Radonifying operators and infinitely divisible Wiener integrals
aa r X i v : . [ m a t h . P R ] J un Radonifying operatorsandinfinitely divisible Wiener integrals
Markus Riedle ∗ Department of MathematicsKing’s College LondonLondon WC2R 2LSUnited KingdomJune 16, 2018
Dedicated to the 50-th anniversary of theDepartment of Theory of Random ProcessesInstitute of Mathematics National Academy of Sciences of Ukraine
Abstract
In this article we illustrate the relation between the existence of Wiener integralswith respect to a L´evy process in a separable Banach space and radonifying operators.For this purpose, we introduce the class of ϑ -radonifying operators, i.e. operators whichmap a cylindrical measure ϑ to a genuine Radon measure. We study this class ofoperators for various examples of infinitely divisible cylindrical measures ϑ and highlightthe differences from the Gaussian case. Starting with the work by Gel’fand [8], Gross [10] and Segal [23] the canonical Gaussiancylindrical measure has gained much attention in different areas of mathematics and appli-cations. It is not only of interest from a theoretical point of view but it is also of importancein various applications such as filtering problems in Bensoussan [2], small ball probabilitiesin Li and Linde [12], interest rate models in Carmona and Tehranchi [6] and stochasticintegration in Banach spaces in van Neerven, Veraar and Weis [26]. ∗ The author acknowledges the EPSRC grant EP/I036990/1
1n his seminal work [10] Gross studies norms on a Hilbert space H such that the canonicalGaussian cylindrical measure γ extends to a σ -additive probability measure on the comple-tion of H with respect to the norm. This directly leads to the class R ( γ ) of γ -radonifyingoperators, which consists of linear and bounded operators T from H to a Banach space V such that the cylindrical image measure γ ◦ T − extends to a σ -additive probability mea-sure. The space R ( γ ) is known to have many desirable properties, such as the completenessunder an appropriate norm, the ideal property and close relations to absolutely summingoperators.Recently, the space of γ -radonifying operators plays a fundamental role in the theoryof stochastic integration in Banach spaces. In [26], van Neerven, Veraar and Weis developa theory of stochastic integration for random operator-valued integrands with respect tocylindrical Wiener processes in UMD Banach spaces. Their approach is strongly based onthe corresponding Wiener integrals for deterministic integrands introduced in [5] and [27],and those existence is naturally closely related to the class of γ -radonifying operators.In our work [20], we extend the approach in [27] to Wiener integrals for deterministicintegrands with respect to martingale-valued measures, in particular to L´evy processes.However, this work [20] was accomplished under the constraint not being able to use anyof the fundamental properties of the space of γ -radonifying operators since the analoguetheory was not developed in a non-Gaussian setting. It became apparent, that if one wouldlike to develop a theory of stochastic integration for random integrands similarly to theone in [26] but for L´evy processes, one needs to study an analog class of operators as γ -radonifying operators but radonifying an infinitely divisible cylindrical measure. This isthe main motivation of this work where we show that one can introduce such a space ofoperators although it lacks many of the fundamental properties of γ -radonifying operators.The canonical Gaussian cylindrical measure γ is distinguished among all Gaussian cylin-drical measures by its characteristic function, which most often serves also as its definition.Equivalently, starting from a Gaussian cylindrical random variable X in an arbitrary Banachspace V with covariance operator Q , one can explicitly construct a cylindrical random vari-able Θ in the reproducing kernel Hilbert space of Q whose cylindrical distribution equals thecanonical Gaussian cylindrical distribution γ . This construction is based on the Karhunen-Lo`eve expansion of X . We show in the first part of this work, that this construction of acanonically Gaussian distributed cylindrical random variable Θ on the reproducing kernelHilbert space can be mimicked for each cylindrical random variable with second moments.Denoting the cylindrical distribution of Θ by ϑ , this construction motivates us to define theclass R ( ϑ ) of ϑ -radonifying operators in analogy to γ -radonifying operators as the spaceof operators T such that the image cylindrical