exp
−1
2z
2σ2+iγz+
Z
❘
g(z,x)ν(dx)
for allz∈❘,
whereg(z,x) =eiz x−1−iz x1D(x),D= [−1, 1], and(σ2,ν,γ)is the characteristic triplet ofµ, that is, σ2 ≥0, ν is a Lévy measure on ❘andγ ∈❘. For t ≥ 0 andµ ∈ID,µt denotes the distribution in ID withµct =µbt.
ForS∈ B(❘k)letB
b(S)be the set of bounded Borel sets inS. LetΛ ={Λ(A):A∈ Bb(S)}denote a family of (real valued) random variables indexed byBb(S), the set of bounded Borel set inS. Following[9]we callΛan independently scattered random measure on Sif the following conditions are satisfied:
(i) Λ(A1), . . . ,Λ(An)are independent wheneverA1, . . . ,An∈ Bb(S)are disjoint. (ii) Λ(S∞n=1An) =
P∞
n=1Λ(An)a.s. whenever A1,A2, . . . ∈ Bb(S)are disjoint with S∞
n=1An ∈
(iii) L(Λ(A))∈ID for allA∈ Bb(S).
If in addition there is a µ∈ID such thatL(Λ(A)) =µLeb(A)for allA∈ B
b(S)thenΛis calleda homogeneous Lévy basis on S andΛis said to beassociated withµ.
Note that in (ii) above there is a null set depending on the sequence A1,A2, . . .. Thus, ifΛis an independently scattered random it is generallynottrue that forωoutside a set of probability zero the mappingA7→Λ(A)(ω)is a (signed) measure whenAis in aσ-field included inBb(S). Thus,
Λis not a random measure in the sense of[8]. The problem of finding large subsetsA ofBb(S)
such that the mapping A ∋A7→ Λ(A)(ω)is regular (in some sense) forωoutside a null set is studied in[1,3].
LetΛdenote an independently scattered random measure onS. Recall from[9]that there exists a control measureλforΛand a family of characteristic triplets(σ2s,νs(dx),γs)s∈S, measurable in s, such that, forA∈ Bb(S),L(Λ(A))has characteristic triplet(σ2(A),ν(A)(dx),γ(A))given by
σ2(A) =
Z
A
σ2sλ(ds), ν(A)(dx) =
Z
A
νs(dx)λ(ds), γ(A) = Z
A
γsλ(ds). (2.1)
If Λ is a homogeneous Lévy basis associated withµ then λ equals Leb and (σ2
s,νs(dx),γs) =
(σ2,ν(dx),γ)where(σ2,ν,γ)is the characteristic triplet ofµ.
As an example letk =1. IfΛis a homogeneous Lévy basis on❘associated withµthen for all s ∈❘the process{Λ((s,s+t]): t ≥0} is a Lévy process in law in the sense of[12], p. 3. In particular it has a càdlàg modification which is a Lévy process and L(Λ((s,s+1])) =µ for all s ∈❘. Conversely, if Z = {Zt : t ∈ ❘} is a Lévy process indexed by ❘(i.e. it is càdlàg with stationary independent increments) thenΛ ={Λ(A):A∈ Bb(❘)}defined asΛ(A) =R1AdZsis a homogeneous Lévy basis. Similarly, a so-called natural additive process induces an independently scattered random measure; see[13].
Since an independently scattered random measure Λon S does not in general induce a usual measure an ω-wise definition of the associated integral is not possible. Therefore, integration with respect toΛwill always be understood in the sense developed in[9]. Recall the definition cf. page 460 in[9]:
A function f :S→❘is called simple if there is ann≥1,αi∈❘andAi∈ Bb(S)such that f = Pn
i=1αi1Ai. In this case define R
Af dΛ = Pn
i=1αiΛ(Ai∩A). In general, iff :S→❘is a measurable function then f is calledΛ-integrable if there is a sequence(fn)n≥1of simple functions such that
(i) fn→ f λ-a.s., where λ is the control measure, and (ii) the sequence R
AfndΛconverges in probability for everyA∈ B(S). In this case the limit in (ii) is called the integral of f overAand is denotedR
AfdΛ. The integral is well-defined, i.e., it does not dependent on the approximating sequence.
