Multiparameter processes with stationary increments: Spectral representation and integration

In this article,  a class of multiparameter processes with wide-sense  stationary increments 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 measure 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 completeness of the class  are given, and this property is linked to a certain martingale representation property.


Introduction
Let d ≥ 1 be an integer which is fixed throughout.In this article we consider a class of real-valued processes X = {X u : u ∈ R d } indexed by R d with wide-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. the d-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 predomains.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 the L 2 -distance between corresponding integrals.If completeness is present, a predomain is referred to as a domain.In the one-dimensional case * Aarhus University, Denmark.Email: basse,matseg,jan@imf.au.dk d = 1 several 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 distributions 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 integrand ϕ belonging to any of the predomains considered is given by where F denotes the Fourier transform and Z the random spectral measure of X.
As is obvious from (1.1) the integral is closely linked to the spectral representation of X. Therefore we study the spectral representation of X in 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 processes X = {X u : u = (t, x) ∈ R 1+d } where t ∈ R is time and x ∈ R d a spatial component.We assume that X is white in time and colored in space.A martingale integral with respect to X 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, when d = 1, X could be fractional in space with Hurst exponent H in (0, 1) in which case X only induces a martingale measure as in [2] when H > 1/2.We show that the integral of a predictable integrand ϕ t (x) with respect to X is where F denotes the Fourier transform in the space variable, and for fixed t, Z t (•) is the random spectral measure of X((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 to X. Definitions and notation: For any measure µ, L 2 C (µ) denotes the set of complex- valued µ-square integrable functions and L 2 R (µ) the subset hereof taking values in R.
Likewise, for any A ⊆ L 2 C (µ), sp C A is the closed complex linear span and sp R A the corresponding closed real linear span of A. Observe that sp R A coincides with the real-valued elements in sp C A if all elements in A are real-valued.According to usual notation the space of tempered distributions, that is the dual of the Schwartz space For the general theory of distributions and especially tempered distributions we refer to Schwartz [9].
Let λ d denote Lebesgue measure on R d .The Fourier transform F maps S C (R d ) onto S C (R d ) and with the usual identification of a locally integrable function with its corresponding tempered distribution when it exists, we have for f ∈ L 1 Here, •, • is the canonical inner product on R d with corresponding norm • .The notation differs from the one used e.g. in [9] where, for f ∈ L 1 as the Fourier transform of f .But apart from a constant (2π) d appearing in Parseval's identity and the explicit form of the inverse F −1 , all results from the general theory of distributions remain valid with the definition given above.When d = 1 we also use the notation F 1 instead of F.
All random variables are defined on a probability space (Ω, F, P) which is fixed throughout.Equality in distribution is denoted

