El e c t ro nic J
o f
Pr
ob a bi l i t y
Electron. J. Probab.17(2012), no. 74, 1–21.
ISSN:1083-6489 DOI:10.1214/EJP.v17-2287
Multiparameter processes with stationary increments:
Spectral representation and integration
Andreas Basse-O’Connor
∗Svend-Erik Graversen
∗Jan Pedersen
∗Abstract
In this article, a class of multiparameter processes with wide-sense stationary incre- ments is studied. The content is as follows. (1) The spectral representation is derived;
in particular, necessary and sufficient conditions for a measure to be a spectral mea- sure is given. The relations to a commonly used class of processes, studied e.g. by Yaglom, is discussed. (2) Some classes of deterministic integrands, here referred to as predomains, are studied in detail. These predomains consist of functions or, more generally, distributions. Necessary and sufficient conditions for completeness of the predomains are given. (3) In a framework covering the classical Walsh-Dalang theory of a temporal-spatial process which is white in time and colored in space, a class of predictable integrands is considered. Necessary and sufficient conditions for com- pleteness of the class are given, and this property is linked to a certain martingale representation property.
Keywords: Multiparameter processes; stationary increments; spectral representation; inte- gration.
AMS MSC 2010: 60G51; 60G12; 60H05.
Submitted to EJP on November 16, 2011, final version accepted on August 29, 2012.
1 Introduction
Letd≥1be an integer which is fixed throughout. In this article we consider a class of real-valued processesX ={Xu :u∈Rd}indexed byRd withwide-sense stationary increments. We refer to Section 2 for the precise definition so for now it suffices to say that this class is large and contains e.g. thed-parameter fractional Brownian sheet and a well-known example from the theory of stochastic partial differential equations cf. e.g.
Dalang [2], p. 5–6; see also Example 2.6. The main purpose is to study different kinds of integrals with respect to such processes, focusing in particular on completeness of various sets of integrands.
In Section 3 we discuss classes of deterministic integrands, referred to as predo- mains. Predomains are not necessarily sets of functions but the corresponding integral takes values in the set of square-integrable random variables. On predomains we use the metric induced by theL2-distance between corresponding integrals. If complete- ness is present, a predomain is referred to as adomain. In the one-dimensional case
∗Aarhus University, Denmark. Email: basse,matseg,jan@imf.au.dk
d= 1several predomains have been studied for processes with stationary increments. A key reference in the case of fractional Brownian motion is Taqqu and Pipiras [7] where various (pre)domains consisting of functions are analyzed. These authors show that many natural predomains studied in the literature are in fact not complete and hence not domains. To remedy this, Jolis [5] introduced a larger predomain consisting of distri- butions in the case of a continuous processes with stationary increments. In particular she showed that this will often lead to a domain. In Section 3 we follow [5] and study predomains containing functions as well as distributions. Generalizing results of [5, 7], necessary and sufficient conditions on the spectral measure for a predomain to be a domain are given. Moreover, we show that the integral of an integrandϕbelonging to any of the predomains considered is given by
Z
Rd
ϕ(u)X(du) = Z
Rd
Fϕ(z)Z(dz), (1.1)
whereFdenotes the Fourier transform andZthe random spectral measure ofX. As is obvious from (1.1) the integral is closely linked to the spectral representation of X. Therefore we study the spectral representation ofXin detail in Section 2. Moreover, a comparison to the class of processes studied e.g. by Yaglom [13] is given.
Finally, in Section 4 we add a temporal component and thus consider Gaussian pro- cessesX ={Xu : u= (t, x)∈ R1+d} wheret ∈Ris time andx∈Rd a spatial compo- nent. We assume thatX iswhite in time and colored in space. A martingale integral with respect toX is constructed akin to the classical papers by Walsh [12] and Dalang [2] although it should be noticed that in the present situation, unlike these papers, X does in general not induce a martingale measure. For example, whend = 1, X could be fractional in space with Hurst exponentH in(0,1)in which case X only induces a martingale measure as in [2] whenH >1/2. We show that the integral of a predictable integrandϕt(x)with respect toX is
Z ∞
0
Z
Rd
ϕt(x)X(d(t, x)) = Z ∞
0
Z
Rd
Fϕt(z)dZt(x),
whereFdenotes the Fourier transform in the space variable, and for fixedt,Zt(·)is the random spectral measure ofX((0, t]× ·)in the space variable. Necessary and sufficient conditions for completeness for a class of integrands are given and in particular this property is linked to a martingale representation property with respect toX.
Definitions and notation: For any measure µ, L2C(µ) denotes the set of complex- valuedµ-square integrable functions andL2R(µ)the subset hereof taking values inR. Likewise, for any A ⊆ L2C(µ), spCA is the closed complex linear span and spRA the corresponding closed real linear span of A. Observe that spRA coincides with the real-valued elements inspCA if all elements inA are real-valued. According to usual notation the space of tempered distributions, that is the dual of the Schwartz space SC(Rd) consisting of complex-valued C∞–functions on Rd of rapid decrease, is de- notedSC0(Rd). The subspace ofSC(Rd)consisting of real-valued functions is denoted SR(Rd), and likewiseSR0(Rd)is the set of elementsΨinSC0(Rd)such thatΨ(φ)∈ R for allϕ∈SR(Rd). Similarly,DC(Rd)(resp.DR(Rd)) denotes the set of complex-valued (resp. real-valued)C∞-functions on Rd of compact support. The class of non-negative elements inDR(Rd)is denoted byDR(Rd)+. For the general theory of distributions and especially tempered distributions we refer to Schwartz [9].