measure ϑ ◦ T − extends to a σ -additiveprobability measure on the Borel σ -algebra.The class R ( ϑ ) of ϑ -radonifying operators is only well studied if ϑ equals the canonicalGaussian cylindrical measure γ or a canonical stable cylindrical measure. In this work weshow that for an arbitrary cylindrical measure ϑ the linear space R ( ϑ ) can be equippedwith a certain norm, introduced in this work, such that it becomes complete. However,already in the case of a canonical stable cylindrical measure ϑ , which might be consideredas a non-Gaussian cylindrical measure most similar to the canonical Gaussian cylindricalmeasure γ , it is known that the space R ( ϑ ) lacks many of the desirable properties of thespace of γ -radonifying operators. We study the linear space R ( ϑ ) for different examples ofinfinitely divisible cylindrical measures ϑ and compare it to the Gaussian situation.In the last part of this work, we illustrate the relation of ϑ -radonifying operators and theexistence of Wiener integrals with respect to a L´evy process. Although this is the underlyingidea in the work [27] and to some extent in the generalisation [20], we are able to illustratethis relation more explicitly by defining a cylindrical integral, which in the case of stochastic2ntegrability is induced by a genuine Banach space valued random variable. In particularfor L´evy driven integrals, this rigorous relation between stochastic integrability and Banachspace valued operators is novel, and it signficantly improves the description of integrableoperators in [20]. Throughout this paper, V is a separable Banach space with dual V ∗ and dual pairing h· , ·i .The Borel σ -algebra is denoted by B ( V ). If U is another separable Banach space the spaceof bounded and linear operators is denoted by L ( U, V ) equipped with the uniform operatornorm k·k U → V . An operator T ∈ L ( U, V ) is called p -absolutely summing if there exists aconstant c > n ∈ N and u , . . . , u n ∈ U it obeys n X k =1 k T u k k p c p sup k u ∗ k n X k =1 |h u k , u ∗ i| p . (2.1)The space of all p -absolutely summing operators is denoted by Π p ( U, V ) and it is a Banachspace under the norm k T k Π p := π p ( T ) where π p ( T ) is the smallest constant c satisfying(2.1).For a measurable space ( S, S , m ) and p > L pm ( S ; V ). A probability space is denoted by (Ω , A , P ) and L P (Ω; R ) denotes the spaceof equivalence classes of measurable functions equipped with the topology of convergence inprobability.For every v ∗ , . . . , v ∗ n ∈ V ∗ and n ∈ N we define a linear map π v ∗ ,...,v ∗ n : V → R n , π v ∗ ,...,v ∗ n ( v ) = (cid:0) h v, v ∗ i , . . . , h v, v ∗ n i (cid:1) . For n ∈ N and B ∈ B ( R n ), sets of the form C ( v ∗ , . . . , v ∗ n ; B ) : = { v ∈ V : ( h v, v ∗ i , . . . , h v, v ∗ n i ) ∈ B } = π − v ∗ ,...,v ∗ n ( B )are called cylindrical sets . If D is a subset of V ∗ then Z ( V, D ) := n π − v ∗ ,...,v ∗ n ( B ) : v ∗ , . . . , v ∗ n ∈ D, B ∈ B ( R n ) , n ∈ N o , defines the cylindrical algebra generated by D . The generated σ -algebra is denoted by C ( V, D )and it is called the cylindrical σ -algebra with respect to ( V, D ). If D = V ∗ we write Z ( V ) := Z ( V, D ) and C ( V ) := C ( V, D ).A function η : Z ( V ) → [0 , ∞ ] is called a cylindrical measure on Z ( V ) if for each finitesubset D ⊆ V ∗ the restriction of η to the σ -algebra C ( V, D ) is a measure. A cylindricalmeasure η is called finite if η ( V ) < ∞ and a cylindrical probability measure if η ( V ) = 1.The characteristic function ϕ η of a finite cylindrical measure η is defined by ϕ η : V ∗ → C , ϕ η ( v ∗ ) := Z V e i h v,v ∗ i η ( dv ) . We will always assume that the characteristic function is continuous, in which case thecylindrical measure η is called continuous . A cylindrical measure η has p -th weak momentsif Z V |h v, v ∗ i| p η ( dv ) < ∞ for all v ∗ ∈ V ∗ . η is of cotype p if it has p -th weak moments and for each sequence( v ∗ n ) n ∈ N ⊆ V ∗ the condition Z V |h v, v ∗ n i| p η ( dv ) → n → ∞ , implies that k v ∗ n k → cylindrical random variable Z in V is a linear and continuous map Z : V ∗ → L P (Ω; R ) . The cylindrical random variable Z has weak p -th moments if E [ | Zv ∗ | p ] < ∞ for all v ∗ ∈ V ∗ .In this case, the closed graph theorem implies that Z : V ∗ → L pP (Ω; R ) is continuous. Thecharacteristic function of a cylindrical random variable Z is defined by ϕ Z : V ∗ → C , ϕ Z ( v ∗ ) = E (cid:2) exp( iZv ∗ ) (cid:3) . By defining for each cylindrical set C = C ( v ∗ , . . . , v ∗ n ; B ) ∈ Z ( V ) the mapping η Z ( C ) := P (cid:0) ( Zv ∗ , . . . , Zv ∗ n ) ∈ B (cid:1) , we obtain a cylindrical probability measure η Z , which is called the cylindrical distributionof Z . The characteristic functions ϕ η Z and ϕ Z of η Z and Z coincide. Conversely, for everycylindrical probability measure η on Z ( V ) there exist a probability space (Ω , A , P ) and acylindrical random variable Z : V ∗ → L P (Ω; R ) such that η is the cylindrical distribution of Z ; see [24, VI.3.2].A cylindrical random variable Z : V ∗ → L P (Ω; R ) is called induced by a random variablein L pP (Ω; V ) if there exists Y ∈ L pP (Ω; V ) such that h Y, v ∗ i = Zv ∗ for all v ∗ ∈ V ∗ . This is equivalent to the fact that the cylindrical distribution of Z extends to a probabilitymeasure on B ( V ); see Theorem IV.2.5 in [24]. The class of infinitely divisible cylindrical probability measures is introduced in [17]. Acylindrical probability measure η on Z ( V ) is called infinitely divisible if for each k ∈ N thereexists a cylindrical probability measure η k such that η = η ∗ kk . Theorem 3.13 in [17] showsthat a cylindrical probability measure η is infinitely divisible if and only if η ◦ π − v ∗ ,...,v ∗ m is infinitely divisible on B ( R m ) for all v ∗ , . . . , v ∗ m ∈ V ∗ and m ∈ N .In this equivalent description it is not sufficient only to take n = 1 as it is shown even inthe case V = R in [14] and [9].Let X be an infinitely divisible cylindrical random variable, that is its cylindrical distri-bution is infinitely divisible, and assume that X has weak second moments and E [ Xv ∗ ] = 0for all v ∗ ∈ V ∗ . Define the covariance operator by Q : V ∗ → V ∗∗ , h Qv ∗ , w ∗ i = E (cid:2) ( Xv ∗ )( Xw ∗ ) (cid:3) . The range of Q , i.e. the continuity of Qv ∗ : V ∗ → R , follows from the Cauchy-Schwarzinequality and the continuity of X : V ∗ → L P (Ω; R ).4or the following we assume that Q is V -valued. This is guaranteed for example if X is agenuine random variable (see Theorem III.2.1 in [24]), in which case we set Xv ∗ = h X, v ∗ i .Other examples of a V -valued covariance operator will be seen later in Section 5. As thecovariance operator Q : V ∗ → V is positive and symmetric, it follows that there exists aHilbert space H and j ∈ L ( H, V ) such that Q = jj ∗ and j ( H ) is dense in V . Moreover, theHilbert space H is unique up to isomorphism and separable as V is separable; see SectionIII.1.2 in [24].Since the range of j ∗ is dense in H we can choose an orthonormal basis ( e k ) k ∈ N of H with e k ∈ j ∗ ( V ∗ ). Thus, there exist some elements v ∗ k ∈ V ∗ obeying j ∗ v ∗ k = e k for all k ∈ N .It follows that Xv ∗ = ∞ X k =1 h je k , v ∗ i Xv ∗ k for all v ∗ ∈ V ∗ , (3.1)where the sum converges in L P (Ω; R ). This representation is the Karhunen-Lo`eve represen-tation and is in this form established in [1]. Define a cylindrical random variable byΘ X : H → L P (Ω; R ) , Θ X h = ∞ X k =1 h e k , h i Xv ∗ k . (3.2)The fact, that Θ X is well defined and is a cylindrical random variable follows from thefollowing lemma where we collect some simple properties of Θ X and its cylindrical probabilitydistribution ϑ X . Lemma 3.1.
For a cylindrical random variable X : V ∗ → L P (Ω; R ) let ϑ X denote thecylindrical distribution of Θ X defined in (3.2) . Then we have: (a) E h | Θ X h | i = k h k for all h ∈ H . (b) the cylindrical distribution of X equals ϑ X ◦ j − . (c) ϑ X is of cotype 2. (d) ϑ X is infinitely divisible.Proof. (a) The identity E [( Xv ∗ k )( Xv ∗ ℓ )] = h Qv ∗ k , v ∗ ℓ i = h e k , e ℓ i (3.3)implies that Xv ∗ k and Xv ∗ ℓ are uncorrelated for k = ℓ and E [ | Xv ∗ k | ] = 1. Thus, part (a)follows from (3.2). This also shows that Θ X is a well defined cylindrical random variable.(b) Due to (3.1) and (3.2) we have Θ X ( j ∗ v ∗ ) = Xv ∗ for all v ∗ ∈ V ∗ which establishes theclaim. Part (c) follows from part (a).(d) For h , . . . , h m ∈ H and n ∈ N define u ( n ) j := n X k =1 h e k , h j i v ∗ k for j = 1 , . . . , m. It follows from (3.2) for each j = 1 , . . . , m thatΘ X h j = lim n →∞ Xu ( n ) j in L P (Ω; R ) , X h , . . . , Θ X h m ) = lim n →∞ (cid:16) Xu ( n )1 , . . . , Xu ( n ) m (cid:17) in probability in R m .The probability distribution of the random vector on the right hand side is infinitely divisibleas it is given by η ◦ π − u ( n )1 ,...,u ( n ) m , where η denotes the cylindrical distribution of the infinitelydivisible cylindrical random variable X . Consequently, the random vector on the left handside is infinitely divisible, which shows that Θ X is an infinitely divisible cylindrical randomvariable. Example 3.2.