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.
Remark 2.2.Let Z be a random measure as above.From (1) and (3) it follows that To see this, notice that by (2) in Definition 2.1, Hence, (2.1)-(2.3)follow by symmetry of the measure F and Definition 2.1(3).
Let Z be an L 2 C (P)-valued random measure with control measure F .As usual, inte- gration with respect to Z can be defined starting with simple functions and extending to L 2 C (F ) using the isometry condition Definition 2.1(3).Thus, the integral ϕ → ϕ dZ maps L 2 C (F ) linearly isometrically onto a closed subset of L 2 C (P) consisting of zero-mean random variables, and satisfies, for A ∈ B b (R d ) and ϕ, ψ ∈ L 2 C (F ), , and E ϕ dZ ψ dZ = ϕψ dF.
Denoting by R C (Z) the set of integrals ϕ dZ, ϕ ∈ L 2 C (F ), R R (Z) refers to the real- valued elements in R C (Z).With L2 C (F ) denoting the set of functions in L 2 C (F ) satisfying Indeed, the inclusion "⊇" follows from Definition 2.1(2) and "⊆" from the fact that for all ϕ ∈ L 2 C (F ) with integral equal to the real part of ϕ dZ. where Notice that H((u, v]) = 0 if u ≤ v and u < v.
where u ∧ 0 = (min(u 1 , 0), . . ., min(u d , 0)), u ∨ 0 = (max(u 1 , 0), . . ., max(u d , 0)) and n u is the number of coordinates in u that are strictly less than 0. Then µ((u, v]) = H((u, v]) To show the first claim it is enough to show that for all u ≤ v in R d , k = 1, . . ., d and r ∈ (u k , v k ) we have This follows, however, directly from definition (2.4).To show the last claim let µ be a finitely additive set function on R and {H u : u ∈ R d } be given by (2.5).The function H vanishes on the axes, i.e.H u = 0 for all u = (u 1 , . . ., u d ) satisfying u j = 0 for some j = 1, . . ., d.Hence, by definition of the increments H((0 ∧ u, 0 ∨ u]), u ∈ R d , there is at most one non-zero term in the sum (2.4), namely when c( ) = u, and in this case .= n u .
That is, (2.6) It is enough that (2.6) holds for all h ∈ R d + for X to have wide-sense stationary increments.To see this assume that (2.6) holds for all h ∈ R d + and let h ∈ R d be given.Choose h ∈ R d such that h ≤ 0 and h ≤ h.An application of (2.6) with h replaced by which shows that (2.6) holds for general h ∈ R d .
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 = {H u : u ∈ R d } could be as 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 intervals are independent and X((u, v]) See also Example 2.6 for another example.However, when d = 1 the 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 as X v − X u , 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 on u, X u is given by ) one can define X(ϕ) for any bounded Borel function ϕ : R d → R with compact support by L 2 3) then the increment over any interval (u, v] 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 for u, v ∈ R d with u < v, where the right-hand side should be understood by continuity if z j = 0 for some j, i.e. the j'th factor equals v j − u j for z j = 0.
Theorem 2.7.Let X = {X u : u ∈ R d } be a real-valued process.Then X has wide-sense stationary increments and the mapping (u ∈ R d and an L 2 The measures F and Z are uniquely determined by X.In addition, R C The measure F above is called the spectral measure of X and Z is the random spectral measure of X.The last statement in Theorem 2.7 shows that Z is Gaussian if X is Gaussian. If X vanishes on the axes then by Remark 2.3 X u = (−1) nu X((0∧u, 0∨u]) for u ∈ R d .
In particular, in this case (2.13) implies that Proof.The "if" part: Let F satisfy (2.12) and Z be an L 2 C (P)-valued random measure with control measure F .By (2.11) F1 (u,v] ∈ L 2 C (F ) thus making the right-hand side of (2.12) well-defined for all u < v in R d .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 to Z. Hence, since for arbitrary h ∈ R d and 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 ∈ R d The "only if" part: In the case d = 1 the 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 processes X(•), X (1) (•) and X 1 (•) as well as Z and F .Then we establish the fundamental formula (2.20) below and finally we prove (2.12)-(2.13).