Let λd denote Lebesgue measure on Rd. The Fourier transformF maps SC0(Rd) ontoSC0(Rd)and with the usual identification of a locally integrable function with its corresponding tempered distribution when it exists, we have forf ∈L1C(λd)that
Ff(z) = Z
Rd
eihz,·if(·)dλd= Z
Rd
eihz,uif(u)du, forz∈Rd.
Here, h·,·i is the canonical inner product on Rd with corresponding norm k·k. The notation differs from the one used e.g. in [9] where, forf ∈L1C(λd), Ff(−2π·)is used as the Fourier transform off. But apart from a constant(2π)d appearing in Parseval’s identity and the explicit form of the inverseF−1, all results from the general theory of distributions remain valid with the definition given above. Whend= 1we also use the notationF1instead ofF.
All random variables are defined on a probability space (Ω,F,P) which is fixed throughout. Equality in distribution is denotedD=. Finally,Bb(Rd)is the class of bounded Borel sets inRd.
2 Spectral representation
In Definition 2.4 the class of processes with wide-sense stationary increments is defined and the spectral representation is given in Theorem 2.7. This representation is stated in terms of the following class of random measures.
Definition 2.1. Let F be a symmetric Borel measure on Rd finite on compacts. A set functionZ: Bb(Rd)→ L2C(P)is said to be an L2C(P)-valued random measure with control measureF if
(1) Z(A∪B) =Z(A) +Z(B)P-a.s. wheneverA, B∈ Bb(Rd)are disjoint;
(2) Z(A) =Z(−A)P-a.s. forA∈ Bb(Rd); (3) E[Z(A)Z(B)] =F(A∩B)forA, B∈ Bb(Rd); (4) E[Z(A)] = 0forA∈ Bb(Rd).
Remark 2.2. Let Z be a random measure as above. From (1) and (3) it follows that Z(∪∞n=1An) =P∞
n=1Z(An)inL2C(P)for any disjoint sequence(An)n≥1inBb(Rd)satis- fying∪∞n=1An∈ Bb(Rd).
DecomposeZasZ(A) =Z1(A) +iZ2(A)forA∈ Bb(Rd); that is,Z1is the real part of Z,Z2the imaginary part, andZ1(A), Z2(A)∈L2R(P)forA∈ Bb(Rd). ForA, B ∈ Bb(Rd) we have
E[Z1(A)Z2(B)] = 0, (2.1)
E[Z1(A)Z1(B)] = 12
F(A∩B) +F(A∩(−B))
, (2.2)
E[Z2(A)Z2(B)] = 12
F(A∩B)−F(A∩(−B)
. (2.3)
To see this, notice that by (2) in Definition 2.1, Z1(A) = 12
Z(A) +Z(−A)
and Z1(B) = 12
Z(B) +Z(−B)], Z2(A) = 2i1
Z(A)−Z(−A)
and Z2(B) = −12i
Z(B)−Z(−B)].
Hence,(2.1)–(2.3)follow by symmetry of the measureF and Definition 2.1(3).
LetZ be anL2C(P)-valued random measure with control measureF. As usual, inte- gration with respect toZ can be defined starting with simple functions and extending toL2C(F)using the isometry condition Definition 2.1(3). Thus, the integralϕ7→R
ϕ dZ mapsL2C(F)linearly isometrically onto a closed subset ofL2C(P)consisting of zero-mean random variables, and satisfies, forA∈ Bb(Rd)andϕ, ψ∈L2C(F),
Z
1AdZ =Z(A), and EhZ ϕ dZ
Z ψ dZi
= Z
ϕψ dF.
Denoting by RC(Z)the set of integrals R
ϕ dZ, ϕ ∈ L2C(F), RR(Z) refers to the real- valued elements inRC(Z). WithL˜2C(F)denoting the set of functions inL2C(F)satisfying ϕ(x) =ϕ(−x)for allx∈Rdwe have
RR(Z) =nZ
ϕ dZ :ϕ∈L˜2C(F)o .
Indeed, the inclusion ”⊇” follows from Definition 2.1(2) and ”⊆” from the fact that for allϕ∈L2C(F), 12(ϕ+ϕ(−·))is inL˜2C(F)with integral equal to the real part ofR
ϕ dZ. For u= (u1, . . . , ud)and v = (v1, . . . , vd)in Rd writeu≤ v if uj ≤ vj for allj, and u < v if uj < vj for all j. Let (u, v] = {y ∈ Rd : u < y ≤ v}. Consider a family H ={Hu:u∈Rd}withHu∈C. Foru≤vinRd definethe increment ofH over(u, v], H((u, v]), as
H((u, v]) = X
=(1,...,d)∈{0,1}d
(−1).H(c1(1),...,cd(d)), (2.4) where.=1+· · ·+d,cj(0) =vj andcj(1) = uj. That is,H((u, v]) =Hv−Huifd= 1 and
H((u, v]) =H(v1,v2)+H(u1,u2)−H(u1,v2)−H(v1,u1) ifd= 2.
Notice thatH((u, v]) = 0ifu≤vandu6< v. Later we shall occasionally write4hH(u) forH((u, u+h])foru∈Rdand anyh∈Rd+.