Assume that X is a Gaussian cylindrical random variable that is Xv ∗ isGaussian for all v ∗ ∈ V ∗ . In this case, it follows from (3.3) that ( Xv ∗ k ) k ∈ N is a sequence ofindependent, Gaussian random variables with E [ | Xv ∗ k | ] = 1, which yields for the charac-teristic function ϕ Θ X of Θ X : ϕ Θ X ( h ) = ∞ Y k =1 exp (cid:0) − h e k , h i (cid:1) = exp (cid:16) − k h k (cid:17) for all h ∈ H. (3.4)Consequently, the cylindrical random variable Θ X is distributed according to the canonicalGaussian cylindrical measure γ in this case. Let H be a separable Hilbert space, V be a separable Banach space and ϑ be a cylindricalprobability measure on Z ( H ). An operator T ∈ L ( H, V ) is called ϑ -radonifying if the imagecylindrical measure ϑ ◦ T − extends to a probability measure on B ( V ). If the extendedmeasure has finite p -th moments, T is called ϑ -radonifying of order p . We define the space R p ( ϑ ) := R pH,V ( ϑ ) := (cid:8) T ∈ L ( H, V ) : T is ϑ -radonifying of order p (cid:9) . Let Θ denote a cylindrical random variable with cylindrical distribution ϑ . Theorem VI.3.1in [24] guarantees that an operator T ∈ L ( H, V ) is in R p ( ϑ ) if and only if the cylindricalrandom variable T (Θ) defined by T (Θ) : V ∗ → L P (Ω; R ) , T (Θ) v ∗ = Θ( T ∗ v ∗ )is induced by a genuine random variable in L pP (Ω; V ). If S and T are in R p ( ϑ ) and α ∈ R then we obtain for all v ∗ ∈ V ∗ that( αS + T )(Θ) v ∗ = Θ(( αS + T ) ∗ v ∗ ) = αS (Θ) v ∗ + T (Θ) v ∗ . Since S (Θ) and T (Θ) are induced by genuine random variables in L pP (Ω; V ), respectively,it follows that ( αS + T )(Θ) is also induced by a genuine random variable in L pP (Ω; V ), andthus αS + T ∈ R p ( ϑ ). For T ∈ R p ( ϑ ) define k T k p := (cid:18)Z V k v k p ( ϑ ◦ T − )( dv ) (cid:19) /p . Since k T k pp = E [ k T (Θ) k p ] it follows that k·k p defines a semi-norm on R p ( ϑ ). We obtain anorm on R p ( ϑ ) by defining k T k R p := k T k p + k T k H → V . heorem 4.1. For every cylindrical probability measure ϑ on Z ( H ) the space R p ( ϑ ) equippedwith k·k R p is a Banach space for each p > .Proof. Let ( T n ) n ∈ N be a Cauchy sequence in R p ( ϑ ). Denote for each n ∈ N by Y n theinduced random variable in L pP (Ω; V ) with probability distribution ϑ ◦ T − n . Since ( T n ) n ∈ N is also a Cauchy sequence in L ( H, V ) it follows that there exists T ∈ L ( H, V ) such that k T n − T k H → V → n → ∞ . Moreover, the equality k Y m − Y n k L pP = k T m − T n k p impliesthat there exists a random variable Y ∈ L pP (Ω; V ) such that k Y n − Y k L pP → n → ∞ .The continuity of Θ : H → L P (Ω; R ) implies for every v ∗ that we have in L P (Ω; R ):lim n →∞ |h Y n , v ∗ i − Θ( T ∗ v ∗ ) | = lim n →∞ (cid:12)(cid:12) Θ (cid:0) ( T ∗ n − T ∗ ) v ∗ (cid:1)(cid:12)(cid:12) = 0 . Since h Y n , v ∗ i → h Y, v ∗ i in L P (Ω; R ) we obtain h Y, v ∗ i = Θ( T ∗ v ∗ ) for all v ∗ ∈ V ∗ , whichcompletes the proof. Example 4.2.
Let ϑ be given by the canonical Gaussian cylindrical measure γ on Z ( H ).Due to Fernique’s theorem, each γ -radonifying operator is of any order p >
1. Thus, thespace R p ( γ ) coincides with the space of γ -radonifying operators. This class of operators iswell studied, and is recently surveyed in [25].In the special setting of Section 3 we obtain the following simplification of the norm in R p ( ϑ ) for p > Proposition 4.3.
For a cylindrical random variable X in V with weak second moments letthe cylindrical random variable Θ X and its cylindrical distribution ϑ X be defined by (3.2) .If p > then T ∈ R p ( ϑ X ) satisfies k T k H → V k T k p , that is the norm k·k R p is equivalent to k·k p .Proof. For T ∈ R p ( ϑ X ) let Y denote the induced random variable in L pP (Ω; V ) with proba-bility distribution ϑ X ◦ T − on Z ( V ). Lemma 3.1 implies: k T k H → V = sup k v ∗ k k T ∗ v ∗ k = sup k v ∗ k E h | Θ X T ∗ v ∗ | i E " sup k v ∗ k |h Y, v ∗ i| = E h k Y k i (cid:16) E h k Y k p i(cid:17) /p , which completes the proof.The following result is a straightforward conclusion of a result by Schwartz [22] andKwapien [11], but it shows an important class of operators which are ϑ -radonifying. Proposition 4.4. If ϑ is a cylindrical probability measure of weak order p > then we have Π p ( H, V ) ⊆ R pH,V ( ϑ ) . Proof.
The space of p -absolutely summing operators Π p ( H, V ) coincides with the space of p -radonifying operators; see Theorem VI.5.4 in [24] for p > p = 1. The space of p -radonifying operators are operators T ∈ L ( H, V ) such for eachcylindrical measure η on Z ( H ) with weak p -moments the image cylindrical measure η ◦ T − extends to a measure on B ( V ) with finite p -moment.7 roposition 4.5. Let
Θ : H → L P (Ω; R ) be a cylindrical random variable with cylindricaldistribution ϑ . Then for T ∈ L ( H, V ) and p > the following are equivalent: (a) T ∈ R p ( ϑ ) . (b) there exists a random variable Y ∈ L pP (Ω; V ) such that for all (some) orthonormalbasis ( e k ) k ∈ N of H : h Y, v ∗ i = ∞ X k =1 h T e k , v ∗ i Θ e k in L P (Ω; R ) for all v ∗ ∈ V ∗ . Proof.