LetBΛdenote the set ofAinB(S)for which 1AisΛ-integrable. ThenBΛcontainsBb(S)and we can extendΛtoBΛby setting
Λ(A) =
Z
1AdΛ, A∈ BΛ.
Moreover,{Λ(A):A∈ BΛ}is an independently scattered random measure; that is, (i)–(iii) above
are satisfied whenBb(S)is replaced byBΛ. ForA∈ BΛ,Λ(A)still has characteristic triplet given
by (2.1).
Let T ∈ B(❘d)for some d. Given an independently scattered random measureΛ onS and a functionφ:S→ T satisfyingφ−1(B)∈ B
Λ for allB ∈ Bb(T), we can define an independently scattered random measure on T, calledthe image ofΛunderφ, to be denotedΛφ={Λφ(B):B∈ Bb(T)}, as
Λφ(B) = Λ(φ−1(B)), B∈ B
Similarly, ifψ:S→❘is measurable and locally bounded then ˜Λ ={Λ(˜ A):A∈ Bb(S)}, defined is an independently scattered random measure onS.
Keeping this notation we have the following result.
Lemma 2.1. (1) Let g :S → ❘be measurable. Then g is Λ˜-integrable if and only if gψis Λ -integrable and in this caseR
Sgd ˜Λ = φ)dΛ. In general, that is, when(2.2)is not satisfied, only the if-part of the statement is true. Proof. Using[9], Theorem 2.7(iv), and formula (2.1) we see that for everyA∈ Bb(S)the
Similarly, gis ˜Λ-integrable if and only if (2.3) and (2.4) are satisfied and
But noticing that the last integral equals Z
one sees that (2.5) and (2.6) are equivalent. Thus we have (1).
(2): Assume (2.2). ThenL(Λφ(B)),B∈ Bb(T), has characteristic triplet given by
whereλφis the image measure ofλunder the the functionφ. Thus, the first part of (2) follows from[9], Theorem 2.7, using the ordinary transformation rule.
In the general case, without (2.2), f ◦φ is Λ-integrable if and only if there is a sequence of simple functions gnapproximating f ◦φ such thatR
AgndΛconverges in probability for allA∈ integrability that f isΛφ-integrable.
Before continuing we recall a few basic properties of the class of selfdecomposable distributions and the classesLm. See e.g.[6,7,11,14,15,16]for fuller information, and[10]for a nice sum-mary of the results used below. LetL0=L0(❘)denote the class of selfdecomposable distributions on ❘. Recall, e.g. from [12], Theorem 15.3, that a probability measure µis in L0 if and only that the sets Lmare decreasing inmand that the stable distributions are in Lmfor allm.
Let IDlog denote the class of infinitely divisible distributions µ with Lévy measureν satisfying
R
3
Existence and characterizations of the integral
Assume that M = {M(A):A∈ Bb(❘k
+)} is a homogeneous Lévy basis on❘k+ associated with
µ∈ID which has characteristic triplet(σ2,ν,γ).
Let f :❘k
+→❘+ be given by f(t) =t.=
Pk
j=1tj and let g:❘+→❘+be given byg(x) = x
k k!.
Then the image of M under f, Mf = {Mf(B) : B ∈ Bb(❘+)}, is an independently scattered
random measure on❘+andL(Mf((0,x])) =µx k/t!
for all x≥0. Since in particular
L(Mf(g−1([0,y]))) =L(Mf([0,g−1(y)])) =µy for y≥0,
it follows thatMg◦f ={Mg◦f(B):B ∈ Bb(❘+)} is a homogeneous Lévy basis on❘+ associated
withµ. WritingΦ(k)forΦ◦ · · · ◦Φ(ktimes), the main result can be formulated as follows.
Theorem 3.1. (1) The three integrals Z
❘k +
e−t.M(dt), Z
❘+
e−xMf(dx), Z
❘+
e−(k!y)1/kMg◦f(dy) (3.1)
exist at the same time and are identical in case of existence. Assume existence and let µ˜ =
L(R❘k +
e−t.M(dt)). Thenµ˜= Φ(k)(µ)∈L
k−1andµ˜has characteristic triplet(σ˜2, ˜ν, ˜γ)given
by
˜ σ2= σ
2
2k (3.2)
˜
ν(B) = 1 (k−1)!
