Stochastic integration with respect to cylindrical L\'evy processes by p-summing operators

We introduce a stochastic integral with respect to cylindrical L\'evy processes with finite $p$-th weak moment for $p\in [1,2]$. The space of integrands consists of $p$-summing operators between Banach spaces of martingale type $p$. We apply the developed integration theory to establish the existence of a solution for a stochastic evolution equation driven by a cylindrical L\'evy process.


Introduction
Cylindrical Lévy processes are a natural generalisation of cylindrical Brownian motions to the non-Gaussian setting, and they can serve as a model of random perturbation of partial differential equations or other dynamical systems. As a generalised random process, cylindrical Lévy processes do not attain values in the underlying space, and they do not enjoy a Lévy-Itô decomposition in general. Since conventional approaches to stochastic integration rely on either stopping time arguments or a semi-martingale decomposition in the one or other form, a completely novel method for stochastic integration has been introduced in the work [11] by one of us with Jakubowski. This method is purely based on tightness arguments, since typical estimates of the expectation are not available without utilising stopping time arguments or a semi-martingale decomposition. As a consequence, although this method guarantees the existence of the stochastic integral for a large class of random integrands, it does not provide any control of the integrals. Since many applications, such as modelling dynamical systems or control problems, require upper estimates of the stochastic integrals, this method seems to be difficult to use for such applications.
In order to provide a control of the stochastic integral, we develop a theory of stochastic integration for random integrands with respect to cylindrical Lévy processes with finite pth weak moments for p ∈ [1,2] in this work. Our approach enables us to develop the theory on a large class of general Banach spaces. We apply the obtained estimates to establish the existence of an abstract partial differential equations driven by a cylindrical Lévy process with finite p-th weak moments.
Stochastic integration with respect to a cylindrical Wiener process is well developed in Hilbert spaces and various classes of Banach spaces. Typical Banach spaces which permit a development of stochastic integration are martingale type 2 Banach spaces, see e.g. Dettweiler [6,7] or UMD spaces, see e.g. van Neerven, Veraar and Weis [32]. Veraar and Yaroslavtsev [33] extend the approach for UMD spaces in [32] to continuous cylindrical local martingales by utilising the Dambis-Dubins-Schwarz Theorem. Stochastic integration in Hilbert spaces with respect to genuine Lévy processes is for example presented by Peszat and Zabczyk in [17], and with respect to cylindrical Lévy processes the theory is developed in [11]. Stochastic integration with respect to a Poisson random measure in Banach spaces is developed for example by Mandrekar and Rüdiger in [16] and by Brzeźniak, Zhu and Hausenblas [34].
In this work we are faced with the similar problem as in [11]. Conventional approaches to stochastic integration utilise either stopping times or the Lévy-Itô decomposition to show continuity of the integral operator separately: firstly with respect to the martingale part with finite 2-nd moments and secondly with respect to the bounded variation part. However, since these approaches are excluded for cylindrical Lévy processes, we show continuity of the integral operator "in one piece", i.e. without applying the semimartingale decomposition of the integrator. For this purpose, we utilise a generalised form of Pietsch's factorisation theorem, originating from the work of Schwartz [29].
More specifically, the space of admissible integrands are predictable stochastic processes with values in the space of p-summing operators and with integrable p-summing norm for p ∈ [1,2] in this work. Due to results by Kwapień and Schwartz, for p > 1 the space of psumming operators coincides with the space of p-Radonifying operators, which are exactly the operators which map each cylindrical random variable with finite p-th weak moments to a genuine random variable. In this way, stochastic processes with values in the space of p-summing operators are the natural class of integrands, as they map the cylindrical increments of the integrator to the genuine random variables. Furthermore, the class of psumming operators coincides with the class of Hilbert-Schmidt operators in Hilbert spaces, and as such the aforementioned space of admissible integrands is a natural generalisation of the integration theory in Hilbert spaces with respect to genuine Lévy process in e.g. [17]. In typical applications such as the heat equation, the p-summing norm of the opeartors appearing in the equation can be explicitely estimated, see [2].
