Non-standard Skorokhod convergence of Levy-driven convolution integrals in Hilbert spaces
aa r X i v : . [ m a t h . P R ] A ug Non-standard Skorokhod convergence of L´evy-drivenconvolution integrals in Hilbert spaces
Ilya PavlyukevichInstitut f¨ur MathematikFriedrich–Schiller–Universit¨at JenaErnst–Abbe–Platz 207743 JenaGermany [email protected]
Markus RiedleDepartment of MathematicsKing’s CollegeStrandLondon WC2R 2LSUnited Kingdom [email protected]
August 9, 2018
Abstract
We study the convergence in probability in the non-standard M Skorokhodtopology of the Hilbert valued stochastic convolution integrals of the type R t F γ ( t − s ) dL ( s ) to a process R t F ( t − s ) dL ( s ) driven by a L´evy process L . In Banach spaceswe introduce strong, weak and product modes of M -convergence, prove a criterionfor the M -convergence in probability of stochastically continuous c`adl`ag processesin terms of the convergence in probability of the finite dimensional marginals and agood behaviour of the corresponding oscillation functions, and establish criteria forthe convergence in probability of L´evy driven stochastic convolutions. The theory isapplied to the infinitely dimensional integrated Ornstein–Uhlenbeck processes withdiagonalisable generators. AMS (2000) subject classification: ∗ , 60F17, 60G51, 60H05. Key words and phrases: M Skorokhod topology, stochastic convolution integral, L´evyprocess, Hilbert space, Banach space, convergence in probability, Ornstein–Uhlenbeck process,integrated Ornstein–Uhlenbeck process.
In many problems of engineering, physics or finance, the evolution of a random systemcan be described by stochastic convolution integrals of some kernel with respect to anoise process, see e.g. Barndorff–Nielsen and Shephard [2], Elishakoff [7], Pavlyukevichand Sokolov [16].The present work is originally motivated by the paper by Chechkin et al. [5], wherethe authors consider a simple model for the motion of a charged particle in a constantexternal magnetic field subject to α -stable L´evy perturbation. The particle’s position x ∈ R is described by the second-order Newtonian equation¨ x = ˙ x × B − ν ˙ x + ε ˙ ℓ, where B ∈ R is the direction of the magnetic field, ν , ε > ℓ is an isometric three-dimensional α -stable L´evy process with the characteristic function Ee i h u,ℓ ( t ) i = e − t | u | α -
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20 20 40 - - Figure 1: Sample paths of the convolution integrals A j X ( j ) γ ( t ) = R t (1 − e − γA j ( t − s ) ) dL ( s ), j = 1 , . . . ,
4, driven by a 1 . L for large γ > u ∈ R . Denoting the velocity ˙ x = v and the linear operator Av := − v × B + νv ,we obtain that the velocity process v satisfies the linear Ornstein–Uhlenbeck equation˙ v = − Av + ε ˙ ℓ, (1.1)whereas the coordinate is obtained by integration of the velocity v . Assuming that v = x = 0 we solve equation (1.1) explicitly to obtain v ( t ) = ε R t e − A ( t − s ) dℓ ( s ), andFubini’s theorem yields x ( t ) = εA − R t (1 − e − A ( t − s ) ) dℓ ( s ).It is possible to study the dynamics of x in the regime of the small noise perturbationby letting ε →
0. Indeed, performing a convenient time-change t ε − α t , using theself-similarity of α -stable processes, i.e. (cid:0) εℓ ( t/ε α ) : t > (cid:1) D = (cid:0) ℓ ( t ) : t > (cid:1) , and taking for convenience another copy L = ℓ of the driving process ℓ , we transfer thesmall noise amplitude into the large friction coefficient; that is the stochastic processes X and V , defined by X ( t ) := x ( t/ε α ) and V ( t ) := v ( t/ε α ) for all t >
0, satisfy theequations ˙ V = − ε α AV + ˙ L, ˙ X = 1 ε α V. By denoting the large parameter γ := ε − α we obtain the solutions V γ ( t ) = Z t e − γA ( t − s ) dL ( s ) , AX γ ( t ) = Z t (1 − e − γA ( t − s ) ) dL ( s ) . It can be shown (see Lemma 4.1) that if the eigenvalues of A have strictly positive realparts, then AX γ → L in probability with respect to an appropriate metric ( M ) in thesample path space.As an example, consider a two dimensional integrated Ornstein–Uhlenbeck processdriven by an α -stable L´evy process L as well as the corresponding sample paths t A j X ( j ) γ ( t ) of the integrated Ornstein–Uhlenbeck processes for the following matrices A j (see Figure 1): A = (cid:18) (cid:19) , A = (cid:18) (cid:19) , A = (cid:18) (cid:19) , A = (cid:18) − (cid:19) . M topology in infinite dimensional spaces. Theparticular example (1.1) of this introduction in one dimension is considered by Hintzeand Pavlyukevich in [8].As one of four topologies, the M topology in the path space D ([0 , , R ), the spaceof c`adl`ag functions f : [0 , T ] → R , was introduced in the seminal paper by Skorokhod[22]. An excellent account on convergence in the M topology in a multi-dimensionalsetting can be found in Whitt [28]. To the best of our knowledge, the M topology hasnot yet been considered in an infinite dimensional setting. Note, that in the M topologyit is possible that a continuous function, as the sample paths of AX , converges to adiscontinuous function, as the sample paths of the L´evy process L .In the present paper we study the following aspects of the M topology. First, wenotice that in a typical setting, such as considered in Whitt [28], one often obtainsconvergence in the M topology not only in the weak sense but also in probability.Second, we generalise the finite-dimensional setting of Skorokhod and Whitt to stochasticprocesses with values in separable Banach spaces. Here it turns out, that in addition tothe two kinds of M topologies in multi-dimensional spaces, a third kind of M topologyarises in infinite dimensional spaces.The second part of our work is concerned with convergence of stochastic convolutionintegrals in Hilbert spaces in the M topology. By considering stochastic convolution in-tegrals we may abandon the semimartingale setting. It is known, see Basse and Pedersen[3] and Basse–O’Connor and Rosi´nski [4], that even one-dimensional convolution inte-grals R t F ( t − s ) dL ( s ) define a semimartingale if and only if F is absolutely continuouswith sufficiently regular density.As a specific example, the case of integrated Ornstein–Uhlenbeck processes in aHilbert space is considered in the last section of this paper. It turns out that onlyin the case of a diagonalisable operator, convergence can be established, and then, onlyin the weakest sense. This result corresponds to the two-dimensional example above,where only in cases j = 1 and j = 2 the stochastic convolution integrals converge in the M topology. Notation:
For two values a, b ∈ R we denote a ∧ b := min { a, b } and a ∨ b := max { a, b } .The Euclidean norm in R d , d ≥
1, is denoted by | · | . A partition ( t i ) mi =1 of an interval[0 , T ] is a finite sequence of numbers t i ∈ [0 , T ] satisfying t < · · · < t m . For functions f : [0 , T ] → S , where S is a linear space with a norm k · k S , we define the supremum norm k f k ∞ := sup t ∈ [0 ,T ] k f ( t ) k S . The 2-variation of a function f : [0 , T ] → S is defined by k f k T V := sup m X k =0 k f ( t k ) − f ( t k − ) k , where the supremum is taken over all partitions of [0 , T ].Let U be a separable Banach space with norm k·k . The dual space is denoted by U ∗ with dual pairing h u, u ∗ i . The Borel σ -algebra in U is denoted by B ( U ). For anotherseparable Banach space V the space of bounded, linear operators from U to V is denotedby L ( U, V ) equipped with the norm topology k·k U → V .Let (Ω , A , P ) be a probability space. The space of equivalence classes of measurablefunctions f : Ω → U is denoted by L P (Ω; U ) and it is equipped with the topology ofconvergence in probability. The space of equivalence classes of measurable functionswhose p -th power has finite integral is denoted by L pP (Ω; U ) for p > cknowledgements: The authors thank the King’s College London and FSU Jena forhospitality. The second named author acknowledges the EPSRC grant EP/I036990/1.
In this section, we introduce the Skorokhod space and some of its topologies. Let V denote a separable Banach space. For a fixed time T >
0, the space of V -valued c`adl`agfunctions is denoted by D ([0 , T ]; V ). For each f ∈ D ([0 , T ]; V ) we define the set ofdiscontinuities by J ( f ) := { t ∈ (0 , T ] : f ( t − ) = f ( t ) } . The set J ( f ) is countably finite. The jump size at t is defined by (∆ f )( t ) = f ( t ) − f ( t − ).For two elements v , v ∈ V we define the segment as the straight line between v and v : [[ v , v ]] := { v ∈ V : v = αv + (1 − α ) v for α ∈ [0 , } . In order to define a metric on D ([0 , T ]; V ), the so-called (strong) M metric , we definefor each f ∈ D ([0 , T ]; V ) the extended graph of f byΓ( f ) := { ( t, v ) ∈ [0 , T ] × V : v ∈ [[ f ( t − ) , f ( t )]] } , where f (0 − ) := f (0). The projection of Γ( f ) to its spatial component in V is given by π (Γ( f )) := { v ∈ V : ( t, v ) ∈ Γ( f ) for some t ∈ [0 , T ] } . A total order relation on Γ( f ) is given by( t , v ) ( t , v ) ⇔ ( t < t or t = t and k f ( t − ) − v k k f ( t − ) − v k . . A parametric representation of the extended graph of f is a continuous, non-decreasing,surjective function( r, u ) : [0 , → Γ( f ) , ( r, u )(0) = (0 , f (0)) , ( r, u )(1) = ( T, f ( T )) . Let Π( f ) denote the set of all parametric representations of f . M topology For f , f ∈ D ([0 , T ]; V ) we define d M ( f , f ) := inf n | r − r | ∞ ∨ k u − u k ∞ : ( r i , u i ) ∈ Π( f i ) , i = 1 , o . As in the finite dimensional situation, cf. [28, Theorem 12.3.1], it follows that d M is a metric on D ([0 , T ]; V ), and we call it the strong M metric . The metric space (cid:0) D ([0 , T ]; V ) , d M (cid:1) is separable but not complete.Convergence of a sequence of functions in the metric d M can be described by quanti-fying the oscillation of the functions. For v, v , v ∈ V the distance from v to the segmentbetween v and v is defined by M ( v , v, v ) := inf α ∈ [0 , k v − ( αv + (1 − α ) v ) k . M obeys for every v, v , v , v ′ , v ′ , v ′ ∈ V the inequality M ( v , v, v ) M ( v ′ , v ′ , v ′ ) + k v − v ′ k + k v − v ′ k + k v − v ′ k , (2.1)and, instead of a triangular inequality, it satisfies M ( v + v ′ , v + v ′ , v + v ′ ) M ( v , v, v ) + k v ′ k + (cid:0) k v ′ k ∨ k v ′ k (cid:1) . (2.2)For functions f, g ∈ D ([0 , T ]; V ) and 0 t t t T it follows from (2.2) that M (cid:0) f ( t ) + g ( t ) , f ( t ) + g ( t ) , f ( t ) + g ( t ) (cid:1) M (cid:0) f ( t ) , f ( t ) , f ( t ) (cid:1) + 2 k g k ∞ , (2.3)and if t − t δ then M (cid:0) f ( t ) + g ( t ) , f ( t ) + g ( t ) , f ( t ) + g ( t ) (cid:1) M (cid:0) f ( t ) , f ( t ) , f ( t ) (cid:1) + sup s ,s ∈ [0 ,T ] | s − s | δ k g ( s ) − g ( s ) k . (2.4)Define for f ∈ D ([0 , T ]; V ) and δ > M ( f ; δ ) := sup n M (cid:16) f ( t ) , f ( t ) , f ( t ) (cid:17) : 0 t < t < t T and t − t δ o . Lemma 2.1.