The operator T is in R p ( ϑ ) if and only if the cylindrical random variable T (Θ) isinduced by a V -valued random variable Y in L pP (Ω; V ), that is Θ( T ∗ v ∗ ) = h Y, v ∗ i for all v ∗ ∈ V ∗ . The continuity of Θ : H → L P (Ω; R ) implies in L P (Ω; R ): h Y, v ∗ i = Θ( T ∗ v ∗ ) = Θ ∞ X k =1 h T ∗ v ∗ , e k i e k ! = ∞ X k =1 h v ∗ , T e k i Θ e k for all v ∗ ∈ V ∗ . Remark 4.6. If ϑ is the canonical Gaussian cylindrical measure γ on H , then the randomvariables (Θ e k ) k ∈ N are independent and symmetric. Thus, Itˆo-Nisio’s Theorem guaranteesthat part (b) in Proposition 4.5 is equivalent to(c) there exists a random variable Y ∈ L pP (Ω; V ) such that for all (some) orthonormalbasis ( e k ) k ∈ N of H : Y = ∞ X k =1 T e k Θ e k in L P (Ω; V ) . Often this property is taken as a definition of γ -radonifying operators in the literature. Theorem 4.7. If V is a separable Hilbert space and the cylindrical measure ϑ has weak p -thmoments for some p > and is of finite cotype then it follows: R p ( ϑ ) = { T ∈ L ( H, V ) : T is Hilbert-Schmidt } Proof.
It is well known that in Hilbert spaces the class of p -radonifying operators (see proofof Proposition 4.4) coincides with the space of Hilbert-Schmidt operators. Thus, the classof Hilbert-Schmidt operators is a subset of R p ( ϑ ).Let T be in R p ( ϑ ). Then the cylindrical measure ϑ ◦ T − extends to a Radon measureof order p and ϑ ◦ ( T ∗∗ ) − is a σ ( V ∗∗ , V ∗ )-Radon measure. Denote the cotype of ϑ by q ∈ [0 , ∞ ). If q p then ϑ is also of cotype p , and thus, we can assume that q >
1. TheoremVI.5.9 in [24] implies that T ∗ is q -absolutely summing, which is equivalent to the fact that T ∗ is Hilbert-Schmidt, since H and V are Hilbert spaces. Remark 4.8. If ϑ is a genuine probability measure with p -th moments then each operatorin L ( H, V ) is in R p ( ϑ ) and Theorem 4.7 cannot be true. This case is excluded since in thiscase ϑ cannot be of finite cotype: if ( h n ) n ∈ N is a sequence in H which converges sequentiallyweakly to 0 then Lebesgue’s theorem implieslim n →∞ Z H h h n , h i p ϑ ( dh ) = 0 . But if H is infinite dimensional then we can choose k h n k = 1 for all n ∈ N .8 table cylindrical measures: A cylindrical measure ϑ on Z ( H ) is called stable oforder α ∈ (0 ,
2] if there exists a measure space ( S, S , m ) and a linear bounded operator F : H → L αm ( S ; R ) such that the characteristic function ϕ ϑ of ϑ obeys ϕ ϑ : H → C , ϕ ϑ ( h ) = exp (cid:16) − k F h k αL αm (cid:17) . (4.1)The characteristic function of ϑ ◦ T − for an arbitrary operator T ∈ L ( H, V ) is given by ϕ ϑ ◦ T − : V ∗ → C , ϕ ϑ ◦ T − ( v ∗ ) = exp (cid:16) − k ( F T ∗ ) v ∗ k αL αm (cid:17) . It follows that T is ϑ -radonifying if and only if F T ∗ ∈ Λ α ( V ∗ , L αm ), whereΛ α ( V ∗ , L αm ) := n R ∈ L ( V ∗ , L αm ) : v ∗ exp( − k Rv ∗ k αL αm )is the characteristic function of a Radon measure on V o . By taking into account that an α -stable measure has finite r -th moments for all r < α according to Theorem 3.2 in [7], we obtain for each p < α that T is in R p ( ϑ ) if onlyif F T ∗ ∈ Λ α ( V ∗ , L αm ). The spaces Λ α ( V ∗ , L αm ) are surveyed in [13, Se.7.8], however anexplicit description is only known in a few case. A case which can be easily described is thefollowing: Example 4.9.
Assume that V = ℓ q for some q ∈ [2 , ∞ ) and L αm ( S ; R ) = ℓ α for some α < q ′ where q ′ := q/ ( q − ϑ be an α -stable measure on the Hilbert space H withcharacteristic function of the form (4.1). Then an operator T ∈ L ( H, V ) is in R p ( ϑ ) for p < α if and only if ∞ X k =1 ∞ X j =1 |h F T ∗ e j , e k i| q ′ αq ′ < ∞ , where ( e k ) k ∈ N denotes the canonical Schauder basis for the spaces of sequences. Example 4.10.