In Section 2, we recall the concepts of cylindrical measures and cylindrical Lévy processes. In Section 3, we present the generalised Pietsch's factorisation theorem due to Schwartz, and derive a result on the strong convergence of p-summing operators, which is needed in the proof of the stochastic continuity of the stochastic convolution. Section 4 is devoted to the construction of the stochastic integral. This is done in two steps as in the article [21] by the second author. Firstly we Radonify the increments of the cylindrical Lévy process by random p-summing operators. Secondly, we define the integral for simple integrands and extend it by continuity to the general ones. We also present some examples of the processes covered by our theory. In Section 5 we apply our results to establish existence and uniqueness solution to the evolution equation driven by a cylindrical Lévy noise with finite p-th weak moments for p ∈ [1, 2].

Preliminaries
Let E and F be Banach spaces with separable duals E * and F * . The operator norm of an operator u : E → F is denoted with u L(E,F ) or simply u . We write B E for the closed unit ball in E. The Borel σ-field is denoted with B(E).
Fix a probability space (Ω, F, P ) with a filtration (F t ). We denote the space of equivalence classes of real-valued random variables equipped with the topology of convergence in probability by L 0 (Ω, F, P ; R). The Bochner space of equivalence classes of E-valued, random variables with finite p-th moment is denoted with L p (Ω, F, P ; E). In case the codomain is not separable we take only separably valued random variables.
Cylindrical sets are sets of the form . . x * n (x)) ∈ B} for x * 1 , . . . , x * n ∈ E * and B ∈ B(R n ). For Γ ⊆ E * we denote with Z(E, Γ) the collection of all cylindrical sets with x * 1 , . . . , x * n ∈ Γ, B ∈ B(R n ) and n ∈ N. If Γ = E * , we write Z(E) to denote the collection of all cylindrical subsets of E. Note that Z(E) is an algebra and that if Γ is finite, then Z(E, Γ) is a σ-algebra. A function µ : Z(E) → [0, ∞] is called a cylindrical measure if its restriction to Z(E, Γ) is a measure for every finite subset Γ ⊆ E * . If µ(E) = 1 we call it a cylindrical probability measure. A cylindrical random variable is a linear and continuous mapping The cylindrical distribution of a cylindrical random variable Y is defined by which defines a cylindrical probability measure on Z(E). The characteristic function of a cylindrical random variable (resp. cylindrical probability measure) is given by for x * ∈ E * . We say that a cylindrical random variable Y (resp. cylindrical measure µ) is of weak order p or has finite p-th A family of cylindrical random variables (L(t) : t ≥ 0) is called a cylindrical Lévy process if, for every x * 1 , . . . , x * n ∈ E * and n ∈ N, we have that is a Lévy process in R n with respect to the filtration (F t ). We say that L is weakly p- The characteristic function of L(1) can be written in the form where p : E * → R is a continuous function with p(0) = 0, q : E * → R is a quadratic form, and ν is a finitely additive set function on cylindrical sets of the form C(x * 1 , · · · , x * n ; B) for x * 1 , . . . , x * n ∈ E * and B ∈ B(R n \{0}), such that for every x * ∈ E * it satisfies Cylindrical Lévy processes are introduced in [1] and further details on the characteristic function can be found in [20]. An operator u : E → F is called p-summing if there exists a constant c such that for all x 1 , . . . , x n ∈ E and n ∈ N; see [8]. We denote with π p (u) its p-summing norm, which is the smallest possible constant c in (1). The space of p-summing operators is denoted with Π p (E, F ). If E and F are Hilbert spaces, this space coincides with the space of Hilbert-Schmidt operators denoted by L HS (E, F ) with the norm · L HS (E,F ) ; see [8,Th. 4.10 and Cor. 4.13]. Moreover, the p-summing norms and the Hilbert-Schmidt norm in L HS (E, F ) are equivalent.