Let f be in D ([0 , T ]; V ) and let t t t T with ( t i , v i ) ∈ Γ( f ) for some v i ∈ V and i = 1 , , . If t − t δ for some δ > then M ( v , v , v ) M ( f ; δ ) . Proof.
Follows as Lemma 12.5.2 in Whitt [28].
Lemma 2.2. If f ∈ D ([0 , T ]; V ) then lim δ ց M ( f ; δ ) = 0 . Proof.
Follows as Lemma 12.5.3 in [28].
Lemma 2.3.
Let f γn , f γ , f n , f , n ∈ N , γ > , be functions in D ([0 , T ]; R ) satisfying (i) lim n →∞ (cid:18) k f n − f k ∞ + lim sup γ →∞ k f γn − f γ k ∞ (cid:19) = 0 ; (ii) lim γ →∞ f γn = f n in ( D ([0 , T ]; R ) , d M ) for all n ∈ N .Then it follows that lim γ →∞ f γ = f in (cid:0) D ([0 , T ]; R ) , d M (cid:1) .Proof. Fix ε > n ∈ N such that k f n − f k ∞ ε and lim sup γ →∞ k f γn − f γ k ∞ ε for all n > n . Thus, there exists γ = γ ( n ) such that (cid:13)(cid:13) f γn − f γ (cid:13)(cid:13) ∞ ε for all γ > γ . D ⊆ [0 , T ] including 0 and T such that for each t ∈ D there exists a γ = γ ( t, n ) > (cid:12)(cid:12) f γn ( t ) − f n ( t ) (cid:12)(cid:12) ε for all γ > γ , (2.5)and that there exists a δ = δ ( n ) > γ →∞ M ( f γn , δ ) ε for all δ δ . (2.6)Consequently, we can conclude from (2.5) for each t ∈ D and γ > max { γ , γ } that | f γ ( t ) − f ( t ) | (cid:12)(cid:12) f γ ( t ) − f γn ( t ) (cid:12)(cid:12) + (cid:12)(cid:12) f γn ( t ) − f n ( t ) (cid:12)(cid:12) + | f n ( t ) − f ( t ) | ε. Thus we have shown that lim γ →∞ f γ ( t ) = f ( t ) for all t ∈ D. (2.7)It follows from (2.6) for each δ δ by inequality (2.3) thatlim sup γ →∞ M ( f γ , δ ) lim sup γ →∞ M ( f γn , δ ) + 2 lim sup γ →∞ (cid:13)(cid:13) f γn − f γ (cid:13)(cid:13) ε. (2.8)By (2.7) and (2.8) a final application of Theorem 12.5.1 in [28] completes the proof.The metric space ( D ([0 , T ]; V ) , d M ) is not complete. However, one can define anothermetric ˆ d M on D ([0 , T ]; V ) such that ( D ([0 , T ]; V ) , ˆ d M ) is complete and the two topologicalspaces ( D ([0 , T ]; V ) , d M ) and ( D ([0 , T ]; V ) , ˆ d M ) are homeomorphic, that is there existsa bijective function i : ( D ([0 , T ]; V ) , d M ) → ( D ([0 , T ]; V ) , ˆ d M ) such that both i and itsinverse are continuous, see [28, Theorem 12.8.1]. The last property, i.e. the existence of ahomeomorphic mapping between the metric spaces, is called the topological equivalence of( D ([0 , T ]; V ) , d M ) and ( D ([0 , T ]; V ) , ˆ d M ). In particular, this means that open, closed andcompact sets are the same in both spaces but also the spaces of real-valued, continuousfunctions on D ([0 , T ]; V ) coincide for both metrics. Moreover, since ( D ([0 , T ]; V ) , d M )is separable, the space ( D ([0 , T ]; V ) , ˆ d M ) is also separable, and thus it is Polish, i.e. atopological space which is metrisable as a complete separable space. M topology For the product M topology we assume that the Banach space V has a Schauder basis e = ( e k ) k ∈ N and that ( e ∗ k ) k ∈ N denotes the bi-orthogonal functionals. Instead of equipping D ([0 , T ]; V ) with the strong M topology we can consider the space as the Cartesianproduct space Q ∞ k =1 D ([0 , T ]; R ) and equip it with the product metric d eM ( f, g ) := ∞ X k =1 k d M ( h f, e ∗ k i , h g, e ∗ k i )1 + d M ( h f, e ∗ k i , h g, e ∗ k i ) for all f, g ∈ D ([0 , T ]; V ) . The metric on the right hand side refers to the metric on the space D ([0 , T ]; R ) intro-duced in the previous Section 2.1. Clearly, convergence in d eM depends on the chosen6chauder basis e of V . Alternatively, we can use the topological equivalent metric ˆ d M on D ([0 , T ]; R ) to define the product metricˆ d eM ( f, g ) := ∞ X k =1 k ˆ d M ( h f, e ∗ k i , h g, e ∗ k i )1 + ˆ d M ( h f, e ∗ k i , h g, e ∗ k i ) for all f, g ∈ D ([0 , T ]; V ) . Since ( D ([0 , T ]; R ) , ˆ d M ) is a Polish space it follows that ( D ([0 , T ]; V ) , ˆ d eM ) is Polish, too.Analogously, we obtain that ( D ([0 , T ]; V ) , d eM ) is topological equivalent to ( D ([0 , T ]; V ) , ˆ d eM ).Recall, that the product topology is the topology of point-wise convergence , i.e. a sequence( f n ) n ∈ N converges to f in ( D ([0 , T ]; V ) , d eM ) if and only if for any k ∈ N lim n →∞ d M (cid:0) h f n , e ∗ k i , h f, e ∗ k i (cid:1) = 0 . M topology In an infinite dimensional Banach space V there is a third mode of convergence in the M sense. A sequence ( f n ) n ∈ N ⊆ D ([0 , T ]; V ) is said to converge weakly to f ∈ D ([0 , T ]; V )if for all v ∗ ∈ V ∗ we havelim n →∞ h f n , v ∗ i = h f, v ∗ i in (cid:0) D ([0 , T ]; R ) , d M (cid:1) . Note, that if V is infinite dimensional the induced topology is not metrisable. The threedifferent modes of convergence are related as shown by the following diagram:strong M ⇒ weak M ⇒ product M .The first implication follows from the fact that f , f ∈ D ([0 , T ]; V ) obey the inequality d M ( h f , v ∗ i , h f , v ∗ i ) ( k v ∗ k ∨ d M ( f , f ) for all v ∗ ∈ V ∗ . Since the product topology is the point-wise convergence it is immediate that it is impliedby weak convergence.If V is finite dimensional it is known that the weak and strong topology coincide,see Theorem 12.7.2 in [28]. In the infinite dimensional situation the situation differs asillustrated by the following example. Example 2.4.
Let V be an arbitrary Hilbert space with orthonormal basis ( e k ) k ∈ N .The functions f n : [0 , T ] → V can be chosen as f n ( t ) := e n for all t ∈ [0 , T ] and n ∈ N .It follows that sup t ∈ [0 ,T ] |h f n ( t ) , v i| = |h e n , v i| → n → ∞ for all v ∈ V, and thus, the sequence ( f n ) n ∈ N converges weakly to 0 in D ([0 , T ]; V ). However, since k f n ( t ) k = 1 for all n ∈ N and t ∈ [0 , T ] it does not converge strongly. Example 2.5.