Assume that V is given by some L q space for q ∈ [2 , ∞ ) and ϑ is an α -stable cylindrical measure on the Hilbert space H with characteristic function of the form(4.1). Then an operator T ∈ L ( H, V ) is in R p ( ϑ ) for p < α if and only if F T ∗ is r -absolutelysumming for any r ∈ (0 , q ′ ), i.e. F T ∗ ∈ Π r ( L q ′ , L αm ); see Proposition 7.8.7 in [13].Stable cylindrical probability measures might be considered as a subset of infinitely di-visible cylindrical measures with elements which are the most similar ones to the canonicalGaussian cylindrical measure γ . Nevertheless, many properties known for γ -radonifying op-erators do not hold for radonifying operators of stable cylindrical measures. One of these isthe ideal property which is true for γ -radonifying operators: let H and H ′ be Hilbert spacesand V and V ′ be Banach spaces. Then if T ∈ R H,V ( γ ), S ∈ L ( H ′ , H ) and S ∈ L ( V, V ′ )then S T S ∈ R H ′ ,V ′ ( γ ). This result can be found in [25]. However, already for stablecylindrical measures it is known that the ideal property is not satisfied any more: for q > T ∈ R pL p ,L q ( ϑ ) and S ∈ L ( L p , L p ) such that T S is not in R pL p ,L q ( ϑ );see [13] for this result. 9 ompound Poisson cylindrical measures: a compound Poisson cylindrical distri-bution (see Example 3.5 in [1]) is an infinitely divisible cylindrical measure ϑ on Z ( H ) withcharacteristic function ϕ ϑ : H → C , ϕ ϑ ( h ) = exp (cid:18) c Z H (cid:16) e i h h,g i − (cid:17) ν ( dg ) (cid:19) , (4.2)where ν is a cylindrical probability measure on Z ( H ) and c > ϑ has weak secondmoments if and only if ν has weak second moments.Equivalently, one can introduce a compound Poisson cylindrical distribution by cylin-drical random variables. Let X , X , . . . be independent, cylindrical random variables in H with identical cylindrical distribution ν and let N be an independent, integer-valued Pois-son distributed random variable with intensity c >
0, all defined on the probability space(Ω , A , P ). Then Y : H → L P (Ω; R ) , Y h := ( , if N = 0 ,X h + · · · + X N h, else,defines a cylindrical random variable Y with a characteristic function which is of the form(4.2). Theorem 4.11.
For a compound Poisson cylindrical distribution ϑ with characteristic func-tion (4.2) it follows for each p > that R p ( ϑ ) = R p ( ν ) . Proof.
Let T ∈ R p ( ϑ ). Then µ := ϑ ◦ T − is an infinitely divisible measure on B ( V ) with p -th moment. If ξ denotes the L´evy measure of µ then it follows that ξ = c ( ν ◦ T − ) on Z ( V )due to the uniqueness of cylindrical L´evy measures. Thus, the image cylindrical measure ν ◦ T − extends to the probability measure c − ξ . Let Y be a V -valued random variablewith distribution µ and ( X k ) k ∈ N a family of independent, V -valued random variables withdistribution c − ξ . It follows that E [ k X k p ] e c c ∞ X k =1 E [ k X + · · · + X k k p ] c k k ! e − c = e c c E [ k Y k p ] < ∞ , i.e. the Radon measure c − ξ has moments of order p which shows T ∈ R p ( ν ).If we assume T ∈ R p ( ν ) then ν ◦ T − is a probability measure on B ( V ) and µ : B ( V ) → [0 , , µ ( C ) := e − c ∞ X k =0 c k ( ν ◦ T − ) ∗ k ( C ) k !defines a probability measure on B ( V ) with characteristic function ϕ µ : V ∗ → C , ϕ µ ( v ∗ ) = exp (cid:18) c Z V (cid:16) e i h v,v ∗ i − (cid:17) ( ν ◦ T − )( dv ) (cid:19) , see [13, Pro.5.3.1]. Since ϕ µ = ϕ ϑ ◦ T − it follows that ϑ ◦ T − extends to the Radon measure µ on B ( V ). As before, let Y and X , X , . . . denote independent random variables with10istributions µ and ν ◦ T − . The measure µ has p -th moments since Minkowski’s inequalityimplies E [ k Y k p ] = ∞ X k =1 E [ k X + · · · + X k k p ] c k k ! e − c ∞ X k =1 k p E [ k X k p ] c k k ! e − c < ∞ , which shows that T ∈ R p ( ϑ ). Example 4.12. (Cylindrical normally distributed jumps)Models of share prices perturbed by a discontinuous noise with normally distributed jumpsare considered in Financial Mathematics from its very early times; see for example thework [15] by Merton. Accordingly, let ν be the canonical Gaussian cylindrical measure γ and let c > ϑ withcharacteristic function (4.2) obeys R p ( ϑ ) = R p ( γ ) . In this section we apply the theory of radonifying operators developed above in order tointroduce Wiener integrals with respect to a L´evy process L with weak second momentson a separable Banach space U . In fact, the same approach can be applied if L is only acylindrical L´evy process, but we want to avoid any more technical complications here; see[19] for details.Recall that the L´evy process L can be decomposed into L ( t ) = b + W ( t ) + M ( t ) for all t >