Remark 2.3. The set function H defined in (2.4)on the semi-ring R:= {(u, v] : u≤ vinRd}is finitely additive. Conversely, letµbe a finitely additive set function onRand set
Hu= (−1)nuµ((u∧0, u∨0]), foru∈Rd, (2.5) whereu∧0 = (min(u1,0), . . . ,min(ud,0)),u∨0 = (max(u1,0), . . . ,max(ud,0))andnu is the number of coordinates inuthat are strictly less than0. Thenµ((u, v]) = H((u, v]) for allu≤vinRd.
To show the first claim it is enough to show that for allu≤v inRd,k= 1, . . . , dand r∈(uk, vk)we have
H((u, v]) =H((u1, v1]× · · · ×(uk−1, vk−1]×(uk, r]×(uk+1, vk+1]× · · · ×(ud, vd]) +H((u1, v1]× · · · ×(uk−1, vk−1]×(r, vk]×(uk+1, vk+1]× · · · ×(ud, vd]).
This follows, however, directly from definition (2.4). To show the last claim letµbe a finitely additive set function onRand{Hu : u∈ Rd}be given by (2.5). The function H vanishes on the axes, i.e.Hu = 0for all u= (u1, . . . , ud)satisfyinguj = 0for some j= 1, . . . , d. Hence, by definition of the incrementsH((0∧u,0∨u]),u∈Rd, there is at most one non-zero term in the sum(2.4), namely whenc() =u, and in this case.=nu. That is,
H((0∧u,0∨u]) = (−1)nuHu=µ((0∧u,0∨u]).
Since any half-open interval(u, v]inRd can be expressed in terms of a finite number of intervals of the form(w∧0, w∨0], w∈Rd, using elementary set operations it follows by finite additivity ofH andµthatH((u, v]) =µ((u, v]).
Definition 2.4. A real-valued processX ={Xu : u∈ Rd} is said to have wide-sense stationary incrementsifX((u, v])∈L2R(P)for allu≤vinRd withE[X((u, v])] = 0and
E
X (u1+h, v1+h]
X (u2+h, v2+h]
=E
X (u1, v1]
X (u2, v2]
(2.6) for allh∈Rdandu1≤v1, u2≤v2inRd.
It is enough that (2.6) holds for all h ∈ Rd+ for X to have wide-sense stationary increments. To see this assume that (2.6) holds for allh∈Rd+and leth∈Rd be given.
Choose˜h∈ Rd such that ˜h≤0 andh˜ ≤h. An application of (2.6) with hreplaced by h−˜h∈Rd+and−h˜∈Rd+yields
E
X (u1+h, v1+h]
X (u2+h, v2+h]
=E
X (u1+ ˜h, v1+ ˜h]
X (u2+ ˜h, v2+ ˜h]
=E
X (u1, v1]
X (u2, v2] which shows that (2.6) holds for generalh∈Rd.
Remark 2.5. In this article we let increments be defined as in (2.4). However, when d≥ 2 an alternative way of defining an increment of H = {Hu : u ∈Rd} could be as Hv −Hu foru ≤ v, and this leads to the very different kind of wide-sense stationary increments studied e.g. by Yaglom [13]. In this context, notice that in contrast to the set function(u, v] 7→ H((u, v])we have in the cased≥2 that the set function(u, v] 7→
Hv−Huis only finitely additive whenH is constant.
Let us compare Yaglom’s definition to the one given above. A real-valued process X ={Xu :u∈Rd}for whichXv−Xu ∈L2R(P)andE[Xv−Xu] = 0for allu≤vinRd is said to have wide-sense stationary increments in Yaglom’s sense if
E[(Xv1+h−Xu1+h)(Xv2+h−Xu2+h)]
=E[(Xv1−Xu1)(Xv2−Xu2)], for allh∈Rdandu1≤v1, u2≤v2inRd. It is easily seen that this implies that X has wide-sense stationary increments in the sense of Definition 2.4. But conversely there are many processes with wide-sense stationary increments that do not have wide-sense stationary increments in Yaglom’s sense. One such example is the Brownian sheet, where increments over disjoint in- tervals are independent and X((u, v]) =D N(0, λd((u, v])) foru ≤ v, in the case d ≥ 2. See also Example 2.6 for another example. However, when d = 1the two definitions coincide.
The term ”wide-sense” refers to a property of the covariance function. In [1] (resp.
in [13], Definition 8.1.2) a process is said to have strict-sense stationary increments, where increments are defined asXv−Xu, if the finite dimensional distributions of the increments are invariant under translations (resp. under translations and rotations). In the following we only consider ”wide-sense” stationary increments.
Assume that X has wide-sense stationary increments in Yaglom’s sense. Yaglom [13], Remark 3, p. 295, shows that, up to addition of a random variable not depending onu,Xuis given by
Xu= Z
Rd
(eihz,ui−1) ˜Z(dz) +hV, ui, foru∈Rd, (2.7) where Z˜ = {Z(A) :˜ A ∈ Bb(Rd)} is an L2C(P)-valued random measure with control measureF˜ satisfying
Z
Rd
(kzk2∧1) ˜F(dz)<∞
andV is a random vector inRd. After a few calculations it follows that whend≥2, X((u, v]) =
Z
Rd
F1(u,v](z)Z(dz), foru < v, (2.8) whereZ(dz) =idz1· · ·zdZ(dz)˜ . That is, the control measureFofZisF(dz) =Qd
j=1z2jF˜(dz) which satisfies
Z
Rd
1∧ kzk2 Qd
j=1zj2F(dz)<∞.