Z
❘ Z∞
0
sk−11B(e−sy)dsν(dy), B∈ B(❘) (3.3)
˜
γ=γ+ 1 (k−1)!
Z∞
0
sk−1e−s Z
1<|y|≤es
ν(dy)ds. (3.4)
(2) A necessary and sufficient condition for the existence of the integrals in(3.1)is that Z
|x|>2
(log|x|)kν(dx)<∞. (3.5)
(3) Let k≥2and assume that the integrals in(3.1)exist. Then Z
❘k +
e−t.M(dt) =
Z
❘+
e−xΛ(dx), (3.6)
whereΛ ={Λ(B):B∈ Bb(❘+)}is given as
Λ(B) =
Z
B×❘k−1
+
e−Pkl=2tlM(dt), B∈ B
b(❘+). (3.7)
Moreover,Λis a homogeneous Lévy basis on❘+. The distribution associated withΛisΦ(k−1)(µ)
(4) Conversely, to every distributionµ˜ ∈ Lk−1 there is a distribution µ ∈IDwith characteristic
triplet(σ2,ν,γ)withνsatisfying(3.5)such thatµ˜is given byµ˜=L(R
❘k +
e−t.M(dt))where M is a homogeneous Lévy basis on❘k
+associated withµ.
Proof. We can apply the first part of Lemma 2.1(2) to the first two integrals in (3.1) sinceΛis homogeneous. Likewise, the lemma applies to the last two integrals sincegis one-to-one. It hence follows immediately that the three integrals in (3.1) exist at the same time and are identical in case of existence. The remaining assertions, except (3), follow from Theorem 49 and Remark 58 of Rocha-Arteaga and Sato[10]. First of all, by Remark 58 a distribution is inLk−1if and only if
it is representable as the law of the last integral in (3.1). That ˜µ= Φ(k)(µ)in case of existence
follows from Remark 58 combined with Theorem 49. Using this, the result in (2) is equation (2.44) in Theorem 49, and the representation of the characteristic triplet in (1) is (2.47)–(2.49) in Theorem 49.
To prove (3) assume t 7→ e−t. is M-integrable and let ˜M(A) =R
Aψ(t)M(dt) forA∈ Bb(❘ k
+)
whereψ(t) =e−Pkl=2tl. Since forB∈ B
b(❘+)there is a constantc>0 such thatψ(t)1B×❘k−1
+ (t)≤ ce−t.1
B×❘k−1
+ (t)we have by Lemma 2.1(1) thatB×❘ k−1
+ ∈ BM˜. Thus,Λin (3.7) is well-defined and a homogeneous Lévy basis. Moreover, Λ = (M˜)φ where φ : ❘+k → ❘+ is φ(t) = t1.
Since, with f :❘+ → ❘+ given by f(x) =e−x, the mapping f ◦φ(t) = e−t1 is ˜M-integrable
by Lemma 2.1(1) it follows from the last part of Lemma 2.1(2) that f isΛ-integrable and we have (3.6). That is, ˜µin (1) is of the form ˜µ=L(R∞
0 e
−sdZ
s)whereZ={Zt:t≥0}is the Lévy process in law given by Zt= Λ((0,t]). As previously noted this means that ˜µ= Φ(L(Z1)). But since, by
(1), ˜µ= Φ(k)(µ)it follows thatL(Z
1) = Φ(k−1)(µ), i.e.L(Λ((0, 1])) = Φ(k−1)(µ).
4
Multiparameter Ornstein-Uhlenbeck processes
For a= (a1, . . . ,ak)∈❘k and b = (b1, . . . ,bk)∈❘k write a≤ b ifaj ≤ bj for all janda< b if aj < bj for all j. Define the half-open interval (a,b]as (a,b] = {t ∈❘k:a< t ≤ b} and let[a,b] ={t∈❘k:a≤ t ≤ b}. Further, letA = {t ∈❘k
+:tj = 0 for some j}, and forR =
(R1, . . . ,Rk)whereRjis either≤or>writeaRbifajRjbjfor all j.