It is well known that in a finite dimensional Hilbert space V convergencein the product topology does not imply strong convergence in the M metric. Since inthe finite dimensional situation, the strong M topology coincides with the topology ofweak convergence in D ([0 , T ]; V ), this is also an example that convergence in the producttopology does not imply weak convergence.7inally let us remark that our notion of the modes of convergence in the M sensedoes not coincide with the one in the literature such as [28]. There the product topologyis called weak topology, whereas our weak topology does not have a name as it coincideswith the strong topology in finite dimensional spaces. Since addition is not continuousin D ([0 , T ]; V ) our notion of weak convergence can not be confused with the usual weaktopology in a linear topological vector space. Let B ( D ) denote the Borel- σ -algebra generated by open sets in ( D ([0 , T ]; V ) , d M ). Asin the finite dimensional situation, see [28, Theorem 11.5.2], it can be shown that B ( D )coincides with the σ -algebra, generated by the coordinate mappings π t ,...,t n : D ([0 , T ]; V ) → V n , π t ,...,t n ( f ) = ( f ( t ) , . . . , f ( t n ))for each t , . . . , t n ∈ [0 , T ] and n ∈ N . The analogous result for the Skorokhod J topologyin separable metric spaces can be found in [9]. On the other hand, the Borel- σ -algebragenerated by open sets in ( D ([0 , T ]; V ) , d eM ) equals the product of Borel- σ -algebras in( D ([0 , T ]; R ) , d M ). Consequently, both Borel- σ -algebras, generated by open sets withrespect to d M and d eM coincide.Let (Ω , A , P ) be a probability space and let X := ( X ( t ) : t ∈ [0 , T ]) be a V -valuedstochastic process with c`adl`ag paths. Since the Borel- σ -algebra B ( D ) is generated by thecoordinate mappings, it follows that X is a D ([0 , T ]; V )-valued random variables. In this section we consider the convergence in probability in the strong metric d M ofstochastic processes ( X n ) n ∈ N to a stochastic process X in the space D ([0 , T ]; V ). If thestochastic processes X and X n have c`adl`ag paths then X and X n are D ([0 , T ); V )-valuedrandom variables. Since (cid:0) D ([0 , T ); V ) , d M (cid:1) is separable, convergence in probability is welldefined in the sense that ( X n ) n ∈ N converges to X in probability in (cid:0) D ([0 , T ); V ) , d M (cid:1) iflim n →∞ P (cid:16) d M ( X n , X ) > ε (cid:17) = 0 for all ε > . Lemma 3.1. A V -valued stochastically continuous stochastic process ( X ( t ) : t ∈ [0 , T ]) with c`adl`ag trajectories obeys lim δ ց sup t ∈ [0 ,T ] P sup s ∈ [0 ,T ] | s − t | δ k X ( t ) − X ( s ) k > ε = 0 for all ε > . Proof.
Define for each δ > t ∈ [0 , T ] the random variable Z ( t, δ ) := sup s ∈ [0 ,T ] | s − t | δ k X ( t ) − X ( s ) k and assume for a contradiction that there exist ε , ε >
0, a sequence ( δ n ) n ∈ N ⊆ R + converging to 0, and a sequence ( t n ) n ∈ N ⊆ [0 , T ] such thatlim n →∞ P (cid:0) Z ( t n , δ n ) > ε (cid:1) > ε .
8y passing to a subsequence if necessary we can assume that t n → t for some t ∈ [0 , T ].Then, for each δ > n ( δ ) ∈ N such that[ t n − δ n , t n + δ n ] ⊆ [ t − δ, t + δ ] for all n > n ( δ ) , which implies by the definition of Z that Z ( t n , δ n ) Z ( t , δ ) for all n > n ( δ ) . Consequently, we obtain for every δ > n ( δ ) ∈ N such that ε E (cid:2) { Z ( t n ,δ n ) > ε } (cid:3) E (cid:2) { Z ( t ,δ ) > ε } (cid:3) for all n > n ( δ ) . (3.1)On the other hand, since Z ( t , δ ) → | ∆ X ( t ) | P -a.s. as δ ց
0, Lebesgue’s theorem ofdominated convergence implies that0 = P (cid:0) | ∆ X ( t ) | > ε (cid:1) = E (cid:20) lim δ ց { Z ( t ,δ ) > ε } (cid:21) = lim δ ց E (cid:2) { Z ( t ,δ ) > ε } (cid:3) , which contradicts (3.1). Theorem 3.2.
For V -valued, stochastically continuous stochastic processes ( X ( t ) : t ∈ [0 , T ]) and ( X n ( t ) : t ∈ [0 , T ]) , n ∈ N , with c`adl`ag trajectories the following are equiva-lent: (a) X n → X in probability in (cid:0) D ([0 , T ]; V ) , d M (cid:1) as n → ∞ . (b) the following two conditions are satisfied: (i) for every t ∈ [0 , T ] we have lim n →∞ X n ( t ) = X ( t ) in probability (ii) for every ε > the oscillation function obeys lim δ ց lim sup n →∞ P (cid:0) M ( X n , δ ) > ε (cid:1) = 0 . (3.2) Proof. (a) ⇒ (b) To establish property (i), let t ∈ [0 , T ] and ε , ε > δ > E ( ε , δ ) := (cid:26) ω ∈ Ω : sup s ∈ [0 ,T ] | s − t | δ k X ( t )( ω ) − X ( s )( ω ) k ε (cid:27) , satisfy P (cid:0) E ( ε , δ ) (cid:1) > − ε . By the assumed condition (a) there is n ∈ N such that P (cid:0) d M ( X n , X ) < ( ε ∧ δ ) (cid:1) > − ε n > n . Consequently, the set F ( ε , δ, n ) := E ( ε , δ ) ∩ { d M ( X n , X ) < ( ε ∧ δ ) } satisfies P (cid:0) F ( ε , δ, n ) (cid:1) > − ε for every n > n . Define for ω ∈ F ( ε , δ, n ) the functions f n := X n ( · )( ω ) and f := X ( · )( ω ). It follows that there are parametric representations( r, u ) ∈ Π( f ) and ( r n , u n ) ∈ Π( f n ) satisfying | r − r n | ∞ ∨ k u − u n k ∞ ( ε ∧ δ ) for all n > n . (3.3)9or every t ∈ [0 , T ] denote τ, τ n ∈ [0 ,
1] for n > n such that( t, f ( t )) = ( r ( τ ) , u ( τ )) and ( t, f n ( t )) = ( r n ( τ n ) , u n ( τ n )) . Since u ( τ n ) ∈ [[ f ( r ( τ n ) − ) , f ( r ( τ n ))]] for every n > n , there is α n ∈ [0 ,
1] such that u ( τ n ) = α n f ( r ( τ n ) − ) + (1 − α n ) f ( r ( τ n )) . Since t = r n ( τ n ) and | r ( τ n ) − r n ( τ n ) | δ for all n > n by (3.3), we have | r ( τ n ) − t | δ .Another application of inequality (3.3) implies k u ( τ n ) − u ( τ ) k = k α n f ( r ( τ n ) − ) + (1 − α n ) f ( r ( τ n )) − f ( t ) k sup s ∈ [0 ,T ] | s − t | δ sup α ∈ [0 , k αf ( s − ) + (1 − α ) f ( s ) − f ( t ) k = sup s ∈ [0 ,T ] | s − t | δ sup α ∈ [0 , (cid:13)(cid:13) α (cid:0) f ( s − ) − f ( t ) (cid:1) + (1 − α ) (cid:0) f ( s ) − f ( t ) (cid:1)(cid:13)(cid:13) sup s ∈ [0 ,T ] | s − t | δ k f ( s − ) − f ( t ) k + sup s ∈ [0 ,T ] | s − t | δ k f ( s ) − f ( t ) k ε . (3.4)Inequalities (3.3) and (3.4) imply for every n > n that k f ( t ) − f n ( t ) k = k u ( τ ) − u n ( τ n ) k k u ( τ ) − u ( τ n ) k + k u ( τ n ) − u n ( τ n ) k ε , which establishes Condition (i) in (b).In order to show Condition (ii) fix some ε , ε >
0. Lemma 2.2 guarantees that thereexists δ > G ( ε , δ ) := { ω ∈ Ω : M ( X ( ω ) , δ ) ε } satisfies P ( G ( ε , δ )) > − ε for all δ ∈ [0 , δ ]. For each δ > n ∈ N by theassumed condition (a) such that P (cid:0) d M ( X n , X ) < ( ε ∧ δ ) (cid:1) > − ε n > n . Together we obtain for every δ ∈ [0 , δ ]lim inf n →∞ P (cid:0) G ( ε , δ ) ∩ { d M ( X n , δ ) < ( ε ∧ δ ) } (cid:1) > − ε . Fix ω ∈ G ( ε , δ ) ∩ { d M ( X n , δ ) < ( ε ∧ δ ) } and define f := X ( · )( ω ) and f n := X n ( · )( ω ).It follows that there are parametric representations ( r, u ) ∈ Π( f ) and ( r n , u n ) ∈ Π( f n )satisfying | r − r n | ∞ ∨ k u − u n k ∞ ( ε ∧ δ ) for all n > n . For every 0 t t t denote τ i , τ i,n ∈ [0 ,
1] such that ( t i , f ( t i )) = ( r ( τ i ) , u ( τ i )) and( t i , f n ( t i,n )) = ( r n ( τ i,n ) , u n ( τ i,n )) for i = 1 , ,