Example 2.6. In some situations one can define X(A)not only forA= (u, v]but also for arbitrary bounded Borel sets inRd. In this case, if the mappingX:Bb(Rd)→L2R(P) isσ-additive one can in fact defineX(ϕ)for a large class of Borel functionsϕ:Rd→R, including all bounded Borel functions with compact support.
An important example of this appears in the theory of stochastic partial differential equations and is presented in Dalang [2], p. 5–6. LetX = {X(ϕ) :ϕ ∈DR(Rd)}be a centered Gaussian process with covariance function
E[X(ϕ)X(ψ)] = Z
Rd
Z
Rd
ϕ(x)ψ(y)g(x−y)dx dy, (2.9) whereg:Rd→[0,∞]is a locally integrable function such thatg=FµinSR0(Rd)for a tempered measureµ. By approximating with a sequence inDR(Rd)in the norm
kφkg=Z
Rd
Z
Rd
|φ(x)φ(y)|g(x−y)dx dy1/2
(2.10) one can defineX(ϕ)for any bounded Borel functionϕ:Rd→Rwith compact support by L2R(P)-continuity. PuttingX(A) = X(1A)forA ∈ Bb(Rd)it follows easily that the mapping(A ∈ Bb(Rd))7→X(A)∈L2R(P)isσ-additive. Finally, if we let {Xu :u∈Rd} be defined asXu= (−1)nuX((0∧u,0∨u])(cf. Remark 2.3) then the increment over any interval(u, v]with u < vin Rd is preciselyX((u, v]). The process {Xu : u ∈ Rd} has wide-sense stationary increments since
E[X(A+h)X(B+h)] = Z
A+h
Z
B+h
g(x−y)dx dy
= Z
A
Z
B
g(x−y)dx dy=E[X(A)X(B)]
for anyA, B∈ Bb(Rd)and anyh∈Rd.
In the next result we give the spectral representation of processes with wide-sense stationary increments. In this case it is natural to look for a representation as in (2.8) rather than (2.7). Recall that foru, v∈Rd withu < v,
F1(u,v](z) =
d
Y
j=1
(eivjzj−eiujzj
izj ), forz= (z1, . . . , zd)∈Rd, (2.11) where the right-hand side should be understood by continuity ifzj = 0for somej, i.e.
thej’th factor equalsvj−ujforzj= 0.
Theorem 2.7. LetX ={Xu:u∈Rd}be a real-valued process. ThenXhas wide-sense stationary increments and the mapping(u∈Rd+)7→X((0, u])is continuous inL2R(P)if and only if there is a symmetric measureF onRdsatisfying
Z
Rd d
Y
j=1
1
1 +zj2F(dz)<∞, (wherez= (z1, . . . , zd)), (2.12) and anL2C(P)-valued random measureZ with control measureF such that
X((u, v]) = Z
F1(u,v]dZ, foru < v. (2.13)
If this is the case then foru1< v1andu2< v2, E[X((u1, v1])X((u2, v2])] =
Z
F1(u1,v1]F1(u2,v2]dF. (2.14) The measuresFandZare uniquely determined byX. In addition,RC(Z) = spC{X((u, v]) : u≤v}andRR(Z) = spR{X((u, v]) :u≤v}.
The measure F above is called the spectral measure of X and Z is the random spectral measure ofX. The last statement in Theorem 2.7 shows thatZ is Gaussian if X is Gaussian.
IfXvanishes on the axes then by Remark 2.3Xu= (−1)nuX((0∧u,0∨u])foru∈Rd. In particular, in this case (2.13) implies that
Xu= Z
F1(0,u]dZ, for0< uinRd.
Proof. The ”if” part: Let F satisfy (2.12) andZ be anL2C(P)-valued random measure with control measureF. By (2.11)F1(u,v] ∈Le2C(F)thus making the right-hand side of (2.12) well-defined for allu < v inRd. Assume that the increments of X are given by (2.13) and notice that we have (2.14) as well by definition of the integral with respect toZ. Hence, since for arbitraryh∈Rdandui≤viinRdfori= 1,2,
F1(u1+h,v1+h]F1(u2+h,v2+h]=F1(u1,v1]F1(u2,v2],
it follows from (2.14) that (2.6) holds, that is,X has wide-sense stationary increments.
From (2.11) and (2.14) it follows that(u∈Rd+)7→X((0, u])is continuous inL2R(P). The ”only if” part: In the cased= 1the result goes back to [11] and [6]; see also Itô [4], Theorem 6.1. In the general case we follow Itô’s approach closely. More specifically, we first define three processesX(·), X(1)(·) and X1(·)as well as Z and F. Then we establish the fundamental formula (2.20) below and finally we prove (2.12)–(2.13).
Assume thatX has wide-sense stationary increments and the mapping(u∈Rd+)7→
X((0, u])is continuous inL2R(P). Define{X(ϕ) :ϕ∈DC(Rd)}as X(ϕ) =
Z
Rd
Xuϕ(u)du, forϕ∈DC(Rd),
where the integral is constructed in theL2C(P)-sense using thatu7→Xuϕ(u)isL2C(P)- continuous with compact support. Clearly,{X(ϕ) :ϕ∈DC(Rd)} constitutes a random distribution in the sense of Itô [4] or Yaglom [13].
Denote by D the differential operator ∂d/∂u1· · ·∂ud and define {X(1)(ϕ) : ϕ ∈ DC(Rd)}according to
X(1)(ϕ) = (−1)d Z
Rd
XuDϕ(u)du, forϕ∈DC(Rd).