0, where b ∈ U and W is a Wiener process with a covariance operator C ∈ L ( U ∗ , U )and M is a L´evy process with weak second moments and a L´evy measure µ . The Wienerintegrals with respect to the Wiener process W are developed in the publications [5] and [27]with many sophisticated refinements and applied to the stochastic Cauchy problem. Ourrather simplified presentation below for integration with respect to W illustrates the coreidea in the approach developed in [27]. In our work [20], we extend the approach in [27]to develop a Wiener integral with respect to a martingale-valued measure. By the theorydeveloped here, we are able to relate this integral to ϑ -radonifying operators.We begin with defining a Wiener integral with respect to the discontinuous martingale M with L´evy measure µ . For ρ := λ ⊗ µ , where λ denotes the Lebesgue measure on [0 , T ],define H M := L ρ ([0 , T ] × U ; R ). Let V denote another separable Banach space and let F : [0 , T ] → L ( U, V ) be a function satisfying h F ( · ) · , v ∗ i ∈ H M for all v ∗ ∈ V ∗ . (5.1)Then one can define a cylindrical random variable by I M : V ∗ → L P (Ω; R ) , I M v ∗ = Z T F ∗ ( s ) v ∗ dM ( s ) . Since the integrand is U ∗ -valued and M has weak second moments, the integral can beeasily defined by following an Itˆo approach, see e.g. [18], or by the approach of M´etivier andPellaumail in [16], or as introduced by Rosi´nski in [21]. In all cases, it follows that I M is aninfinitely divisible cylindrical random variable with weak second moments. The covarianceoperator of I M is given by Q M : V ∗ → V, h Q M v ∗ , w ∗ i = Z [0 ,T ] × U h F ( s ) u, v ∗ ih F ( s ) u, w ∗ i ρ ( ds, du ) . Q M is V -valued and not only V ∗∗ -valued, since Pettis’ measurability theoremguarantees due to (5.1) that ( t, u ) F ( t ) u is strongly measurable. The covariance operator Q M can be factorised by j M : H M → V, h j M f, v ∗ i := Z [0 ,T ] × U h F ( s ) u, v ∗ i f ( s, u ) ρ ( ds, du ) . Since the adjoint operator is given by j ∗ M w ∗ = h F ( · ) · , w ∗ i = F ∗ ( · ) w ∗ for all w ∗ ∈ V ∗ itfollows that Q M = j M j ∗ M , and thus H M = L ρ ([0 , T ] × U ; R ) is established as the reproducingkernel Hilbert space of Q M . Define the cylindrical random variableΘ M : H M → L P (Ω; R ) , Θ M f = Z [0 ,T ] × U f ( s, u ) M ( ds, du ) , and let ϑ M denote the cylindrical distribution of Θ M . Let ( f k ) k ∈ N be an orthonormal basisof H M and choose v ∗ k ∈ V ∗ such that f k = j ∗ M v ∗ k . By continuity of Θ M it follows for all v ∗ ∈ V ∗ that we have in L P (Ω; R ): I M v ∗ = Θ M ( j ∗ M v ∗ ) = Θ M ∞ X k =1 h f k , j ∗ M v ∗ i f k ! = ∞ X k =1 h j M f k , v ∗ i Θ M f k = ∞ X k =1 h j M f k , v ∗ i I M v ∗ k . In summary, we have explicitly derived the setting of Section 3: for the cylindrical randomvariable I M we derived the reproducing kernel Hilbert space H M of its covariance operator Q M with embedding j M : H M → V . In addition, we constructed the cylindrical randomvariable Θ M in H M , which is based on the Karhunen-Lo`eve representation of I M accordingto (3.1) and which satisfies j M (Θ M ) = I M .It follows from Lemma 3.1 that Θ M is an infinitely divisible cylindrical random variable,and by approximating f ∈ H M by step functions, the characteristic function of Θ M is givenby ϕ Θ M : H M → C , ϕ Θ M ( f ) = exp (cid:18)Z H M (cid:16) e i h f,g i − − i h f, g i (cid:17) ν ( dg ) (cid:19) , where ν is a cylindrical measure on Z ( H M ) satisfying ν ◦ ( h· , f i ) − = ρ ◦ f − for all f ∈ H M .Since the cylindrical distribution of I M equals ϑ M ◦ j − M according to Lemma 3.1, thereexists a random variable Y M ∈ L P (Ω; V ) obeying h Y M , v ∗ i = Z T F ∗ ( s ) v ∗ M ( ds ) for all v ∗ ∈ V ∗ , (5.2)if and only if j M : H M → V is ϑ M -radonifying.The same approach can be applied to introduce the stochastic integral with respect to theWiener process W with covariance operator C . Let K denote the reproducing kernel Hilbertspace of C with embedding i C : K → U , i.e. C = i C i ∗ C . For functions F : [0 , T ] → L ( U, V )satisfying i ∗ C F ∗ ( · ) v ∗ ∈ L ([0 , T ]; K ) for all v ∗ ∈ V ∗ , (5.3)one can define a cylindrical random variable by I W : V ∗ → L P (Ω; R ) , I W v ∗ = Z T F ∗ ( s ) v ∗ dW ( s ) . I W is given by Q W : V ∗ → V, h Q W v ∗ , w ∗ i = Z T h i ∗ C F ∗ ( s ) v ∗ , i ∗ C F ∗ ( s ) w ∗ i ds. The covariance operator Q W can be factorised through the Hilbert space H W := L ([0 , T ]; K ),and the embedding is given by j W : H W → V, h j W f, v ∗ i = Z T h i ∗ C F ∗ ( s ) v ∗ , f ( s ) i ds, with adjoint operator j ∗ W v ∗ = i ∗ C F ∗ ( · ) v ∗ . If γ denotes the canonical Gaussian cylindricalmeasure on H W it follows that γ ◦ j − W is a cylindrical Gaussian distribution with covarianceoperator j W j ∗ W , that is γ ◦ j − W coincides with the cylindrical distribution of I W . Conse-quently, we obtain that there exists a random variable Y W ∈ L P (Ω; V ) obeying h Y W , v ∗ i = Z T F ∗ ( s ) v ∗ dW ( s ) for all v ∗ ∈ V ∗ , if and only if j W : H W → V is γ -radonifying.Finally, let F : [0 , T ] → L ( U, V ) be a function obeying (5.1), (5.3) and h F ∗ ( · ) v ∗ , b i ∈ L ([0 , T ]; R ) for all v ∗ ∈ V ∗ . (5.4)Then one can define for each A ∈ B ([0 , T ]) a cylindrical random variable I A : V ∗ → L P (Ω; R )by I A v ∗ = Z T A ( s ) h F ∗ ( s ) v ∗ , b i ds + Z T A ( s ) F ∗ ( s ) v ∗ dW ( s ) + Z T A ( s ) F ∗ ( s ) v ∗ dM ( s ) . (5.5)The function F is called stochastically integrable with respect to L if and only if for each A ∈ B ([0 , T ]) there exists a random variable Y A ∈ L P (Ω; V ) such that h Y A , v ∗ i = I A v ∗ for all v ∗ ∈ V ∗ . (5.6)By the derivation above one obtains the following result: Theorem 5.1.