3. Inequality (2.1) and Lemma 2.1 implyfor every n > n that M (cid:0) f n ( t ) , f n ( t ) , f n ( t ) (cid:1) = M (cid:0) u n ( τ ,n ) , u n ( τ ,n ) , u n ( τ ,n ) (cid:1) M (cid:0) u ( τ ,n ) , u ( τ ,n ) , u ( τ ,n ) (cid:1) + 3 k u − u n k ∞ M ( f, δ ) + 3 k u − u n k ∞ ε + 3 ε = 4 ε , ⇒ (b).(b) ⇒ (a). Let ε , ε > n ∈ N and δ > G ( ε , δ ) := n ω ∈ Ω : M ( X ( ω ) , δ ) < ε o ,G n ( ε , δ ) := n ω ∈ Ω : M ( X n ( ω ) , δ ) < ε o . Condition (ii) guarantees that there exist δ > n ∈ N such thatlim sup n →∞ P (cid:0) G cn ( ε , δ ) (cid:1) ε δ ∈ [0 , δ ] . Consequently, for each δ ∈ [0 , δ ] there exists n = n ( δ ) such thatsup n > n P (cid:0) G cn ( ε , δ ) (cid:1) ε , (3.5)whereas Lemma 2.2 implies that there exist δ > P (cid:0) G ( ε , δ ) (cid:1) > − ε δ ∈ [0 , δ ] . (3.6)Define for c > B ( c ) := n ω ∈ Ω : k X ( ω ) k ∞ c − o . Since X is a random variable with values in D ([0 , T ]; V ) there exists c > P (cid:0) B ( c ) (cid:1) > − ε . (3.7)Choose a partition π = ( t i ) mi =0 of the interval [0 , T ] such that0 = t < t < · · · < t m = T and max i ∈{ ,...,m } | t i − t i − | min { δ , δ , ε } , and define the set F n ( ε , π ) := n ω ∈ Ω : max i =1 ,...,m k X n ( t i )( ω ) − X ( t i )( ω ) k < ε o . Condition (i) guarantees that there exists n ∈ N such thatsup n > n P (cid:16) F cn ( ε , π ) (cid:17) ε . (3.8)It follows from (3.5) to (3.8) that for δ := δ ∧ δ the set E n ( ε , δ, c, π ) := G n ( ε , δ ) ∩ G ( ε , δ ) ∩ B ( c ) ∩ F n ( ε , π ) obeys P (cid:16) E n ( ε , δ, c, π ) (cid:17) > − ε for all n > n ∨ n . For ω ∈ E n ( ε , δ, c, π ) define f ( · ) := X ( · )( ω ) and f n ( · ) := X n ( · )( ω ). Let N denote theintegers { n ∨ n , . . . } and N the union N ∪{ } . For n ∈ N and i ∈ { , . . . , m } let Γ ni bethe graph of f n between ( t i − , f n ( t i − )) and ( t i , f n ( t i )). By defining d i to be the smallest11nteger larger than k f ( t i − ) − f ( t i ) k ε we can divide the segment [[ f ( t i − ) , f ( t i )]] inequidistant points ξ i,j := f ( t i − ) + α i,j ( f ( t i ) − f ( t i − )) for α i,j := jd i , j = 0 , . . . , d i . We claim that for each i ∈ { , . . . , m } the balls B i,j := (cid:8) h ∈ V : k h − ξ i,j k < ε (cid:9) coverseach of the graphs Γ in for n ∈ N , i.e. π (Γ in ) ⊆ d i [ j =0 B i,j for all n ∈ N . (3.9)Indeed, let ( t, h ) ∈ Γ ni be of the form h = αf n ( t − ) + (1 − α ) f n ( t ) for some α ∈ [0 , t ∈ [ t i − , t i ] and t i − t i − δ it follows from the definition of M ( f n , δ ) that thereexists ℓ n , r n ∈ [[ f n ( t i − ) , f n ( t i )]] such that k f n ( t − ) − ℓ n k ε and k f n ( t ) − r n k ε for all n ∈ N . Since u n := αℓ n + (1 − α ) r n ∈ [[ f n ( t i − ) , f n ( t i )]] we have M ( f n ( t i − ) , u n , f n ( t i )) = 0.Inequality (2.1) implies that M ( f ( t i − ) , u n , f ( t i )) M ( f n ( t i − ) , u n , f n ( t i )) + k f ( t i − ) − f n ( t i − ) k + k f ( t i ) − f n ( t i ) k i ∈{ ,...,m } k f n ( t i ) − f ( t i ) k < ε . Consequently, there exists u ∈ [[ f ( t i − ) , f ( t i )]] such that k u n − u k ε . (If n = 0we can choose u = u n .) Since u ∈ [[ f ( t i − ) , f ( t i )]] we can choose the closest node ξ i,j for some j = 0 , . . . , d i such that k u − ξ i,j k ε . It follows k h − ξ i,j k = (cid:13)(cid:13)(cid:0) αf n ( t − ) + (1 − α ) f n ( t ) (cid:1) − ξ i,j (cid:13)(cid:13) α k f n ( t − ) − ℓ n k + (1 − α ) k f n ( t ) − r n k + k u n − u k + k u − ξ i,j k ε
512 + 2 ε
512 + ε ε , which shows (3.9).In the following, we define for each i ∈ { , . . . , m } and n ∈ N an ordered sequenceof points (cid:0) ( r ni, , z ni, ) , . . . , ( r ni,m i , z ni,m i ) (cid:1) ∈ (cid:0) Γ ni × · · · × Γ ni (cid:1) , for some m i ∈ N , independent of n , such that they satisfy for every j = 1 , . . . , m i :sup ( z,r ) ∈ Γ ni,j max (cid:8)(cid:13)(cid:13) z − z ni,j − (cid:13)(cid:13) , (cid:13)(cid:13) z − z ni,j (cid:13)(cid:13) , (cid:12)(cid:12) r − r ni,j − (cid:12)(cid:12) , (cid:12)(cid:12) r − r ni,j (cid:12)(cid:12)(cid:9) ε , (3.10)where Γ ni,j := { ( r, z ) ∈ Γ ni : ( r ni,j − , z ni,j − , ) ( r, z ) ( r ni,j , z ni,j ) } .If d i = 1 we define m i = 1 and for every n ∈ N the points:( r ni, , z ni, ) := ( t i − , f n ( t i − )) , ( r ni, , z ni, ) := ( t i , f n ( t i )) .
12t follows from (3.9) that for each ( r, z ) ∈ Γ ni there is k ∈ { , } such that z ∈ B i,k . For k = 0 this results in (cid:13)(cid:13) z − z ni, (cid:13)(cid:13) ∨ (cid:13)(cid:13) z − z ni, (cid:13)(cid:13) k z − ξ i, k + k ξ i, − ξ i, k + k ξ i, − f n ( t i ) k ε
25 + ε
16 + ε N ( n ) ε , (3.11)and analogously for k = 1. Since each r ∈ [ r ni, , r ni, ] satisfies (cid:12)(cid:12) r − r ni,j (cid:12)(cid:12) (cid:12)(cid:12) r ni, − r ni, (cid:12)(cid:12) ε for j ∈ { , } we obtain the inequality (3.10).If d i = 2 we define m i = 3 but we distinguish two cases. Firstly, assume that π (Γ ni ) ⊆ B i, ∪ B i, . Then we define for each n ∈ N the points( r ni, , z ni, , ) := ( t i − , f n ( t i − )) , ( r ni, , z ni, ) := ( t i , f n ( t i ))and we choose z ni, , z ni, ∈ π (Γ ni ) ∩ B i, ∩ B i, and r ni, , r ni, ∈ [0 ,
1] such that( r ni, , z ni, ) < ( r ni, , z ni, ) < ( r ni, , z ni, ) < ( r ni, , z ni, ) . In the case π (Γ in ) B i, ∪ B i, we define for every n ∈ N the points( r ni, , z ni, ) := ( t i − , f n ( t i − )) , ( r ni, , z ni, ) := inf { ( r, z ) ∈ Γ in : ( r, z ) > ( r ni, , z ni, ) and z ∈ ∂B i, } , ( r ni, , z ni, ) := inf { ( r, z ) ∈ Γ in : ( r, z ) > ( r ni, , z ni, ) and z ∈ ∂B i, } , ( r ni, , z ni, ) := ( t i , f n ( t i )) . If d i > m i = d i + 1. Since k ξ i,j − ξ i,j − k > d i − d i ε > ε we have B i,j ∩ B i,j +2 = ∅ for all j = 1 , . . . , d i . Thus, we can define the following increasingsequence:( r ni, , z ni, ) := ( t i − , f n ( t i − )) , ( r ni,j , z ni,j ) := inf { ( r, z ) ∈ Γ ni : ( r, z ) > ( r ni,j − , z ni,j − ) and z ∈ ∂B i,j } , j = 1 , . . . , d i , ( r ni,m i , z ni,m i ) := ( t i , f n ( t i )) . In both cases for d i = 2 and in the case d i > n ∈ N that (cid:13)(cid:13) z ni, − ξ i, (cid:13)(cid:13) = k f n ( t i − ) − f ( t i − ) k < ε , (cid:13)(cid:13) z ni,m i − ξ i,d i (cid:13)(cid:13) = k f n ( t i ) − f ( t i ) k < ε . (3.12)Consequently, z ni, ∈ B i, and z ni,m i ∈ B i,d i and thus z ni, , z ni, ∈ ¯ B i, and z ni,m i − , z ni,m i ∈ ¯ B i,d i for every n ∈ N . Since z ni,j − , z ni,j ∈ ¯ B i,j − for all j = 2 , . . . , m i − (cid:13)(cid:13) z ni,j − − z ni,j (cid:13)(cid:13) ε
25 for all j ∈ { , . . . , m i } , n ∈ N . (3.13)If ( r, z ) ∈ Γ ni,j for some j ∈ { , . . . , m i } and n ∈ N then M ( z ni,j − , z, z ni,j ) < ε since (cid:12)(cid:12) r ni,j − − r ni,j (cid:12)(cid:12) | t i − − t i | δ . Thus, there exists z ∈ [[ z ni,j − , z ni,j , ]] such that k z − z k ε . Together with (3.13) it follows for each ( r, z ) ∈ Γ ni,j and k ∈ { j − , j } for j = 1 , . . . , m i that (cid:13)(cid:13) z − z ni,k (cid:13)(cid:13) k z − z k + (cid:13)(cid:13) z − z ni,k (cid:13)(cid:13) k z − z k + (cid:13)(cid:13) z ni,j − − z ni,j (cid:13)(cid:13) ε
512 + 2 ε ε . (cid:12)(cid:12) r − r ni,k (cid:12)(cid:12) | t i − t i − | ε , we obtain (3.10).The constructed sequence exhibits a further property: since for every i ∈ { , . . . , m } and n ∈ N the points z i,j and z ni,j are in the same closed ball ¯ B i,j for j ∈ { , m i } by(3.12) and for j ∈ { , . . . , m i − } by construction , it follows thatsup i ∈{ ,...,m } j ∈{ ,...,m i } (cid:8)(cid:13)(cid:13) z i,j − z ni,j (cid:13)(cid:13)(cid:9) ε . Since (cid:12)(cid:12) r i,j − r ni,j (cid:12)(cid:12) | t i − t i − | ε we obtain thatsup i ∈{ ,...,m } j ∈{ ,...,m i } max (cid:8)(cid:13)(cid:13) z i,j − z ni,j (cid:13)(cid:13) , (cid:12)(cid:12) r i,j − r ni,j (cid:12)(cid:12)(cid:9) ε . for all n ∈ N. (3.14)By gluing together we obtain for each n ∈ N an ordered sequence (cid:0) ( r n , , z n , ) , . . . , ( r n ,m , z n ,m ) , ( r n , , z n , ) , . . . , ( r nm,m m , z nm,m m ) (cid:1) ∈ (Γ n × · · · × Γ n ) , satisfying the inequalities (3.10) and (3.14). It follows as in the proof of the implication( vi ) ⇒ ( i ) of Theorem 12.5.1 in [28], that one can define for every parametric represen-tation ( r, u ) ∈ Π( f ) a parametric representation ( r n , u n ) ∈ Π( f n ) such that | r − r n | ∞ ∨ k u − u n k ∞ ε ε
25 for all n ∈ N. Thus, we have shown that for each ω ∈ E n ( ε , δ, c, π ) we have d M ( X ( ω ) , X n ( ω )) ε ε