A Banach space E is of martingale type p ∈ [1,2] if there exists a constant C p such that for all finite E-valued martingales (M k ) n k=1 the following inequality is satisfied: where we use the convention that M 0 = 0; see [10]. We use the notation u(µ) for the push forward cylindrical measure µ • u −1 for a continuous linear function u : E → F and a cylindrical measure µ. An operator u : E → F is called p-Radonifying for some p ≥ 0 if for every cylindrical measure µ on E of weak order p, the measure u(µ) extends to a Radon measure on F with finite p-th strong moment. Equivalently, the mapping u is p-Radonifying if for every cylindrical random variable Y on E * with finite weak p-th moment, the cylindrical random variable Y • u * is induced by an F -valued random variable with finite p-th strong moment; see [31,Prop. VI.5.2].
A Banach space E has the approximation property if for every compact set K ⊆ E and for every ε > 0 there exists a finite rank operator u : A Banach space E has the Radon-Nikodym property if for any probability space (Ω, F, P ) and vector-valued measure µ : F → E, which is absolutely continuous with respect to P , there exists f ∈ L 1 (Ω, F, P ; E) such that It is well known that every reflexive Banach space has the Radon-Nikodym property; see [

Some results on p-suming operators
Our approach to stochastic integration with respect to a cylindrical Lévy process is based on a generalisation of Pietsch's factorisation theorem, which is due to Schwartz; see [30, p. 23-28] and [28]. For a measure µ on B(E) and p ∈ [1, 2] we define and we say that µ is of weak order p if µ * p < ∞.
Remark 2. Pietsch's factorisation theorem states that if u : E → F is a p-summing map then there exists a regular probability measure ρ on B E * such that If X is a genuine random variable X : Ω → E with probability distribution µ on B(E), Pietsch's factorisation theorem immediately implies For this reason, we refer to Theorem 1 as generalised Pietsch's factorisation theorem.
For establishing the stochastic continuity of the stochastic convolution in Section 5, we need a result on the convergence of p-summing operators between Banach spaces. In the case of Hilbert spaces, this convergence result can easily be seen: suppose that U and H are separable Hilbert spaces and let ψ : U → H be a Hilbert-Schmidt operator. If (ϕ n ) is a sequence of operators ϕ n : H → H converging strongly to 0 as n → ∞, then the composition ϕ n ψ converges to 0 in the Hilbert-Schmidt norm. Indeed, take (e n ) an orthonormal basis of U and calculate Every term in the above sum converges to 0 as n → ∞ due to the strong convergence of ϕ n . By Lebesgue's dominated convergence theorem we obtain ϕ n ψ 2 L HS (U,H) → 0. The following result extends this conclusion in Hilbert spaces to the Banach space setting by approximating p-summing operators with finite rank operators.
Theorem 3. Suppose that E is a reflexive Banach space or a Banach space with separable dual and that E * * has the approximation property. If ψ : E → F is a p-summing operator and (ϕ n ) is a sequence of operators ϕ n : F → F converging strongly to 0 then we have Proof. We first prove the assertion for finite rank operators ψ : E → F , in which case we can assume that ψ = N k=1 x * k ⊗y k for some x * k ∈ E * and y k ∈ F . Then ϕ n ψ = N k=1 x * k ⊗(ϕ n y k ) and since π p (x * ⊗ y) = x * y by a simple argument (see [8, p. 37]), we estimate Consider now the case of a general p-summing operator ψ. Under the assumptions on E and F , by Corollary 1 in [26], the finite rank operators are dense in the space of p-summing operators. That is, there exists a sequence of finite rank operators (ψ k ) such that π p (ψ k − ψ) → 0 as k → ∞. It follows that for all k, n ∈ N.