Since, withe= (1, . . . ,1)∈Rddenoting the vector of ones, Dϕ(u) = lim
↓0ϕ((u−e, u])/d= lim
↓04eϕ(u−e)/d, foru∈Rd andϕ∈DC(Rd), we get, using the assumptions and linear change of variables together with the formula
Z
Rd
f(u)4hg(u)du= (−1)d Z
Rd4hf(u−h)g(u)du, h∈Rd+, (2.15) that forϕ∈DC(Rd)
(−1)d Z
Rd
XuDϕ(u)du= lim
↓0−d Z
Rd
X((u, u+e])ϕ(u)du, inL2C(P). (2.16) A key point is thatX(1)is stationary in the sense that
E[τhX(1)(ϕ)τhX(1)(ψ)] =E[X(1)(ϕ)X(1)(ψ)], forh∈Rd, ϕ, ψ∈DC(Rd),
where
τhX(1)(ϕ) =X(1)(ϕ(· −h)), forh∈Rd, ϕ∈DC(Rd).
To see this, leth∈Rd andϕ, ψ∈DC(Rd). Using (2.16) it follows that E[τhX(1)(ϕ)τhX(1)(ψ)]
= lim
↓0−2dE[ Z
Rd
X((u, u+e])ϕ(u−h)du Z
Rd
X((v, v+e])ψ(v−h)dv]
= lim
↓0−2dE[ Z
Rd
X((u+h, u+h+e])ϕ(u)du Z
Rd
X((v+h, v+h+e])ψ(v)dv]
= lim
↓0−2d Z
R2dE[X((u+h, u+h+e])X((v+h, v+h+e])]ϕ(u)ψ(v)dudv
= lim
↓0−2d Z
R2dE[X((u, u+e])X((v, v+e])]ϕ(u)ψ(v)dudv
=E[X(1)(ϕ)X(1)(ψ)].
Applying [13], Theorem 3, there exists an L2C(P)-valued random measure Z with symmetric control measureF satisfying
Z
Rd
1
(1 +kzk2)q F(dz)<∞, for someq≥1, (2.17) such that
X(1)(ϕ) = Z
Rd
Fϕ(z)Z(dz), forϕ∈DC(Rd).
LetDCp(Rd)denote the set ofϕ∈DC(Rd)of the form ϕ(z) =
d
Y
j=1
gj(zj) (2.18)
wheregj∈DC(R). Notice that for such aϕ,Fϕ(z) =Qd
j=1(F1gj)(zj). Following Itô [4], set
X1(ϕ) = Z
Rd
Gϕ(z)Z(dz), forϕ∈DCp(Rd), (2.19) where
Gϕ(z) = Z
Rd d
Y
j=1
eiujzj −1{|zj|≤1}
izj
ϕ(u)du.
Observe that forϕas in (2.18)Gϕ(z) =Qd
j=1G1gj(zj)where G1gj(zj) =
Z
R
eiujzj −1{|zj|≤1}
izj gj(uj)duj. SinceG1gj(zj)is bounded and
G1gj(zj) =F1gj(zj) izj
, for|zj|>1,
and thus tends to zero faster than any polynomial it follows, withqgiven in (2.17), that sup
z∈Rd
|Gϕ(z))|2(1 +kzk2)q <∞.
Hence, by (2.17), Gϕbelongs to L2C(F)making (2.19) well-defined. Maintaining the definition of the differential operatorD from above and using integration by parts we getG(Dϕ) = (−1)dFϕforϕ∈DCp(Rd), implying that
X(Dϕ) = (−1)dX(1)(ϕ) =X1(Dϕ), forϕ∈DCp(Rd), or equivalently,
X1(ϕ) =X(ϕ), forϕ∈D0p(Rd),
whereD0p(Rd)is the subspace ofDCp(Rd)consisting ofϕon the form (2.18) where, for j= 1, . . . , d,R
Rgj(zj)dzj = 0.
Forϕ∈DCp(Rd)andh∈Rd+we have4hϕ∈D0p(Rd)and thus X(4hϕ) =
Z
Rd
G(4hϕ)(z)Z(dz)
implying, since
G(4hϕ)(z) =Fϕ(z)
d
Y
j=1
e−ihjzj −1 izj ,
that
X(4hϕ) = Z
Rd
Fϕ(z)
d
Y
j=1
e−ihjzj−1 izj
Z(dz). (2.20)
SplittingRdinto disjoint sets according to the coordinates being numerically greater than or less than2πwe see, using thatF is finite on bounded sets, that equation (2.12) is equivalent to that
Z
CI
Y
j∈I
1
z2j F(dz)<∞ (2.21)
for each non-emptyI⊆ {1, . . . , d}, where
CI ={z∈Rd:|zj|>2πforj∈Iand|zj| ≤2πforj /∈I}.
To show (2.21) for a givenIwe argue as follows. By (2.20) we have forϕ∈DCp(Rd)and h∈Rd+that
kX(4hϕ)k2L2 C(P)=
Z
Rd
|Fϕ(z)|2
d
Y
j=1
|1−e−ihjzj izj
|2F(dz)
≥ Z
CI d
Y
j=1
|1−e−ihjzj|2|Fϕ(z)|2 F(dz) Qd
j=1z2j .