25 for all n ∈ N, which completes the proof. In this part we equip the space D ([0 , T ]; V ) with the product topology d eM for a fixedSchauder basis e := ( e k ) k ∈ N of V with bi-orthogonal sequence ( e ∗ k ) k ∈ N , and we considerthe convergence in probability of stochastic processes ( X n ) n ∈ N to a stochastic process X . For stochastic process X and ( X n ) n ∈ N with c`adl`ag trajectories we say that ( X n ) n ∈ N converges to X in probability in (cid:0) D ([0 , T ]; V ) , d eM (cid:1) iflim n →∞ P (cid:16) d eM ( X n , X ) > ε (cid:17) = 0 for all ε > . Since the product topology corresponds to point-wise convergence, the stochastic pro-cesses ( X n ) n ∈ N converges to X in probability in (cid:0) D ([0 , T ]; V ) , d eM (cid:1) if and only if for every k ∈ N lim n →∞ P (cid:16) d M (cid:0) h X n , e ∗ k i , h X, e ∗ k i (cid:1) > ε (cid:17) = 0 for all ε > , (3.15)see [10, Lemma 4.4.4]. Consequently, we obtain as an analogue of Theorem 3.2:14 orollary 3.3. Let ( e k ) k ∈ N be a Schauder basis of V with bi-orthogonal sequence ( e ∗ k ) k ∈ N .For V -valued, stochastically continuous stochastic processes ( X ( t ) : t ∈ [0 , T ]) and ( X n ( t ) : t ∈ [0 , T ]) , n ∈ N , with c`adl`ag trajectories the following are equivalent: (a) X n → X in probability in (cid:0) D ([0 , T ]; V ) , d eM (cid:1) as n → ∞ ; (b) the following two conditions are satisfied for every k ∈ N : (i) for every t ∈ [0 , T ] we have lim n →∞ h X n ( t ) , e ∗ k i = h X ( t ) , e ∗ k i in probability; (ii) for every ε > the oscillation function obeys lim δ ց lim sup n →∞ P (cid:0) M ( h X n , e ∗ k i , δ ) > ε (cid:1) = 0 . Proof.
Follows immediately from Theorem 3.2 and (3.15).
Recall that the weak M topology in an infinite dimensional Hilbert space is not metris-able. A sequence ( X n ) n ∈ N of stochastic processes ( X n ) n ∈ N with trajectories in D ([0 , T ]; V )is said to converge weakly in M in probability to a process X with trajectories in D ([0 , T ]; V ) if for all v ∗ ∈ V ∗ we havelim n →∞ h X n , v ∗ i = h X, v ∗ i in probability in (cid:0) D ([0 , T ]; R ) , d M (cid:1) .Equivalently, by using the metric d M in D ([0 , T ]; R ), this convergence takes place if andonly if for each v ∗ ∈ V ∗ we havelim n →∞ P (cid:16) d M (cid:0) h X n , v ∗ i , h X, v ∗ i (cid:1) > ε (cid:17) = 0 for all ε > . (3.16)By comparing (3.16) with (3.15) one can colloquially describe the difference betweenconvergence in the weak sense and in the product topology by testing the one-dimensionalprojections either with all elements, i.e. h X n , v ∗ i for all v ∗ ∈ V ∗ , or only with the bi-orthogonal elements of V ∗ , i.e. h X n , e ∗ k i for all k ∈ N . Clearly, the first one is independentof the chosen basis. Corollary 3.4.
For V -valued, stochastically continuous stochastic processes ( X ( t ) : t ∈ [0 , T ]) and ( X n ( t ) : t ∈ [0 , T ]) , n ∈ N , with c`adl`ag trajectories the following are equiva-lent: (a) X n → X weakly in M in probability in D ([0 , T ]; V ) as n → ∞ ; (b) the following two conditions are satisfied for every v ∗ ∈ V ∗ : (i) for every t ∈ [0 , T ] we have lim n →∞ h X n ( t ) , v ∗ i = h X ( t ) , v ∗ i in probability; (ii) for every ε > the oscillation function obeys lim δ ց lim sup n →∞ P (cid:0) M ( h X n , v ∗ i , δ ) > ε (cid:1) = 0 . Proof.
Follows immediately from Theorem 3.2 and (3.16).15 emark 3.5.
In this part we always require that the considered stochastic processeshave c`adl`ag paths in the Hilbert space V . If V is infinite dimensional this might be atoo restrictive assumption. In fact the definition of weak convergence only requires thatthe stochastic processes have cylindrical c`adl`ag trajectories , that is (cid:0) h X ( t ) , v ∗ i : t ∈ [0 , T ] (cid:1) , (cid:0) h X n ( t ) , v ∗ i : t ∈ [0 , T ] (cid:1) , n ∈ N , have c`adl`ag trajectories for all v ∗ ∈ V ∗ . Since Corollary 3.4 is just proved by theapplication of Theorem 3.2 to these real-valued stochastic processes we could easily softenour assumption on the path regularities of the considered stochastic processes accordingly.The same comment applies to convergence in the product topology ( D ([0 , T ]; V ) , d eM )for a Schauder basis e = ( e k ) k ∈ N of V with bi-orthogonal sequence e = ( e ∗ k ) k ∈ N . Here itis sufficient to require that the considered stochastic processes have D -cylindrical c`adl`agtrajectories for D = { e ∗ , e ∗ , . . . } , that is (cid:0) h X ( t ) , e ∗ k i : t ∈ [0 , T ] (cid:1) , (cid:0) h X n ( t ) , e ∗ k i : t ∈ [0 , T ] (cid:1) , n ∈ N , have c`adl`ag trajectories for all k ∈ N . The notions of cylindrical c`adl`ag and D -cylindricalc`adl`ag paths can be found in [19].In order to have a clearer presentation of our paper, and not at least since our focusis rather on the different modes of convergence instead of the subtle issue of temporalregularity, we require stochastic processes to have c`adl`ag trajectories in the underlyingBanach space. However, if necessary, it should be obvious how to extend our results tostochastic processes with c`adl`ag trajectories only in the cylindrical sense. In this section we apply our results of Section 3 to the convergence of stochastic convo-lution integrals with respect to L´evy process. Although it would be possible to continuewith the general setting of Banach spaces with a Schauder basis, we restrict ourselveshere to Hilbert spaces in order to make use of standard integration theory as in [6]. Inthis case, we identify the dual spaces U ∗ and V ∗ with the separable Hilbert spaces U and V .Let ξ be an infinitely divisible Radon measure on B ( U ). Then the characteristicfunction of ξ is given by ϕ ξ : U → C , ϕ ξ ( u ) = exp (cid:0) Ψ( u ) (cid:1) , where the L´evy symbol ψ : U → C is defined byΨ( u ) = i h a, u i − h Qu, u i + Z U (cid:16) e i h u,r i − − i h u, r i B U ( r ) (cid:17) ν ( dr ) , where a ∈ U , Q : U → U is the covariance operator of a Gaussian Radon measure on B ( U ) and ν is a σ -finite measure on B ( U ) with ν ( { } ) = 0 and Z U (cid:0) k r k ∧ (cid:1) ν ( dr ) < ∞ . Consequently, the triplet ( a, Q, ν ) characterises the distribution of the Radon measure ξ and thus, it is called it the characteristics of ξ . If X is an U -valued random variablewhich is infinitely divisible then we call the characteristics of its probability distribution16he characteristics of X . The L´evy symbol Ψ : U ∗ → C is sequentially weakly continuousand satisfies | Ψ( u ) | c (1 + k u k ) for all u ∈ U, (4.1)for a constant c > {F t } t > be a filtration for the probability space (Ω , A , P ). An adapted stochasticprocess L := ( L ( t ) : t >
0) with values in U is called a L´evy process if L (0) = 0 P -a.s., L has independent and stationary increments and L is continuous in probability. It followsthat there exists a version of L with paths which are continuous from the right and havelimits from the left (c`adl`ag paths). In the sequel we always assume that a L´evy processhas c`adl`ag paths. Clearly, the random variable L (1) is infinitely divisible and we call itscharacteristics the characteristics of L .In the work [6], Chojnowska–Michalik introduces a theory of stochastic integrationfor deterministic, operator-valued integrands with respect to a U -valued L´evy process.Another approach in a more general setting can be found in [21] but we follow here [6].Let V be another separable Hilbert space and define H ( U, V ) := ( F : [0 , T ] → L ( U, V ) : F is measurable, Z T k F ( s ) k U → V ds < ∞ ) . For F ∈ H ( U, V ) we denote by F ∗ ( t ) the adjoint operator ( F ( t )) ∗ : V → U for each t ∈ [0 , T ]. In [6], the author starts with step functions in H ( U, V ) to define a stochasticintegral and finally shows, that for each element in H ( U, V ) this stochastic integral existsas the limit of the stochastic integrals for step functions in H ( U, V ) in the topology ofconvergence in probability. We denote this stochastic integral for F ∈ H ( U, V ) withrespect to the L´evy process L by I ( F ) := Z T F ( s ) dL ( s ) . If Ψ is the L´evy symbol of L then the stochastic integral I ( F ) is infinitely divisible andhas the characteristic function ϕ I ( F ) : V → C , ϕ I ( F ) ( v ) = exp Z T Ψ( F ∗ ( s ) v ) ds ! . (4.2)By firstly considering step functions and then passing to the limit, one can show that foreach F ∈ H ( U, V ) the stochastic