Consider a familyF={Ft:t∈S}whereSis either❘kor❘+k andFt ∈❘for allt∈S. Fora,b∈S witha≤bdefinethe increment of F over(a,b],∆b
aF, as
∆b aF=
X
ε=(ε1,...,εk)∈{0,1}k
(−1)ε.F
(c1(ε
1),...,ck(εk)),
wherecj(0) =bjandcj(1) =aj. That is,∆abF =Fb−Fa ifk=1, and∆abF =F(b1,b2)+F(a1,a2)−
F(a1,b2)−F(b1,a2)ifk=2. Note that∆
b
aF=0 ifa≤bandb−a∈ A. We say thatF ={Ft:t∈S}islampif the following conditions are satisfied:
(i) for t ∈ ❘k
+ the limit F(t,R) = limu→t,tRuFu exists for each of the 2k relations R =
(R1, . . . ,Rk) whereRj is either ≤ or >. When S = ❘+k let F(t,R) = Ft if there is no u withtRu.
(ii) Ft =F(t,R)forR= (≤, . . . ,≤);
Here lamp stands for limits along monotone paths. See Adler et al. [2] for references to the literature on lamp trajectories. When S = ❘k
Assume thatM={M(A):A∈ Bb(❘k)}is a homogeneous Lévy basis on❘kassociated withµ∈ID which has characteristic triplet(σ2,ν,γ). DefineU={U
t:t∈❘k+}andX={Xt:t∈❘+k}as
Ut =
Z
[0,t]
es.M(ds) and X
t =e−t.Ut fort∈❘k+. (4.1)
Since fora,b∈❘k
+witha≤b,
∆abU=
Z
(a,b]
es.M(ds),
the random variables∆b1
a1U, . . . ,∆
bn
anUare independent whenever(a1,b1], . . . ,(an,bn]are disjoint intervals in❘k
+. Moreover, U is continuous in probability since M is homogeneous. Thus, U is
a Lévy process in the sense of Adler et al.[2], p. 5, and a Lévy sheet in the sense of Dalang and Walsh[4](in the casek=2). It hence follows e.g. from[2], Proposition 4.1, that by modification we may and do assume thatU, and hence alsoX, is lamp. Similarly, we may and do assume that t7→M((0,t])is lamp fort∈❘k
+.
Assuming in addition that (3.5) is satisfied we can define processesV ={Vt :t∈❘k} andY =
{Yt:t∈❘k}as
Vt=
Z
s≤t
es.M(ds) and Y
t=e−t.Vt fort∈❘k. (4.2) For fixed t∈❘k defineφt :❘k →❘k asφt(s) = t−s. By Lemma 2.1 and the fact thatM and Mφt are homogeneous Lévy bases associated withµwe have
Yt=
Z
s≤t
e−(t.−s.)M(ds) = Z
❘k +
e−s.M
φt(ds)
D
=
Z
❘k +
e−s.M(ds) fort∈❘k.
That is,Yt has the same law as the three integrals in (3.1). The same kind of arguments show that Y is stationary in the sense that
(Yt1, . . . ,Ytn)
D
= (Yt+t1, . . . ,Yt+tn) for alln≥1 andt,t1, . . . ,tn∈❘k.
When k = 1, Y is often referred to as an Ornstein-Uhlenbeck process. The above is a natural generalization so we shall callY ak-parameter Ornstein-Uhlenbeck process. There are many nice representations and properties ofY as the next remarks illustrate.
Remark4.1. Denote a generic element in❘k−1by ˜t= (t
1, . . . ,tk−1)and let ˜t.=
Pk−1
j=1tj. A generic elementtin❘kcan then be decomposed as t= (˜t,tk). ForB∈ B(❘)and ˜t∈❘k−1,{˜s≤˜t} ×B is the subset of❘kgiven by
{˜s≤˜t} ×B={s= (˜s,sk): ˜s≤˜tandsk∈B}. Assuming that (3.5) is satisfied,Yt is representable as
Yt=e−tk Z tk
−∞
eskM˜t(dsk) =e−˜t. Z
˜
s≤˜t
HereM˜t ={M˜t(B):B∈ Bb(❘)}andMtk={Mtk(C):C∈ B
b(❘k−1)}are given as
M˜t(B) =e−˜t. Z
{˜s≤˜t}×B
e˜s.M(ds), B∈ B b(❘)
Mtk(C) =e−tk Z
C×(∞,tk]
eskM(ds), C∈ Bb(❘k−1).