In particular this holds for everyϕn ∈DCp(Rd),n≥1, of the formϕn(z) =Qd
j=1gn(zj) forz∈Rd, wheregn ∈DR(R)+satisfies
gn(x) = 0, for|x| ≥1/n, and Z
R
gn(x)dx= 1.
In this case|F1gn(zj)| ≥1/2 for allzj satisfying|zj| ≤ 2πn/16; see [4], p. 221. Hence for arbitrarynandh∈Rd+,
kX(4hϕn)k2L2
R(P)≥(1/2)2d Z
CI
d
Y
j=1
|1−e−ihjzj|21{|zj|≤2πn/16}
zj2 F(dz). (2.22)
Following [4] integrate both sides with respect todh over the cube [0,1]d. Using the definition ofCI and the product structure the integral of the integrand on the right- hand side of (2.22) equals for eachz∈CI andn≥16
Y
j∈I
1 z2j
Z 1
0
|1−e−ihjzj|2dhj1{2π<|zj|≤2πn/16}Y
j /∈I
1 z2j
Z 1
0
|1−e−ihjzj|2dhj1{|zj|≤2π}. (2.23)
Now, there is a constantb >0such that for all|zj| ≤2πand0≤hj≤1/2
|1−e−ihjzj|2
|zj|2 ≥bh2j,
and, as shown on p. 221 in [4], there exists a constantc >0such that Z 1
0
|1−e−ihjzj|2dhj≥c
for all2π < |zj|. Hence, using that we get a smaller value by integrating over[0,1/2]
instead of over[0,1]forj 6∈I, it follows that the expression in (2.23) is greater than or equal to
Y
j∈I
c
|zj|21{2π<|zj|≤2πn/16}Y
j /∈I
b
241{|zj|≤2π}.
Inserting into (2.22) and applying monotone convergence (2.21) follows if sup
n≥1, h∈[0,1]d
kX(4hϕn)k2L2
R(P)<∞.
But using Jensen’s inequality we have, for alln≥1andh∈[0,1]d, kX(4hϕn)k2L2
R(P)=EhZ
Rd
Xu4hϕn(u)du 2i
=EhZ
Rd
(−1)d4hXu−hϕn(u)du 2i
≤ Z
RdE[(4hXu−h)2]ϕn(u)du≤ sup
u∈Rd, h∈[0,1]dE[(4hXu−h)2]
which is finite due to theL2R(P)-continuity and the stationary increments.
Leth=v−u. From (2.20) and (2.15) it follows that Z
Rd
4hXx−hϕn(x)dx= (−1)d Z
Rd
Fϕn(z)
d
Y
j=1
e−ihjzj −1 izj
Z(dz), (2.24)
for alln≥1, where(ϕn)n≥1⊆DCp(Rd)∩DR(Rd)+is a sequence satisfying Z
Rd
ϕn(x)dx= 1forn≥1 and ϕn(x)dx→δvweakly.
Asntends to infinity both sides of (2.24) converge inL2R(P)due to the continuity as- sumption onXand the integrability property (2.12) ofF, giving the identity
X((u, v]) =4v−uXu= Z
Rd
eihz,vi
d
Y
j=1
1−e−i(vj−uj)zj izj Z(dz)
= Z
Rd d
Y
j=1
eivjzj −eiujzj izj
Z(dz) = Z
Rd
F1(u,v](z)Z(dz)
which is (2.13).
To prove the last part notice that X and X(1) are in one-to-one correspondence, that spC{X((u, v]) : u ≤ v} = spC{X(1)(ϕ) : ϕ ∈ DC(Rd)}, and that there is a similar result with subscriptCreplaced byR. By construction (see [13] p. 281),Z is uniquely determined; moreover we have RC(Z) = spC{X(1)(ϕ) : ϕ ∈ DC(Rd)} as well as the corresponding result with subscriptCreplaced byR. This concludes the proof.
In connection with the integrability condition (2.12) on the spectral measureF we have the following inequalities for allz= (z1, . . . , zd)inRd:
1 1 +kzk2
d
≤
d
Y
j=1
1
1 +zj2 ≤ 1
1 +kzk2. (2.25)
This should compared with the integrability condition satisfied by general tempered measures cf. Lemma 3.4 below. The inequalities (2.25) can be shown as follows:
1 +kzk2= 1 +
d
X
j=1
z2j ≤ X
1,...,d∈{0,1}
d
Y
j=1
zj2j =
d
Y
j=1
(1 +zj2)
= X
1,...,d∈{0,1}
d
Y
j=1
zj2j ≤ X
1,...,d∈{0,1}
d
Y
j=1
kzk2j = 1 +kzk2d
.
The following corollary gives necessary and sufficient conditions for a Gaussian pro- cess with stationary increments to be of the form described in Example 2.6.
Corollary 2.8. LetX ={Xu:u∈Rd}be a centered Gaussian process with wide-sense stationary increments, spectral measureF and random spectral measureZ. IfFF is a positive locally integrable function then the process
X(φ) = Z
Fφ dZ, φ∈DR(Rd), (2.26) is of the form (2.9)withµ=F andg=FµinSR0(Rd).