integral I ( F ) obeys h Z T F ( s ) dL ( s ) , v i = Z T F ∗ ( s ) v dL ( s ) P -a.s. for all v ∈ V . (4.3)Here, the right hand side is understood as the same stochastic integral but for the inte-grand F ∗ ( · ) v ∈ H ( U, R ). If F ∈ H ( U, V ) is for some v ∈ V of the special form F ∗ ( t ) v = ϕ ( t ) Gv for all t ∈ [0 , T ] , for a function ϕ : R → R and G ∈ L ( V, U ) then one obtains Z T F ∗ ( s ) v dL ( s ) = Z T ϕ ( s ) dℓ ( s ) , (4.4)17here ℓ denotes the real-valued L´evy process defined by ℓ ( t ) := h L ( t ) , Gv i . If F ∈H ( U, V ) is of the special form F ( · ) = S ( · ) G for some G ∈ L ( U, V ) and S ∈ H ( V, V ),we obtain Z T G ∗ S ∗ ( s ) v dL ( s ) = Z T S ∗ ( s ) v dK ( s ) , (4.5)where K is the L´evy process in V defined by K ( t ) := GL ( t ) for all t > F ∈ H ( U, V ) we define the stochastic convolution integral process F ∗ L := ( F ∗ L ( t ) : t ∈ [0 , T ]) by F ∗ L ( t ) := Z t F ( t − s ) dL ( s ) for all t ∈ [0 , T ] . In this section we apply our results of Section 3 to the convergence of stochastic convo-lution integral processes in the weak and product topology M , that is for functions F , F γ ∈ H ( U, V ), depending on a parameter γ >
0, we establish the convergencelim γ →∞ F γ ∗ L = F ∗ L in probability in the weak and product topology.The study of the limiting behaviour requires that the stochastic processes have c`adl`agpaths in V , or at least in the appropriate cylindrical sense as pointed out in Remark 3.5.There is no condition for regularities of trajectories available covering our rather generalsetting but for numerous specific situations one knows sufficient conditions guaranteeingeither continuous or c`adl`ag trajectories of stochastic convolution integrals. For example,classical results on continuity of Gaussian processes can be found in [14] and [23], and onregularity of infinitely divisible processes in [24]; temporal path regularity of stochasticconvolution integrals are considered in [11] and [13], the infinite-dimensional Ornstein-Uhlenbeck process is treated in [15] and [17]. As our work is focused on the convergencerather than regularities of trajectories we will assume the following in this section: Assumption A:
For all considered functions F , F γ ∈ H ( U, V ), γ >
0, the stochasticprocesses F ∗ L and F γ ∗ L , γ >
0, have c`adl`ag trajectories.Furthermore, if W denotes the Gaussian part of L then the stochastic process F ∗ W has continuous trajectories.We do not assume that F γ ∗ W has continuous paths but only the prospective limit F ∗ W . This is a quite natural assumption for the M topology that only the limit iscontinuous. Lemma 4.1.
Let F , F γ , γ > , be functions in H ( U, V ) satisfying for a subset D ⊆ V and all u ∈ U (i) F ∗ ( · ) v, F ∗ γ ( · ) v ∈ D ([0 , T ]; U ) for all v ∈ D and γ >
0; (4.6)(ii) sup γ> kh F γ ( · ) u, v ik ∞ < ∞ for all v ∈ D ; (4.7)(iii) for each v ∈ D there exists a Lebesgue null set B ∈ B ([0 , T ]) such that lim γ →∞ h (cid:0) F γ ( s ) − F ( s ) (cid:1) u, v i = 0 for all s ∈ B c , u ∈ U. (4.8)18 hen for each t ∈ [0 , T ] and v ∈ D we have lim γ →∞ h F γ ∗ L ( t ) , v i = h F ∗ L ( t ) , v i in probability.Proof. Define for each t ∈ [0 , T ] and γ > X γ ( t ) := Z t (cid:0) F γ ( t − s ) − F ( t − s ) (cid:1) dL ( s ) . By linearity of the stochastic integral and since the Euclidean topology in R n coincideswith the product topology it is sufficient to prove that for each v ∈ D and t ∈ [0 , T ] wehave h X γ ( t ) , v i → R as γ → ∞ .Let Ψ denote the L´evy symbol of L . Due to equality (4.3) we obtain for the characteristicfunction of X γ for β ∈ R that E (cid:2) exp (cid:0) iβ h X γ ( t ) , v i (cid:1)(cid:3) = E (cid:20) exp (cid:18) iβ Z t (cid:0) F ∗ γ ( t − s ) − F ∗ ( t − s ) (cid:1) v dL ( s ) (cid:19)(cid:21) = exp (cid:18)Z t Ψ (cid:0)(cid:0) F ∗ γ ( t − s ) − F ∗ ( t − s ) (cid:1) ( βv ) (cid:1) ds (cid:19) = exp (cid:18)Z t Ψ (cid:0)(cid:0) F ∗ γ ( s ) − F ∗ ( s ) (cid:1) ( βv ) (cid:1) ds (cid:19) . (4.9)Fix v ∈ D and define for each γ > T γ : U → D ([0 , T ] , k·k ∞ ) , T γ u := h u, F ∗ γ ( · ) v i . Condition (4.7) guarantees for each u ∈ U thatsup γ> k T γ u k ∞ = sup γ> (cid:13)(cid:13) h u, F ∗ γ ( · ) v i (cid:13)(cid:13) ∞ < ∞ . Thus, the uniform boundedness principle implies M := sup γ> sup s ∈ [0 ,T ] (cid:13)(cid:13) F ∗ γ ( s ) v (cid:13)(cid:13) U = sup γ> sup s ∈ [0 ,T ] sup k u k (cid:12)(cid:12) h u, F ∗ γ ( s ) v i (cid:12)(cid:12) = sup γ> k T γ k U → D < ∞ . The estimate (4.1) for the L´evy symbol Ψ implies that there exists a constant c > γ > s ∈ [0 , T ] (cid:12)(cid:12) Ψ (cid:0) ( F ∗ γ ( s ) − F ∗ ( s ))( βv ) (cid:1)(cid:12)(cid:12) c (cid:16) (cid:13)(cid:13)(cid:0) F ∗ γ ( s ) − F ∗ ( s ) (cid:1) ( βv ) (cid:13)(cid:13) (cid:17) c (cid:16) | β | (cid:16)(cid:13)(cid:13) F ∗ γ ( s ) v (cid:13)(cid:13) + k F ∗ ( s ) v k (cid:17)(cid:17) c (cid:16) | β | (cid:16) M + k v k k F ( s ) k U → V (cid:17)(cid:17) . Since Condition (4.8) implies by the sequentially weak continuity of the L´evy symbolΨ : U ∗ → C thatlim γ →∞ Ψ (cid:0) ( F ∗ γ ( s ) − F ∗ ( s ))( βv ) (cid:1) = 0 for Lebesgue almost all s ∈ [0 , T ] , Lebesgue’s theorem of dominated convergence enables us to concludelim γ →∞ Z t Ψ (cid:0) ( F ∗ γ ( s ) − F ∗ ( s ))( βv ) (cid:1) ds = 0 , which completes the proof by (4.9). 19 .2 The reproducing kernel Hilbert space In this subsection we fix a L´evy process L in U with characteristics ( a, Q, ν ) and let α be a positive constant. The L´evy process L can be decomposed into L ( t ) = W ( t ) + X α ( t ) + Y α ( t ) for all t >
0, (4.10)where W is a Wiener process with covariance operator Q : U → U , and X α and Y α are U -valued L´evy processes with characteristic functions ϕ X α ( t ) ( u ) = exp − t Z k r k α (cid:16) e i h r,u i − − i h r, u i (cid:17) ν ( dr ) ! ,ϕ Y α ( t ) ( u ) = exp it h b α , u i − t Z α< k r k (cid:16) e i h r,u i − (cid:17) ν ( dr ) ! , for u ∈ U and t >
0. The element b α ∈ U is determined by the characteristics of L andby the constant α . Since X α (1) has finite moments we can define the covariance operator R α : U → U of X α (1) by h R α u , u i = E (cid:2) h X α (1) , u ih X α (1) , u i (cid:3) for all u , u ∈ U. Since R α is positive and symmetric there exists a separable Hilbert space H α with norm k·k H α and an embedding i α : H → U satisfying R α = i α i ∗ α . In particular, we havelim α → k i α k H α → U = 0 , (4.11)which follows from the estimate k i ∗ α u k H α = h R α u, u i = Z k r k α h u, r i ν ( dr ) k u k Z k r k α k r k ν ( dr )for each u ∈ U . One obtains for α β h R α u, u i = Z k r k α h u, r i ν ( dr ) Z k r k β h u, r i ν ( dr ) = h R β u, u i for all u ∈ U. One can deduce from Riesz representation theorem, see [26, Proposition 1.1] or [27,Section 1.1], that H α ⊆ H β and the embedding H α → H β is contractive.Since the range of i ∗ α is dense in H α and H α is separable there exits a basis ( h αk ) k ∈ N ⊆ i ∗ α ( U ). We choose u αk ∈ U such that i ∗ α u αk = h αk for all k ∈ N and we define real-valuedL´evy processes ℓ αk by ℓ αk ( t ) := h X α ( t ) , u αk i for all t ∈ [0 , T ]. The L´evy process X α can berepresented by X α ( t ) = ∞ X k =1 i α h αk ℓ αk ( t ) for all t ∈ [0 , T ] , (4.12)where the sum converges weakly in L P (Ω; U ), i.e. h X α ( t ) , u i = ∞ X k =1 h i α h αk , u i ℓ k ( t ) in L P (Ω; R ) for all t ∈ [0 , T ] and u ∈ U. The representation (4.12) is called
Karhunen–Lo`eve expansion , and it can be derived inthe same way as it is done in Riedle [20] for Wiener processes. Note, that it follows easilyfrom their definition that the L´evy processes ( ℓ k ) k ∈ N are uncorrelated.20 .3 Estimating the small jumps We begin with a generalisation to the Hilbert space setting of a result of Marcus andRosi´nski in [12] on a maximal inequality of convolution integrals. We apply here thedecomposition (4.10) of the L´evy process L for some α > Lemma 4.2.