Assume that X has wide-sense stationary increments and the mapping (u ∈ R d where the integral is constructed in the L 2 C (P)-sense using that u → X u ϕ(u) is L 2 C (P)- continuous with compact support.Clearly, {X(ϕ) : ϕ ∈ D C (R d )} constitutes a random distribution in the sense of Itô [4] or Yaglom [13].
Denote by D the differential operator ∂ d /∂u 1 • • • ∂u d and define {X (1) Since, with e = (1, . . ., 1) ∈ R d denoting the vector of ones, we get, using the assumptions and linear change of variables together with the formula (2.16) A key point is that X (1) is stationary in the sense that where Applying [13], Theorem 3, there exists an L 2 where g j ∈ D C (R).Notice that for such a ϕ, Fϕ(z) = d j=1 (F 1 g j )(z j ).Following Itô [4], set where Observe that for ϕ as in (2.18) Gϕ(z) Since G 1 g j (z j ) is bounded and and thus tends to zero faster than any polynomial it follows, with q given in (2.17), that sup Hence, by (2.17), Gϕ belongs to L 2 C (F ) making (2.19) well-defined.Maintaining the definition of the differential operator D from above and using integration by parts we get G(Dϕ) = (−1) or equivalently, ) consisting of ϕ on the form (2.18) where, for j = 1, . . ., d, R g j (z j ) dz j = 0.
Splitting R d into disjoint sets according to the coordinates being numerically greater than or less than 2π we see, using that F is finite on bounded sets, that equation (2.12) is equivalent to that To show (2.21) for a given I we argue as follows.By (2.20) we have for ϕ ∈ D p In particular this holds for every ϕ n ∈ D p g n (x) = 0, for |x| ≥ 1/n, and R g n (x) dx = 1.
In this case |F 1 g n (z j )| ≥ 1/2 for all z j satisfying |z j | ≤ 2πn/16; see [4], p. 221.Hence for arbitrary n and h (2.22) Following [4] integrate both sides with respect to dh over the cube [0, 1] d .Using the definition of C I and the product structure the integral of the integrand on the righthand side of (2.22) equals for each z ∈ C I and n ≥ 16 Now, there is a constant b > 0 such that for all |z j | ≤ 2π and 0 ≤ h j ≤ 1/2 and, as shown on p. 221 in [4], there exists a constant c > 0 such that But using Jensen's inequality we have, for all n ≥ 1 and h ∈ which is finite due to the L 2 R (P)-continuity and the stationary increments.
Let h = v − u.From (2.20) and (2.15) it follows that for all n ≥ 1, where As n tends to infinity both sides of (2.24) converge in L 2 R (P) due to the continuity as- sumption on X and the integrability property (2.12) of F , giving the identity which is (2.13).
To prove the last part notice that X and X (1) are in one-to-one correspondence, that sp C {X((u, v]) : u ≤ v} = sp C {X (1) (ϕ) : ϕ ∈ D C (R d )}, and that there is a similar result with subscript C replaced by R. By construction (see [13] p. 281), Z is uniquely determined; moreover we have R C (Z) = sp C {X (1) (ϕ) : ϕ ∈ D C (R d )} as well as the corresponding result with subscript C replaced by R.This concludes the proof.
In connection with the integrability condition (2.12) on the spectral measure F we have the following inequalities for all z = (z 1 , . . ., z d ) in R d : 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: The following corollary gives necessary and sufficient conditions for a Gaussian process with stationary increments to be of the form described in Example 2.6.(2.29) R (P).According to (2.28)-( 2.29) we have where in the last equality we have used that Fφ n → Fφ in L 2 C (F ), cf.Lemma 3.6 below.
Example 2.9.Consider the case d = 1 and let X = {X u : u ∈ R} be a fractional Brownian motion with Hurst exponent H ∈ (0, 1).Then X has absolutely continuous spectral measure F with density f : x → C|x| 1−2H for some C > 0. By Corollary 2.8, X is of the form (2.9) if and only if FF = Ff is a positive locally integrable function.For H ∈ ( 1 2 , 1), Ff = (x → K|x| 2H−2 ) for some constant K > 0 and hence X is of the form (2.9).On the other hand, X is not of the form (2.9) when H ∈ (0, 1  2 ] because Ff is not a locally integrable function.Indeed, to show the last claim let r α : x → C|x| α for all α > −1.For the moment assume that Fr α is a locally integrable function.By the scaling property of r α we have for all u ∈ R that Fr α (x) = |u| α+1 (Fr α )(ux) for λ 1 -a.e.
This shows that Fr α is not a locally integrable function when α ≥ 0. In particular, Ff is not a locally integrable function when H ∈ (0, 1  2 ].