Conversely, let {X(φ) : φ ∈ DR(Rd)} be of the form (2.9) and {Xu : u ∈ Rd} be the corresponding process with wide-sense stationary increments constructed in Exam- ple 2.6. Then{Xu:u∈Rd}has spectral measureF =µ, and therefore
Z d Y
j=1
1
1 +zj2µ(dz)<∞. (2.27) Proof. Assume that g = FF is a positive locally integrable function and let {X(φ) : φ∈ DR(Rd)} be given by (2.26). By elementary properties of Fourier transforms and convolutions
E[X(φ)X(ψ)] = Z
FφFψ dF = Z
Rd
Z
Rd
φ(x)ψ(y)g(x−y)dx dy
forφ, ψ∈DR(Rd), which shows the first part.
Conversely, consider the process {Xu : u ∈ Rd} with wide-sense stationary incre- ments constructed in Example 2.6 and let us show thatF = µ. Again by elementary properties of Fourier transforms and convolutions
E[X(φ)X(ψ)] = Z
FφFψ dµ, forφ, ψ∈DR(Rd). (2.28)
LetE be the set of simple functions onRddefined in (3.1) below. By (2.14) and linearity we have that
E[X(φ)X(ψ)] = Z
FφFψ dF, forφ, ψ∈ E. (2.29) For(φj)dj=1⊆DR(R)setφ(z) =Qd
j=1φj(zj)and φn(z) =
d
Y
j=1
φn,j(zj) where φn,j =
∞
X
k=−∞
φj(k−1n )1((k−1)/n,k/n].
Notice that(φn)n∈N ⊆ E andφ ∈ DR(Rd). By continuity ofφwe have thatφn → φin k · kg (see (2.10)) and hence X(φn) → X(φ)in L2R(P). According to (2.28)–(2.29) we have
Z
|Fφ|2dµ=E[X(φ)2] = lim
n→∞E[X(φn)2]
= lim
n→∞
Z
|Fφn|2dF = Z
|Fφ|2dF,
where in the last equality we have used thatFφn→ FφinL2C(F), cf. Lemma 3.6 below.
This proves thatµ=F. Finally, (2.27) follows from Theorem 2.7.
Example 2.9. Consider the case d = 1 and let X = {Xu : u ∈ R} be a fractional Brownian motion with Hurst exponent H ∈ (0,1). ThenX has absolutely continuous spectral measureF with density f: x 7→C|x|1−2H for someC > 0. By Corollary 2.8, X is of the form (2.9)if and only ifFF = Ff is a positive locally integrable function.
ForH ∈ (12,1),Ff = (x7→K|x|2H−2)for some constantK >0 and henceX is of the form (2.9). On the other hand,X is not of the form (2.9)whenH ∈(0,12]becauseFf is not a locally integrable function. Indeed, to show the last claim letrα:x7→C|x|αfor allα > −1. For the moment assume thatFrα is a locally integrable function. By the scaling property ofrα we have for allu∈ Rthat Frα(x) =|u|α+1(Frα)(ux)forλ1-a.e.
x, which implies thatFrα(x) =K0|x|−1−αforλ1-a.e.xand some constantK0 ∈R\ {0}. This shows thatFrαis not a locally integrable function whenα≥0. In particular,Ff is not a locally integrable function whenH ∈(0,12].
3 Deterministic integrands
Let X ={Xu : u∈Rd} be a real-valued process with wide-sense stationary incre- ments having spectral measure F satisfying (2.12) and random spectral measure Z. Assume furthermore thatF is absolutely continuous with respect toλdwith densityf. In the following we study classes of deterministic integrands with respect toX.
LetE be the set of simple functions onRd of the form ϕ=
n
X
j=1
αj1(uj,vj] (3.1)
where{αj} ⊆Rand{uj},{vj} ⊆Rd satisfyuj ≤vj for allj. Forϕ∈ E represented as in (3.1) define thesimple integral as
Z
ϕ dX :=
n
X
j=1
αjX((uj, vj]), (3.2)
and equip E with the normkϕkE := kR
ϕ dXkL2
R(P). Notice that the integral (3.2) is well-defined by finite additivity of the mapping(u, v] 7→X((u, v]), that is, R
ϕ dX does
not depend on the representation (3.1). By Theorem 2.7, Z
Rd
ϕ(u)dXu= Z
Rd
Fϕ(z)dZz and kϕk2E = Z
Rd
|Fϕ|2dF forϕ∈E. (3.3) Definition 3.1. A pseudo normed linear space(Λ,k · kΛ)containingE as a dense sub- space and satisfyingkϕkE =kϕkΛ forϕ∈ E is called a predomain forX. A domain is a complete predomain. Given a predomainΛ, there is a unique continuous linear map- pingR
·dX: Λ→L2R(P), extending the simple integral (3.2). This mapping is called the integral with respect toX.
Notice that Λ is not assumed to be a function space. By definition, a domain is a completion ofE and thus uniquely determined up to an isometric isomorphism. Below we give concrete examples of predomains and domains.
Remark 3.2. Using the completeness ofL2R(P)we see that a predomainΛis a domain if and only if
nZ
ϕ dX :ϕ∈Λo
= spR
X((u, v]) :u, v∈Rd, u≤v . (3.4) This emphasizes why domains are more attractable than predomains since for the latter we only have ”⊆” in(3.4).
For ease of reading we formulate two lemmas. The first generalizes Lemma 3.1 of [5] tod≥2. For the second see [9], Chapter VII, Théorème VII.
Lemma 3.3. Letϕ∈SC0(Rd)be given such thatFϕis a function. Thenϕ∈SR0(Rd)if and only ifFϕ(−x) =Fϕ(x)forλd-a.e. x.