A function f ∈ H ( U, R ) satisfies for each α > the estimate E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t f ( t − s ) dX α ( s ) (cid:12)(cid:12)(cid:12)(cid:12) κ √ T ∞ X k =1 kh f ( · ) , i α h αk ik T V , where κ := 32 √ R p ln(1 /s ) ds .Proof. Since α > X can be represented by X ( t ) = ∞ X k =1 ih k ℓ k ( t ) for all t ∈ [0 , T ] , where the sum converges weakly in L P (Ω; U ). Since the real-valued L´evy processes( ℓ k ) k ∈ N are uncorrelated we obtain E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t f ( t − s ) dX ( s ) (cid:12)(cid:12)(cid:12)(cid:12) = E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k =1 Z t h f ( t − s ) , ih k i dℓ k ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k =1 E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t h f ( t − s ) , ih k i dℓ k ( s ) (cid:12)(cid:12)(cid:12)(cid:12) . In order to estimate the expectation of the real-valued stochastic integrals on the righthand side we follow some arguments by Marcus and Rosi´nski in [13]. Let ℓ ′ k be anindependent copy of ℓ k and define the symmetrisation ˆ ℓ k := ℓ k − ℓ ′ k for each k ∈ N .Since E [ ℓ k ( t )] = 0 for all t > k ∈ N one obtains E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t h f ( t − s ) , ih k i dℓ k ( s ) (cid:12)(cid:12)(cid:12)(cid:12) = E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12) E (cid:20)Z t h f ( t − s ) , ih k i dℓ k ( s ) − Z t h f ( t − s ) , ih k i dℓ ′ ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ℓ k ( T ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) E " sup t ∈ [0 ,T ] E (cid:20)(cid:12)(cid:12)(cid:12)(cid:12)Z t h f ( t − s ) , ih k i d ˆ ℓ k ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ k ( T ) (cid:21) E " E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t h f ( t − s ) , ih k i d ˆ ℓ k ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ k ( T ) = E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t h f ( t − s ) , ih k i d ˆ ℓ k ( s ) (cid:12)(cid:12)(cid:12)(cid:12) . Theorem 1.1 in [12] guarantees for all k ∈ N that E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t h f ( t − s ) , ih k i d ˆ ℓ k ( s ) (cid:12)(cid:12)(cid:12)(cid:12) κ E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T (cid:13)(cid:13) h f ( · ) , ih k i [0 ,T − s ] (cid:13)(cid:13) T V d ˆ ℓ k ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . λ k of ℓ k is given by λ k := ν α ◦ h· , u k i − where ν α denotes theL´evy measure of X we conclude E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T (cid:13)(cid:13) h f ( · ) , ih k i [0 ,T − s ] (cid:13)(cid:13) T V d ˆ ℓ k ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T (cid:13)(cid:13) h f ( · ) , ih k i [0 ,T − s ] (cid:13)(cid:13) T V d ˆ ℓ k ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 2 Z T Z R (cid:13)(cid:13) h f ( · ) , ih k i [0 ,T − s ] (cid:13)(cid:13) T V β λ k ( dβ ) ds = 2 Z T (cid:13)(cid:13) h f ( · ) , ih k i [0 ,T − s ] (cid:13)(cid:13) T V ds Z U h u, u k i ν α ( du )= 2 Z T (cid:13)(cid:13) h f ( · ) , ih k i [0 ,T − s ] (cid:13)(cid:13) T V ds h Ru k , u k i = 2 Z T (cid:13)(cid:13) h f ( · ) , ih k i [0 ,T − s ] (cid:13)(cid:13) T V ds T kh f ( · ) , ih k ik T V , (4.13)which completes the proof.The bound on the right hand side in Lemma 4.2 depends on the regularity of thefunction f and of the covariance structure of the underlying L´evy process. It is a naturalgeneralisation to the infinite dimensional setting of the result in [12].We will later consider the special case, that the integrands of the stochastic convolu-tion integrals can be diagonalised with respect to an orthonormal basis. In this case, wecan improve the analogue estimate of Lemma 4.2. Lemma 4.3.
Let V be a Hilbert space with an orthonormal basis ( e k ) k ∈ N and let F ∈H ( U, V ) be a function of the form F ∗ ( · ) e k = ϕ k ( · ) Ge k for all k ∈ N , for c`adl`ag functions ϕ k : [0 , T ] → R and G ∈ L ( V, U ) . Then it follows for each α > and v ∈ V that E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t F ∗ ( t − s ) v dX α ( s ) (cid:12)(cid:12)(cid:12)(cid:12) κ √ T ∞ X k =1 |h v, e k i| k ϕ k ( · ) k T V Z k u k α |h u, Ge k i| ν ( du ) ! / , where κ := 32 √ R p ln(1 /s ) ds .Proof. For each k ∈ N define the real-valued L´evy process x αk by defining x αk ( t ) =22 X α ( t ) , Ge k i for all t >
0. It follows from (4.4) for each v ∈ V that E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t F ∗ ( t − s ) v dX α ( s ) (cid:12)(cid:12)(cid:12)(cid:12) = E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k =1 h v, e k i Z t ϕ k ( t − s ) Ge k dX α ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k =1 h v, e k i Z t ϕ k ( t − s ) dx αk ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k =1 |h v, e k i| E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t ϕ k ( t − s ) dx αk ( s ) (cid:12)(cid:12)(cid:12)(cid:12) . As in the proof of Lemma 4.2 we obtain E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t ϕ k ( t − s ) dx αk ( s ) (cid:12)(cid:12)(cid:12)(cid:12) κ E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T (cid:13)(cid:13) ϕ k ( · ) [0 ,T − s ] (cid:13)(cid:13) T V d ˆ x αk ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where ˆ x αk := x αk − x α ′ k denotes the symmetrisation of x αk for each k ∈ N by an independentcopy x α ′ k of x αk . Since the L´evy measure λ αk of x αk is given by λ αk = ν α ◦ h· , Ge k i − where ν α denotes the L´evy measure of X α we conclude by a similar calculation as in (4.13) that E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T (cid:13)(cid:13) ϕ k ( · ) [0 ,T − s ] (cid:13)(cid:13) T V d ˆ x αk ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T k ϕ k ( · ) k T V Z k u k α h u, Ge k i ν ( du ) . Summarising the estimates above completes the proof.
In this section we consider the convergence in the weak topology and the product topol-ogy. For the latter we assume that e = ( e k ) k ∈ N is an orthonormal basis of V . Theorem 4.4.
Let
F, F γ ∈ H ( U, V ) , γ > , be functions satisfying for a subset D ⊆ V (i) F ∗ ( · ) v, F ∗ γ ( · ) v ∈ D ([0 , T ]; U ) for all v ∈ D and γ >
0; (4.14)(ii) sup γ> kh F γ ( · ) u, v ik ∞ < ∞ for all u ∈ U, v ∈ D ; (4.15)(iii) lim sup α → sup γ> ∞ X k =1 (cid:13)(cid:13) h F ∗ γ ( · ) v, i α h αk i (cid:13)(cid:13) T V = 0 for all v ∈ D ; (4.16)(iv) lim γ →∞ d M (cid:0) h F γ ( · ) u, v i , h F ( · ) u, v i (cid:1) = 0 for all u ∈ U, v ∈ D. (4.17)(1) If { e , e , . . . } ⊆ D then it follows lim γ →∞ (cid:0) F γ ∗ L ( t ) : t ∈ [0 , T ] (cid:1) = (cid:0) F ∗ L ( t ) : t ∈ [0 , T ] (cid:1) in probability in the product topology (cid:0) D ([0 , T ]; V ) , d eM (cid:1) . (2) If V = D then it follows lim γ →∞ (cid:0) F γ ∗ L ( t ) : t ∈ [0 , T ] (cid:1) = (cid:0) F ∗ L ( t ) : t ∈ [0 , T ] (cid:1) weakly in probability in D ([0 , T ]; V ) . roof. We show that each v ∈ D the stochastic processes X and X γ , γ >