Deterministic integrands
Let X = {X u : u ∈ R d } be a real-valued process with wide-sense stationary increments having spectral measure F satisfying (2.12) and random spectral measure Z.
Assume furthermore that F is absolutely continuous with respect to λ d with density f .In the following we study classes of deterministic integrands with respect to X.
Let E be the set of simple functions on R d of the form where {α j } ⊆ R and {u j }, {v j } ⊆ R d satisfy u j ≤ v j for all j.For ϕ ∈ E represented as in (3.1) define the simple integral as and equip E with the norm ϕ E := ϕ dX L 2 R (P) .Notice that the integral (3.2) is well-defined by finite additivity of the mapping (u, v] → X((u, v]), that is, ϕ dX does not depend on the representation (3.1).By Theorem 2.7, Definition 3.1.A pseudo normed linear space (Λ, • Λ ) containing E as a dense subspace and satisfying ϕ E = ϕ Λ for ϕ ∈ E is called a predomain for X.A domain is a complete predomain.Given a predomain Λ, there is a unique continuous linear mapping • dX : Λ → L 2 R (P), extending the simple integral (3.2).This mapping is called the integral with respect to X.
Notice that Λ is not assumed to be a function space.By definition, a domain is a completion of E and thus uniquely determined up to an isometric isomorphism.Below we give concrete examples of predomains and domains.Remark 3.2.Using the completeness of L 2 R (P) we see that a predomain Λ is a domain if and only if 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] to d ≥ 2. For the second see [9], Chapter VII, Théorème VII.
for some positive integer k ≥ 1.This condition is also necessary if µ is a positive measure.In particular, a real-valued Borel function h is a tempered distribution if, and in case h is non-negative only if, 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.Let ϕ be a tempered distribution.If the Fourier transform Fϕ is a function, then this function is determined up to Lebesgue null sets and hence by absolute continuity of F the following spaces are well-defined: Moreover, let Λ dist and Λ func be equipped with the pseudo norms Theorem 3.5.(1) Λ dist is a predomain for X and the integral on Λ dist is given by ϕ dX = Fϕ dZ, ϕ ∈ Λ dist . ( (2) Λ dist is a domain for X if and only if In particular, Λ dist is a domain for X if there exists k ∈ N such that (3) Λ func is a predomain, and it is a domain if and only if By Lemma 3.6 we further have that Λ dist is complete if and only if F(Λ dist ) = L2 C (F ). Proof.
(1): Lemma 3.6 below implies that E is dense in Λ dist showing together with (3.3) that Λ dist is a predomain for X.The continuous linear mapping ϕ → ϕ dX from Λ dist to L 2 C (P) defined by (3.5) extends the simple integral by (3.3) and is hence the corresponding integral since L 2 R (P) is a closed subspace of L 2 C (P).
(2): Assume that for all g ∈ L 2 R (F ), (3.6) holds for some k and let us show that Λ dist is a domain for X.Let {ϕ n } be a Cauchy sequence in Λ dist .By completeness of L 2 there exists g ∈ L 2 C (F ) with g = g(−•) such that Fϕ n → g in L 2 C (F ).Since we may assume that g = 0 on {f = 0}, (3.6) and Lemma 3.4 show that g ∈ S C (R d ).Hence, using Lemma 3.3, ϕ := F −1 g is in Λ dist and ϕ n → ϕ in Λ dist which shows that Λ dist is complete.
Conversely, assume that Λ dist is complete.For contradiction consider an h ∈ L 2 R (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 ∈ S R (R d ).Let h 1 = 1 2 (h + h(−•)) and h 2 = 1 2 (h − h(−•)) be the even and odd parts of h and set g = h 1 + ih 2 .By linearity, g ∈ L 2 and by construction g = g(−•).Since F is a tempered measure, S R (R d ) is dense in L 2 R (F ) and therefore there exist sequences {g e,n } and {g o,n } in S R (R d ) consisting of even and odd functions approximating h 1 and C (F ).Thus ϕ n := F −1 g n is a Cauchy sequence in Λ dist which does not converge.
The last statement in (2) follows since for any measurable function g : R d → R, we have by the Cauchy-Schwarz inequality that (3): Assume (3.8) and let {ϕ n } be Cauchy in Λ func .As in the proof of (2) there is a g ∈ L 2 C (F ) with g = g(−•) and satisfying g = 0 on {f = 0} such that Fϕ n → g in L 2 C (F ).
Since g ∈ L 2 C (R d ) we have by Lemma 3.3 that ϕ := F −1 g is in Λ func and ϕ n → ϕ in Λ func , showing that the latter space is complete.
Proof.For simplicity assume that supp(φ j ) ⊆ (0, 1) for all j.For all n, j and t ∈ R \ {0} Hence, denoting by K j the total variation of φ j on [0, 1], Furthermore, for all j, pointwise by (3.9), and which by dominated convergence implies that Fφ n → Fφ in L 2