Radonification of increments and Stochastic integral
In this section we fix p ∈ [1, 2] and assume that the cylindrical Lévy process L has finite p-th weak moments and assume either that p > 1 or that F has the Radon-Nikodym for some disjoint sets A 1 , . . . , A m ∈ F s and ψ 1 , . . . , ψ m ∈ Π p (E, F ). The space of simple, F s -measurable random variables is denoted with S := S(Ω, F s ; Π p (E, F )), and it is equipped with the norm Ψ S,p := (E [π p (Ψ) p ]) 1/p .
Since for p > 1 or for p = 1 with F having the Radon-Nikodym property, each psumming operator ψ k : E → F is p-Radonifying, it follows that the cylindrical random variable (L(t) − L(s))ψ * k is induced by a classical, F -valued random variable X ψ k : Ω → F : This enables us to define the operator The following result allows us to extend the operator J s,t to L p (Ω, F s , P ; Π p (E, F )).

Lemma 5. (Radonification of the increments)
For fixed 0 ≤ s < t ≤ T , the operator J s,t defined in (7) is continuous and satisfies and thus J s,t can be extended to J s,t : Proof. Let Ψ be of the form (6). Since the sets A k are disjoint it follows that Using the fact that each A k is F s -measurable and that X ψ k is independent of F s we can calculate further In order to estimate E [ X ψ k p ] we apply Theorem 1 to obtain that Since stationary increments of the real-valued Lévy processes yield it follows that Note, that by the closed graph theorem and the continuity of L(t−s) : E * → L 0 (Ω, F, P ; R), the mapping L(t − s) : E * → L p (Ω, F, P ; R) is continuous. This shows that the last expression in (11) is finite. Applying estimates (10) and (11) to (9) results in which proves (8).
For defining the stochastic integral below, let Λ(Π p (E, F )) denote the space of predictable processes Ψ : where 0 = t 1 < · · · < t N = T , and each Ψ k is an F t k -measurable, Π p (E, F )-valued random variable with E[π p (Ψ k ) p ] < ∞. We denote with Λ S 0 (Π p (E, F )) the space of simple processes of the form (13) where each Ψ k is a simple random variable of the form (6), i.e. taking only a finite number of values.
Since for stochastic processes in Λ S 0 (Π p (E, F )) the Radonification of the increments are defined by the operator J s,t , we can define the integral operator by where Ψ is assumed to be of the form (13).

Theorem 7. (stochastic integration)
Assume that the cylindrical Lévy process L has the characteristics (b, 0, ν) and satisfies for all x * ∈ E * (15) and that F is of martingale type p and has the Radon-Nikodym property if p = 1. Then the integral operator I defined in (14) is continuous and extends to the operator Proof. Let Ψ in Λ S 0 (Π p (E, F )) be given by (13) where Ψ k is of the form for some disjoint sets A k,1 , . . . , A k,m k ∈ F t k and ψ k,1 , . . . , ψ k,m k ∈ Π p (E, F ) for all k ∈ {0, . . . , N − 1}. The cylindrical Lévy process L can be decomposed into For the first integral in (16) we calculate By Hölder's inequality with q = p p−1 and q = ∞ if p = 1 we obtain Since Ψ * (s) L(F * ,E * ) = Ψ(s) L(E,F ) ≤ π p (Ψ(s)) according to [8, page 31], it follows that For estimating the second term in (16) Here, we use that the Lévy measure of M (1)x * is given by ν • (x * ) −1 . It follows that we can consider the map M : On the other hand, continuity of x * for all t ∈ (0, T ] a.s., and the closed graph theorem satisfies that M : E * → R p is continuous. It follows that Let J t k ,t k+1 be the operators defined in (7) with L replaced by M . Since F is of martingale type p here exists a constant C p > 0 such that Lemma 5 and inequality (19) imply Together with (17), this completes the proof.