A function F : [0 , T ] → L ( U, V ) satisfying (5.1) , (5.3) and (5.4) is stochas-tically integrable with respect to L ( · ) = b + W ( · ) + M ( · ) if and only if the following aresatisfied: (i) F ( · ) b : [0 , T ] → V is Pettis integrable; (ii) j W : H W → V is γ -radonifying; (iii) j M : H M → V is ϑ M -radonifying.Proof. If part: for A ∈ B ([0 , T ]) define the cylindrical random variables I Ab : V ∗ → R , I Ab v ∗ = Z A h v ∗ , F ( s ) b i ds,I AW : V ∗ → L P (Ω; R ) , I AW v ∗ = Z T A ( s ) F ∗ ( s ) v ∗ dW ( s ) .I AM : V ∗ → L P (Ω; R ) , I AM v ∗ = Z T A ( s ) F ∗ ( s ) v ∗ dM ( s ) .
13e have to show that these cylindrical random variables are induced by genuine randomvariables in L P (Ω; V ), respectively. Condition (i) implies that there exists v A ∈ V such that h v A , v ∗ i = I Ab v ∗ for all v ∗ ∈ V ∗ . (5.7)The covariance operator of the cylindrical random variable I AW is given by Q AW : V ∗ → V, h Q W v ∗ , w ∗ i = Z A h i ∗ C F ∗ ( s ) v ∗ , i ∗ C F ∗ ( s ) w ∗ i ds. It follows that h Q AW v ∗ , v ∗ i h Q [0 ,T ] W v ∗ , v ∗ i for all v ∗ ∈ V ∗ . Since Condition (ii) guarantees that Q [0 ,T ] W is the covariance operator of a Gaussian measureon B ( V ), Theorem 3.3.1 in [3] implies that there exists a Gaussian measure on B ( V ) withcovariance operator Q AW . Thus, Theorem IV.2.5 in [24] guarantees that there exists a randomvariable Y AW ∈ L P (Ω; V ) such that h Y AW , v ∗ i = I AW v ∗ for all v ∗ ∈ V ∗ . (5.8)From the independent increments of M it follows that the cylindrical distribution of I AM isinfinitely divisible and that its cylindrical L´evy measure ν A is given by ν A : Z ( V ) → [0 , ∞ ] , ν A ( B ) = Z A × U B ( F ( s ) u ) ρ ( ds, du ) . Condition (iii) implies that ν [0 ,T ] is the genuine L´evy measure of an infinitely divisibleprobability measure on B ( V ). Since ν A ( B ) ν [0 ,T ] ( B ) for all B ∈ Z ( V ) , Theorem 3.4 in [19] implies that ν A extends to a genuine L´evy measure on B ( V ). TheoremIV.2.5 in [24] guarantees that there exists a random variable Y AM ∈ L P (Ω; V ) such that h Y AM , v ∗ i = I AM v ∗ for all v ∗ ∈ V ∗ . (5.9)It follows from (5.7), (5.8) and (5.9) that the random variable Y A := v A + Y AW + Y AM satisfies(5.6). Only if part:
Let I b , I W and I M denote the cylindrical random variables defined in thebeginning of the proof for A = [0 , T ]. Stochastic integrability of F implies that there existsa random variable Y ∈ L P (Ω; V ) such that h Y, v ∗ i = I b v ∗ + I W v ∗ + I M v ∗ for all v ∗ ∈ V ∗ . Since I W v ∗ is symmetric and independent of ( I b + I M ) v ∗ for all v ∗ ∈ V ∗ , it follows fromProposition 7.14.51 in [4] that the cylindrical distributions of I W and I b + I M extend toprobability measures on B ( V ). Consequently, j W is γ -radonifying and I b + I M is inducedby a random variable in L P (Ω; V ). Let M ′ denote an independent copy of M . Then I b + I M − I b − I M ′ is induced by a random variable X ∈ L P (Ω; V ) and X is infinitelydivisible. Denoting the L´evy measure of X by ξ it follows that ν [0 ,T ] ( B ) ν [0 ,T ] ( B ) + ν [0 ,T ] ( − B ) = ξ ( B ) for all B ∈ Z ( V ) . Theorem 3.4 in [19] implies that ν [0 ,T ] extends to a genuine L´evy measure on B ( V ), whichyields that j M is ϑ M -radonifying. Acknowledgments:
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