Arguments as in the proof of Theorem 3.1(3) show that M˜t andMtk are well-defined homoge-neous Lévy bases associated with respectivelyΦ(k−1)(µ)andΦ(µ), and we have (4.3).
The first expression in (4.3) shows that for fixed ˜t, {Yt :tk ∈❘} is a one-parameter Ornstein-Uhlenbeck process. By the second expression, {Yt : ˜t ∈❘k−1} is a(k−1)-parameter Ornstein-Uhlenbeck process for fixed tk.
Remark 4.2. (1) AssumeR
❘x2ν(dx) < ∞. By [12], Corollary 25.8, µ has finite second mo-ment. Moreover, denoting the variance by Var, we have Var(M(A)) =Leb(A)(σ2+R
❘x2ν(dx)) for A∈ Bb(❘k). In this case (3.5) is satisfied, implying thatY is well-defined. The character-istic triplet (σ˜2, ˜ν, ˜γ)ofL(Yt), t ∈❘k, is given in (3.2)–(3.4). Since, by (3.3),
R
❘x2ν˜(dx) = 2−kR
❘x2ν(dx)<∞it follows thatYt is square-integrable with Var(Yt) = 1
2k(σ
2+
Z
❘
x2ν(dx)).
Let us find the covariance function of Y. Let, for j = 1, 2, tj = (tj
1, . . . ,t
j k) ∈ ❘
k. Set t =
(t11∧t21, . . . ,t1k∧t2k)∈❘k,Dj={s∈❘k:s≤tj}andD={s∈❘k:s≤t}=D1∩D2. For j=1, 2,
Yt=e−(t.j−t.)Y t+e−t.
Z
Dj\D
es.M(ds). (4.4)
Since D1\DandD2\Dare disjoint the last term on the right-hand side of (4.4) with j =1 is independent of the corresponding term with j=2. Thus,
Cov(Yt1,Yt2) =e−(t 1
.−t.)e−(t2.−t.)Var(Y
t). (4.5)
(2) Now letν andγbe zero. In this case Hirsch and Song[5]defined ak-parameter Ornstein-Uhlenbeck ˜Y ={Y˜t :t∈❘k}as ˜Y
t =e−t.M((0,e2t])wheree2t= (e2t1, . . . ,e2tk). Witht1,t2andt given as under (1) we have
Cov(Y˜t1, ˜Yt2) =σ2e−(t 1
.−t.)e−(t2.−t.).
Thus, from (4.5) and Gaussianity it follows that up to a scaling constantY and ˜Yhave the same dis-tribution; in other words, in the Gaussian case our definition of ak-parameter Ornstein-Uhlenbeck process coincides essentially with that of[5].
Remark4.3. Assume (3.5) is satisfied. We may and do assume thatV, and thus alsoY, is lamp. To see this, note that for arbitrarys= (s1, . . . ,sk)∈❘k andt= (t
1, . . . ,tk)∈❘k+\ A we have
Vs+t = ∆s+t s V−
X
ε∈{0,1}k
ε6=(1,...,1)
(−1)ε.V
(s1+ε1t1,...,sk+εktk).
(i) for arbitrarys∈❘kthe process{∆s+t
s V :t∈❘ k
+}has a lamp modification.
(ii) If at least one coordinate is fixed, thenV has a lamp modification in the remaining coordi-nates. That is, if e.g.tk=0 then ˜t= (t1, . . . ,tk−1)7→V(˜t,0)is a.s. lamp on❘k−1. simplicity the case where tk is fixed at tk = 0 while all other coordinates vary freely. As in Remark 4.1 we have
Proof. Since the proofs are similar we only prove the representation ofY.
SinceRhas independent increments and is continuous in probability we may and do assume that it is lamp, see[2].
Since for fixed t2,{Vt1,t2:t1≥0}is a semimartingale we can apply integration by parts, together
with Lemma 2.1(1) and the fact that all terms are lamp, to obtain
e−t1V
The same kind of argument fort2instead of t1gives
Yt1,t2=e−t2V
Acknowledgment: We thank the referee for providing a detailed list of additional references, and in particular for drawing our attention to the paper by Hirsch and Song[5].
References
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