Lemma 3.4. Letµbe a signed Borel measure on Rd. Thenµis a tempered measure, that isµ∈SC0(Rd)if
Z
Rd
(1 +kuk2)−k|µ|(du)<∞
for some positive integer k ≥ 1. This condition is also necessary if µ is a positive measure. In particular, a real-valued Borel functionhis a tempered distribution if, and in casehis non-negative only if,
Z
Rd
|h(u)|
(1 +kuk2)kdu <∞ for some positive integerk≥1.
In view of (3.3) it is natural to look for predomains consisting of objects for which a Fourier transform can be defined, that is spaces of distributions. Letϕbe a tempered distribution. If the Fourier transformFϕis a function, then this function is determined up to Lebesgue null sets and hence by absolute continuity ofF the following spaces are well-defined:
Λdist =n
ϕ∈SR0(Rd) :Fϕis a function such that Z
Rd
|Fϕ(z)|2F(dz)<∞o ,
Λfunc=n
ϕ∈L2R(Rd) : Z
Rd
|Fϕ(z)|2F(dz)<∞o .
Moreover, letΛdistandΛfuncbe equipped with the pseudo norms kϕk2Λfunc=
Z
Rd|Fϕ(z)|2F(dz), kϕk2Λdist = Z
Rd|Fϕ(z)|2F(dz).
Notice thatSR(Rd)⊆Λfunc⊆Λdist.
Theorem 3.5. (1)Λdistis a predomain forXand the integral onΛdistis given by Z
ϕ dX= Z
Fϕ dZ, ϕ∈Λdist. (3.5) (2)Λdistis a domain forX if and only if
∀g∈L2R(F)∃k∈N: Z
{f >0}
|g(u)|
(1 +kuk2)k du <∞. (3.6) In particular,Λdist is a domain forX if there existsk∈Nsuch that
Z
{f >0}
1
f(u)(1 +kuk2)k du <∞. (3.7) (3)Λfuncis a predomain, and it is a domain if and only if
L2R(F)⊆L2R(1{f(u)>0}du). (3.8) By Lemma 3.6 we further have thatΛdistis complete if and only ifF(Λdist) = ˜L2C(F). Proof. (1): Lemma 3.6 below implies that E is dense in Λdist showing together with (3.3) thatΛdist is a predomain forX. The continuous linear mappingϕ7→R
ϕ dX from Λdist to L2C(P) defined by (3.5) extends the simple integral by (3.3) and is hence the corresponding integral sinceL2R(P)is a closed subspace ofL2C(P).
(2): Assume that for allg ∈L2R(F), (3.6) holds for somekand let us show thatΛdist is a domain forX. Let{ϕn}be a Cauchy sequence inΛdist. By completeness ofL2C(F) there exists g ∈ L2C(F) with g = g(−·)such that Fϕn → g in L2C(F). Since we may assume that g = 0 on{f = 0}, (3.6) and Lemma 3.4 show that g ∈ SC0(Rd). Hence, using Lemma 3.3,ϕ :=F−1g is in Λdist and ϕn → ϕinΛdist which shows thatΛdist is complete.
Conversely, assume thatΛdist is complete. For contradiction consider anh∈L2R(F) which does not satisfy (3.6) with g replaced by h. Without loss of generality we may assume that h ≥ 0 and h = 0 on {f = 0}. By Lemma 3.4, h 6∈ SR0(Rd). Let h1 =
1
2(h+h(−·))andh2= 12(h−h(−·))be the even and odd parts ofhand setg=h1+ih2. By linearity, g ∈ L2C(F) and if g ∈ SC0(Rd) then h1, h2 ∈ SR0(Rd) which implies that h=h1+h2∈SR0(Rd). Thusg∈L2C(F)\SC0(Rd)and by constructiong=g(−·). SinceF is a tempered measure,SR(Rd)is dense inL2R(F)and therefore there exist sequences {ge,n}and{go,n}inSR(Rd)consisting of even and odd functions approximatingh1and h2 inL2R(F). Setting gn = ge,n+igo,n forn ≥ 1 we have {gn} ⊆ SC(Rd) ⊆ SC0(Rd) satisfyinggn =gn(−·)andgn →ginL2C(F). Thusϕn:=F−1gn is a Cauchy sequence in Λdist which does not converge.
The last statement in (2) follows since for any measurable functiong:Rd →R, we have by the Cauchy-Schwarz inequality that
Z
{f >0}
|g(u)|
(1 +kuk2)kdu≤Z
{f >0}
|g(u)|2f(u)du1/2Z
{f >0}
1 f(u)
1
(1 +kuk2)2kdu1/2 .
(3): Assume (3.8) and let {ϕn}be Cauchy in Λfunc. As in the proof of (2) there is a g∈L2C(F)withg=g(−·)and satisfyingg= 0on{f = 0}such thatFϕn →g inL2C(F). Sinceg∈L2C(Rd)we have by Lemma 3.3 thatϕ:=F−1gis inΛfuncandϕn→ϕinΛfunc, showing that the latter space is complete.
Conversely assume that Λfunc is complete. As in the proof of (2), if (3.8) is not satisfied there is a functiong :Rd →Csatisfyingg = 0on{f = 0}andg=g(−·)such thatg ∈L2C(F)\L2C(1{f(u)>0}du). Again as in (2) we can construct a sequence{gn}in L2C(Rd)∩L2C(F)satisfyinggn =gn(−·)such thatgn→ginL2C(F). Thenϕn:=F−1gnis a non-converging Cauchy sequence inΛfunc.