By rewriting Condition (15) as it follows that Condition (15) is equivalent to This is a natural requirement if we want to control the moments, see [15,25] and Remark 9 below. Condition (15) implies in particular that the the Blumenthal-Getoor index of (L(t)x * : t ≥ 0) is at most p. The interplay between the integrability of the Lévy process and its Blumenthal-Getoor index was observed also in [3,4].
Example 8 (Gaussian case). Note that if p < 2, then L cannot have the Gaussian part for the assertion to hold. Indeed, let W be a one-dimensional Brownian motion and suppose for contradiction that for some constant C and every real-valued predictable process Ψ with E T 0 |Ψ(t)| 2 dt < ∞. Choose for each n ∈ N the stochastic process Ψ n (t) = ½ [0,1/n] (t) for t ∈ [0, T ]. By [9,Sec. 3.478] we calculate But on the other side, since E T 0 |Ψ n (t)| p dt = 1 n , solving (20) for n yields which results in a contradiction by taking the limit as n → ∞.
The canonical α-stable cylindrical Lévy process has the characteristic function ϕ L(1) (x * ) = exp(− x * α ) for each x * ∈ E * ; see [23]. It follows that the real-valued Lévy process (L(t)x * : t ≥ 0) is symmetric α-stable with Lévy measure ν • (x * ) −1 (dβ) = c 1 |β| 1+α dβ for a constant c > 0. Condition (15) fails to hold since One can observe in a similar way as in the Gaussian case that the stochastic integral operator with respect to the α-stable cylindrical Lévy process L is not continuous. If Ψ n (t) = ½ [0,1/n] (t), then in the inequality the left-hand side is infinite for p ≥ α. For p < α we use the self-similarity of the stable processes to calculate Solving (21) for n yields which results in a contradiction by taking the limit as n → ∞. Therefore, the stochastic integral mapping with respect to the α-stable process cannot be continuous as a mapping from L p ([0, T ] × Ω, P, dt ⊗ P ; R) to L p (Ω, F T , P ; R) for any p > 0. A moment inequality with different powers on the left and right-hand sides was proven in the case of real-valued integrands and vector-valued integrators in [24]. They prove for any α-stable Lévy process L and p < α that Example 10. In various publications, e.g. [14,18,19,22], specific examples of the following kind of a cylindrical Lévy process has been studied: let E be a Hilbert space with an orthonormal basis (e k ) and let L be given by where (ℓ k ) is a sequence of independent, one-dimensional Lévy processes ℓ k with characteristics (b k , 0, ρ k ). Precise conditions under which the sum converges and related results can be found in [22]. In this case, we claim that Condition (15) is satisfied if and only if It is shown in [22,Lem. 4.2] that the cylindrical Lévy measure ν of L is given by where m e k : R → E is given by m e k (x) = xe k . Condition (15) simplifies to for any y ∈ E. This is equivalent to which results in R |β| p ρ k (dβ) k∈N ∈ ℓ 2/p (R) * = ℓ 2/(2−p) (R).
Example 11. Another example are cylindrical compound Poisson process, see e.g. [1,Ex. 3.5]. These are cylindrical Lévy processes of the form where N is a real-valued Poisson process with intensity λ and Y k are identically distributed, cylindrical random variables, independent of N , and say with cylindrical distribution ρ.
Since the Lévy measure of (L(t)x * : t ≥ 0) is given by λρ • (x * ) −1 , it follows that Condition (15) is satisfied if and only if Remark 12. If p = 2 and E and F are Hilbert spaces, the space of admissible integrands Λ(Π 2 (E, F )) are given by as in the work [21]. This is only suboptimal, as it is known that in this case the space of admissible integrands can be enlarged to predictable processes satisfying where Q is the covariance operator associated to L; see [12]. In this way, the space of integrands depends on the Lévy triplet of the integrator. One can ask if a similar result is possible in our more general setting for p < 2 and for Banach spaces by replacing the covariance operator by the quadratic variation.