C (F ).
Let G be the real linear span of functions φ of the form φ(u) = d j=1 φ j (u j ) where {φ j } ⊆ D R (R).Since (Fφ)(z) = d j=1 (Fφ j )(z j ) it follows by arguments as in [4], Theorem 4.1, that F(G) is dense in L2 C (F ).To show that F(E ) is dense in L2 C (F ) it is hence enough to show that for all φ ∈ G there exists a sequence {φ n } ⊆ E such that Fφ n → Fφ in L 2 C (F ) which, however, follows by the above.

Stochastic integrands for processes white in time and colored in space
In the following we add a temporal component; that is, we consider processes indexed by R 1+d rather than R d .A generic element u ∈ R 1+d will be decomposed as u = (t, x) where t ∈ R is time and x = (x 1 , . . ., x d ) ∈ R d a space variable.Intervals in R 1+d will be written either as (u, v] or (s, t] × (x, y] where (s, t] is an interval in R and (x, y] is an interval in R d .Functions on R 1+d will often be denoted by ϕ t (x) for t ∈ R and x ∈ R d , and Fϕ t (z) denotes the Fourier transform in the space variable for fixed t.
Let F denote a symmetric measure on R d with density f with respect to λ d satisfying f (x) = f (−x) for all x ∈ R d .Assume throughout that F satisfies condition (2.12).The measure λ 1 × F on R 1+d then satisfies (2.12) as well.Consider an L 2 R (P)-continuous Gaussian process X = {X u : u ∈ R 1+d } with wide-sense stationary increments and spectral measure λ 1 × F .By Parseval's identity and (2.14) we have, for s i < t i (in R) Thus, there are independent increments in time, and the correlation in space is determined by F .That is, X is white in time but colored in space.
From now on we consider only time points in R + .Notice that in general X((0, t] × •) may not extend to an L 2 R (P)-valued measure defined on B b (R d ) as in [2].As an example, if d = 1 and X is fractional in space we have f (x) = C|x| 1−2H for x ∈ R (where H ∈ (0, 1)) for some constant C > 0. In this case X extends to a measure in space as in [2] if and only if H > 1/2 cf.Example 2.9.We shall see that one can nevertheless define a martingale integral with respect to X; moreover, we show that the set of integrands forms a complete space if and only if Λ dist is complete.For t ≥ 0 let Z t = {Z t (A) : A ∈ B b (R d )} denote the random spectral measure of X((0, t] × •).Notice that by (4.1) the latter has spectral measure 2πtF .
Define the filtration G = (G t ) t≥0 as where N denotes the set of P-null sets.A standard argument based on the stationary independent increments in X shows that G is right-continuous.For fixed u and v the process {X((0, t] × (u, v]) : t ≥ 0} is a (non-standard) Brownian motion with respect to G. By the last property in Theorem 2. (2) Let A, B ∈ B b (R d ).The processes {Z 1 t (A) : t ≥ 0} and {Z 2 t (B) : t ≥ 0} are independent Brownian motions with respect to G.Moreover, {(Z j t (A), Z j t (B)) : t ≥ 0} is a bivariate Brownian motion with respect to G and E[Z j t (A)Z j t (B)] = c j (A, B)t for j = 1, 2.
Notice that Λ X is Dalang's space P considered in [2], page 9, with a few modifications: We consider the time interval [0, ∞) rather than [0, T ] and as mentioned above our X does in general not induce a martingale measure.For ϕ ∈ Λ X define By the above lemma, the integral is well-defined and maps Λ X into L 2 R (P).On Λ X define the norm • Λ X as The integral with respect to X just defined extends the simple integral since if ϕ is given by (4.2) then, by definition, where the second equality is due to (2.13).Moreover, if ψ : R + → S R (R d ) is a deterministic measurable mapping, then the integral ∞ 0 R d ψ dX exists if and only if ψ ∈ Λ X , that is, Fψ t is a function satisfying R 1+d |Fψ t (x)| 2 dtF (dx) < ∞.Thus, in view of the the first part of Theorem 4.3 below, this improves Theorem 3 in [2].Proof.The first part: Using Lemma 3.3 and Lemma 4.2 it suffices to show that whenever ϕ ∈ L 2 C (Z) is of the form ϕ t (ω, x) = 1 F (ω)1 (s1,s2] (t)ψ(x), where ψ ∈ L2 C (F ) then there is a sequence (ϕ n ) n≥1 in L 2 C (Z) of the form ϕ n,t (ω, x) = 1 F (ω)1 (s1,s2] (t)Fψ n (x) where ψ n ∈ E (see (3.1)) approximating ϕ in Λ X .However, this follows since by Lemma 3.6 the ψ n 's can be chosen such that Fψ n approximates ψ in L 2 C (F ).