Existence and uniqueness of solution
In this section we apply the developed integration theory to derive the existence of an evolution equation in a Banach space under standard assumptions. For this purpose, we consider dX(t) = AX(t) + B(X(t)) dt + G X(t) dL(t), where X 0 is an F 0 -measurable random variable in a Banach space F and the driving noise L is a cylindrical Lévy process in a Banach space E with finite p-th weak moments and finite p-variation. The operator A is the generator of a C 0 -semigroup on F and B : F → F and G : F → Π p (E, F ) are some functions.
Definition 13. A mild solution of (24) is a predictable process X such that for some p ≥ 1, and such that, for all t ∈ [0, T ], we have P -a.s.
We assume Lipschitz and linear growth condition on the coefficients F and G and an integrability assumption on the initial condition.
Theorem 15. Let p ∈ [1,2] and suppose that the Banach spaces E and F satisfy that (a) E is reflexive or has separable dual, E * * has the approximation property, (c) if p = 1, then F has the Radon-Nikodym property.
Proof. We define the spacẽ and a family of seminorms for β ≥ 0: Let H T be the set of equivalence classes of elementsH T relative to · T,0 . Define an operator K : H T → H T by K(X) := K 0 (X) + K 1 (X) + K 2 (X), where The Bochner integral and the stochastic integral above are well defined because X is predictable and for every t ∈ [0, T ] the mappings are continuous. The appropriate integrability condition follows from (26) and (27) below. For applying Banach's fixed point theorem, we first show that K indeed maps to H T . Choose constants m ≥ 1 and ω ∈ R such that S(t) ≤ me ωt for each t ≥ 0. It follows that By Assumption (A1) and Hölder's inequality, we obtain with q = p p−1 that Similarly, we conclude from Assumption (A2) and Theorem 7 that there exists a constant Next, we establish that K is stochastically continuous. For this purpose, let ε > 0. For each t ≥ 0 we obtain Since X(s) ≤ 1 + X(s) p for all s ≥ 0, it follows, for ε → 0, that With the same estimate X(s) ≤ 1 + X(s) p we obtain Since the integrand of I 2 tends to 0 as ε → 0 by the strong continuity of the semigroup, Lebesgue's dominated convergence theorem shows that I 2 tends to 0 as ε → 0. For K 2 we obtain by Theorem 7 that there exists a constant c > 0 such that π p ((Id −S(ε))S(t − s)G(X(s))) p ds .
By Theorem 3 the integrand π p ((Id −S(ε))S(t − s)G(X(s))) p converges to 0 for all t and ω ∈ Ω. Moreover it is bounded by (1 + me |ω| ) p g(t − s) p (1 + X(s) ) p , which is dt ⊗ Pintegrable. Thus, Lebesgue's theorem on dominated convergence implies that J 2 → 0 as ε → 0 which completes the proof of stochastic continuity of K. In particular, stochastic continuity guarantees the existence of a predicable modification of K by [17,Prop. 3.21].
In summary, we obtain that K maps H T to H T . For applying Banach's fixed point theorem it is enough to show that K is a contraction for some β. We have K(X 1 ) − K(X 2 ) p T,β ≤ 2 p−1 K 1 (X 1 ) − K 1 (X 2 ) T,β + K 2 (X 1 ) − K 2 (X 2 ) T,β .
For the part corresponding to the drift we calculate similarly to [17,Th. 9.29] In the following calculation for the part corresponding to the diffusion we use in the first inequality the continuity of the stochastic integral formulated in Theorem 7: where C ′ (β) = c T 0 e −βs g(s) p ds → 0 as β → ∞. Consequently, Banach's fixed point theorem implies that there exists a unique X ∈ H T such that K(X) = X which completes the proof.
Remark 17. For processes of the form (22) the integrability assumption can be relaxed to include for example stable processes in the same way as in [13] where the existence of variational solutions is demonstrated. The details can be found in the PhD thesis of the first author.