Definition 2 . 1 .
Let F be a symmetric Borel measure on R d finite on compacts.A set function Z : B

Example 2 . 6 .
In some situations one can define X(A) not only for A = (u, v] but also for arbitrary bounded Borel sets in R d .In this case, if the mapping X :B b (R d ) → L 2 R (P) is σ-additive one canin fact define X(ϕ) for a large class of Borel functions ϕ : R d → 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.Let X = {X(ϕ) : ϕ ∈ D R (R d )} be a centered Gaussian process with covariance function

Corollary 2 . 8 .
Let X = {X u : u ∈ R d } be a centered Gaussian process with wide-sense stationary increments, spectral measure F and random spectral measure Z.If FF is a positive locally integrable function then the process X(φ) = Fφ dZ, φ ∈ D R (R d ),
Conversely, let {X(φ) : φ ∈ D R (R d )} be of the form (2.9) and {X u : u ∈ R d } be the corresponding process with wide-sense stationary increments constructed in Example 2.6.Then {X u : u ∈ R d } has spectral measure F = µ, and therefore Assume that g = FF is a positive locally integrable function and let {X(φ) :φ ∈ D R (R d )} be given by(2.26).By elementary properties of Fourier transforms and convolutionsE[X(φ)X(ψ)] = FφFψ dF = R d R d φ(x)ψ(y)g(x − y) dx dy for φ, ψ ∈ D R (R d ),which shows the first part.Conversely, consider the process {X u : u ∈ R d } with wide-sense stationary increments constructed in Example 2.6 and let us show that F = µ.Again by elementary properties of Fourier transforms and convolutions E[X(φ)X(ψ)] = Fφ Fψ dµ, for φ, ψ ∈ D R (R d ).

Remark 3 . 7 . 1 .
Theorem 3.5(2)-(3) underlines that it is easier for Λ dist than for Λ func to be a domain.As an illustration, consider the fractional Brownian sheet, which corresponds to X being Gaussian and f (u) = d j=1 c Hj |u j | 1−2Hj , where H 1 , . . ., H d ∈ (0, 1) and c Hj > 0 are constants, see e.g.[10].Since f satisfies (3.7), Theorem 3.5 shows that Λ dist is complete.Moreover, by Theorem 3.5 it follows that Λ func is complete if and only if H 1 = • • • = H d = 1 2 , that is, X is a Brownian sheet.In the case d = 1, where X is a fractional Brownian motion, a quite long proof of the non-completeness of Λ func can be founded in Taqqu and Pipiras [7], Theorem 3.1, and the completeness of Λ dist is shown by Jolis [5], Proposition 4.Remark 3.8.Λ dist is not always a domain.For instance, if d = 1 and f (u) = e −u 2 then g(u) = e u belongs to L 2 (F ) but R |g(u)|(1 + u 2 ) −k du = ∞ for all k ∈ N. Hence, by Theorem 3.5(2) Λ dist is not a domain.
7 it follows thatG t = σ{Z s (A) : s ≤ t, A ∈ B b (R d )} ∨ N .Let us first describe the basic properties of Z = {Z t (A) : t ≥ 0, A ∈ B b (R d )}.For A, B ∈ B b (R d ) let c 1 (A, B) = π[F (A ∩ B) + F (A ∩ (−B))] and c 2 (A, B) = π[F (A ∩ B) − F (A ∩ (−B))].Decompose Z t (A) into the real and imaginary parts as Z t (A) = Z 1 t (A) + iZ 2 t (A).Proposition 4.1.(1) The process Z is Gaussian and has stationary independent increments in the sense that Z t − Z s = {Z t (A) − Z s (A) : A ∈ B b (R d )} is independent of G s and Z t − Z s D = Z t−s for all 0 ≤ s < t.

Theorem 4 . 3 . 2 R
The real linear span of processes given by (4.2) is dense in Λ X .Moreover, the following three statements are equivalent: (a) f satisfies (3.6),(b) Λ X equipped with the norm • Λ X is complete, (c) for every G ∞ -measurable random variable V ∈ L (P) there is a ϕ ∈ Λ X such that V = E[V ] +

Remark 2.3. The set function H defined in (2.4) on the semi-ring
Later we shall occasionally write h H(u) for H((u, u + h]) for u ∈ R d and any h ∈ R d+ .R := {(u, v] : u ≤ v in R d }

Since any half-open interval (u, v] in R d can be expressed in terms of a finite number of intervals of the form
(w ∧ 0, w ∨ 0], w ∈ R d ,

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] → H((u, v]) we have in the case d ≥ 2 that the set function (u, v] → H v − H u is only finitely additive when H is constant.Let us compare Yaglom's definition to the one given above.A real-valued process

1 0
|1 − e −ihj zj | 2 dh j ≥ c for all 2π < |z j |.Hence, using that we get a smaller value by integrating over [0, 1/2] instead of over [0, 1] for j ∈ I, it follows that the expression in (2.23) is greater than or Inserting into (2.22) and applying monotone convergence (2.21) follows if