0, defined by X ( t ) := h Z t F ( t − s ) dL ( s ) , v i , X γ ( t ) := h Z t F γ ( t − s ) dL ( s ) , v i , t ∈ [0 , T ],satisfy the conditions in Theorem 3.2. Note, that for 0 t t T and β ∈ R we have E [exp ( iβ ( X ( t ) − X ( t )))]= exp (cid:18)Z t Ψ (cid:0) ( F ∗ ( t − s ) − F ∗ ( t − s ))( βv ) (cid:1) ds (cid:19) exp (cid:18)Z t t Ψ (cid:0) ( F ∗ ( t − s )( βv ) (cid:1) ds (cid:19) . Since F ∗ ( · )( βv ) is Lebesgue almost everywhere continuous due to (4.14) it follows thatthe stochastic process X and analogously X γ are stochastically continuous.It follows from Condition (4.17) by Theorem 12.5.1 in [28] for each u ∈ U and v ∈ D that lim γ →∞ h F γ ( t ) u, v i = h F ( t ) u, v i for all t ∈ (cid:0) J ( h F ( · ) u, v i ) (cid:1) c . (4.18)Since the set J ( F ∗ ( · ) v ) of discontinuities of F ∗ ( · ) v is a Lebesgue null set by Condition(4.14) and satisfies (cid:0) J ( h F ( · ) u, v i ) (cid:1) ⊆ (cid:0) J ( F ∗ ( · ) v ) (cid:1) for every u ∈ U , Lemma 4.1 guaranteesthat Condition (i) in Theorem 3.2 is satisfied.In order to show Condition (ii) we have to establish for every ε > v ∈ D thatlim δ ց lim sup γ →∞ P (cid:18) M (cid:18) h Z · F γ ( · − s ) dL ( s ) , v i , δ (cid:19) > ε (cid:19) = 0 . (4.19)For this purpose, fix some constants ε , ε > v ∈ D . Condition (4.16) enables usto choose a constant α > γ> ∞ X k =1 (cid:13)(cid:13) h F ∗ γ ( · ) v, i α h αk i (cid:13)(cid:13) T V ε ε κ √ T (4.20)for κ := 32 √ R p ln(1 /s ) ds . As in (4.10) we decompose the L´evy process L into L ( t ) = W ( t ) + X α ( t ) + Y α ( t ) for all t >
0, but we suppress the notion of α in the sequel.Here, W is a Wiener process with covariance operator Q : U → U , and X and Y are U -valued L´evy processes with characteristic functions ϕ X ( t ) ( u ) = exp − t Z k r k α (cid:16) e i h r,u i − − i h r, u i (cid:17) ν ( dr ) ! ,ϕ Y ( t ) ( u ) = exp it h b, u i − t Z α< k r k (cid:16) e i h r,u i − (cid:17) ν ( dr ) ! , for all t > u ∈ U . It follows from (4.3) for every t ∈ [0 , T ] and γ > h Z t F γ ( t − s ) dL ( s ) , v i = Z t F ∗ γ ( t − s ) v dL ( s ) . By the decomposition of L we obtain the representation I γ ( t ) := Z t F ∗ γ ( t − s ) v dL ( s ) = C γ ( t ) + A γ ( t ) + B γ ( t ) , C γ ( t ) := Z t F ∗ γ ( t − s ) v dW ( s ) ,A γ ( t ) := Z t F ∗ γ ( t − s ) v dY ( s ) , B γ ( t ) := Z t F ∗ γ ( t − s ) v dX ( s ) . In the sequel, we will consider the three stochastic integrals separately.1) We show that lim δ ց lim sup γ →∞ P sup t ,t ∈ [0 ,T ] | t − t | δ | C γ ( t ) − C γ ( t ) | > ε = 0 . (4.21)Let the covariance operator Q of W be decomposed into Q = i Q i ∗ Q for some i Q ∈L ( H Q , U ) and for a Hilbert space H Q . Since the characteristic function ϕ W (1) of W (1)is sequentially weakly continuous and is given by ϕ W (1) ( u ) = exp (cid:16) − k i ∗ u k H Q (cid:17) for all u ∈ U, also the function i ∗ : U → H Q is sequentially weakly continuous. Consequently, we canconclude from (4.18) thatlim γ →∞ (cid:13)(cid:13) i ∗ Q (cid:0) F ∗ γ ( s ) v − F ∗ ( s ) v ) (cid:13)(cid:13) H Q = 0 for Lebesgue-a.a. s ∈ [0 , T ] . Due to (4.15) Lebesgue’s theorem of dominated convergence implieslim γ →∞ Z T (cid:13)(cid:13) i ∗ Q (cid:0) F ∗ γ ( s ) v − F ∗ ( s ) v ) (cid:13)(cid:13) H Q ds = 0 . (4.22)Let I := [0 , T ] ∩ Q and define for each γ > G γ := ( G γ ( t ) : t ∈ I )where G γ ( t ) := C γ ( t ) − h F ∗ W ( t ) , v i . Since G γ has independent increments it is areal-valued Gaussian process with a.s. bounded sample paths due to Assumption A. Bydenoting M γ := sup t ∈ I G γ ( t ) it follows for each δ > t ,t ∈ I | t − t | δ | C γ ( t ) − C γ ( t ) | t ∈ I | G γ ( t ) − E [ M γ ] | + sup t ,t ∈ I | t − t | δ |h F ∗ W ( t ) − F ∗ W ( t ) , v i| . (4.23)Borell’s inequality, see [1, Theorem 2.1], (or Borell–Tsirelson–Ibragimov–Sudakov in-equality) implies E [ M γ ] < ∞ and that for every ε > P (cid:18) sup t ∈ I | G γ ( t ) − E [ M γ ] | > ε (cid:19) P (cid:18) sup t ∈ I G γ ( t ) − E [ M γ ] > ε (cid:19) exp (cid:18) − ε σ γ (cid:19) , where σ γ := sup t ∈ I E [ G γ ( t ) ]. Since equation (4.22) guarantees σ γ = Z T (cid:13)(cid:13) i ∗ Q (cid:0) F ∗ γ ( s ) v − F ∗ ( s ) v (cid:1)(cid:13)(cid:13) H Q ds → γ → ∞ ,
25e can conclude lim γ →∞ P (cid:18) sup t ∈ I | G γ ( t ) − E [ M γ ] | > ε (cid:19) = 0 . (4.24)As the stochastic process ( h F ∗ W ( t ) , v i : t ∈ [0 , T ]) is continuous according to AssumptionA it follows lim δ ց sup t ,t ∈ I | t − t | δ |h F ∗ W ( t ) − F ∗ W ( t ) , v i| = 0 P -a.s. (4.25)Equations (4.24) and (4.25) establish (4.21).2) Lemma 4.2 implies by Markov inequality for each γ > P sup t ∈ [0 ,T ] | B γ ( t ) | > ε ! κ √ Tε ∞ X k =1 (cid:13)(cid:13) h F ∗ γ ( · ) v, i α h αk i (cid:13)(cid:13) T V ε . (4.26)3) Let ( N ( t ) : t ∈ [0 , T ]) denote the counting process for Y , i.e. N ( t ) := X s ∈ [0 ,t ] {k ∆ Y ( s ) k >α } for all t ∈ [0 , T ] , and let τ j , j ∈ N ∪{ } , denote the jump times of Y , recursively defined by τ = 0 and τ j := inf (cid:8) t > τ j − : k ∆ Y ( t ) k > α (cid:9) for all j ∈ N with inf ∅ = ∞ . Note, that the jump times of Y are countable in increasing order sincethe jump size of Y is bounded from below by α . Since Y is a pure jump process withdrift b , we obtain A γ ( t ) = Z t F ∗ γ ( t − s ) v dY ( s ) = R γ ( t ) + S γ ( t ) , where we define for every t ∈ [0 , T ] and γ > R γ ( t ) := N ( t ) X j =0 R jγ ( t ) , R jγ ( t ) := h F ∗ γ ( t − τ j ) v, ∆ Y ( τ j ) i [ τ j , ∞ ) ( t ) ,S γ ( t ) := Z t h F ∗ γ ( t − s ) v, b i ds. Analogously, we have A ( t ) = Z t F ∗ ( t − s ) v dY ( s ) = R ( t ) + S ( t ) , where we define for every t ∈ [0 , T ] R ( t ) := N ( t ) X j =0 R j ( t ) , R j ( t ) := h F ∗ ( t − τ j ) v, ∆ Y ( τ j ) i [ τ j , ∞ ) ( t ) ,S ( t ) := Z t h F ∗ ( t − s ) v, b i ds. R j := ( R j ( t ) : t ∈ [0 , T ]) has c`adl`ag pathsfor each j ∈ N ∪{ } and the random set J ( R j ) of its discontinuities satisfies J ( R j ) ⊆ n t ∈ [0 , T ] : t − τ j ∈ J (cid:0) F ∗ ( · ) v (cid:1) , t ∈ ( τ j , T ] o ∪ { τ j } . Consequently, the set of joint discontinuities of R i and R j for i < j satisfies J ( R i ) ∩ J ( R j ) ⊆ n t ∈ [0 , T ] : t = τ j ( ω ) − τ i ( ω ) ∈ J (cid:0) F ∗ ( · ) v (cid:1) − J (cid:0) F ∗ ( · ) v (cid:1) for some ω ∈ Ω o , where we apply the convention ∞ − ∞ = ∞ . Since the deterministic set J (cid:0) F ∗ ( · ) v (cid:1) − J (cid:0) F ∗ ( · ) v (cid:1) = { t ∈ [0 , T ] : t = s − s for some s , s ∈ J (cid:0) F ∗ ( · ) v (cid:1) } is at most countable and the random vector ( τ i , τ j ) is absolutely continuous for every i < j it follows that P (cid:0) R i and R j have joint discontinuities (cid:1) = P (cid:0) J ( R i ) ∩ J ( R j ) (cid:1) P (cid:0) τ j − τ i ∈ J (cid:0) F ∗ ( · ) v (cid:1) − J (cid:0) F ∗ ( · ) v (cid:1)(cid:1) = 0 . Consequently, for the stochastic processes ( R j ) j ∈ N restricted to the complement S c ofthe null set S := ∞ [ i =2 i − [ j =1 (cid:8) ω ∈ Ω : τ i ( ω ) − τ j ( ω ) ∈ J (cid:0) F ∗ ( · ) v (cid:1) ∩ J (cid:0) F ∗ ( · ) v (cid:1)(cid:9) , there does not exist any i = j such that R i and R j have joint discontinuities. SinceCondition (4.17) guarantees that for each ω ∈ Ω h F ∗ γ ( · − τ j ( ω )) v, ∆ Y ( τ j )( ω ) i → h F ∗ ( · − τ j ( ω )) v, ∆ Y ( τ j )( ω ) i in ( D ([0 , T ]; R ) , d M ) it follows that R jγ ( ω ) → R j ( ω ) in ( D ([0 , T ]; R ) , d M ) for all ω ∈ Ωand j ∈ N . Due to the disjoint sets of discontinuities on S c , Corollary 12.7.1 in [28]guarantees that R γ ( ω ) converges to R ( ω ) in ( D ([0 , T ]; R ) , d M ) for all ω ∈ S c . Thedeterministic analogue of Theorem 3.2, i.e. [28, Theorem 12.5.1], implies thatlim δ ց lim sup γ →∞ M ( R γ ( ω ); δ ) = 0 for all ω ∈ S c .By combining with Fatou’s Lemma it follows that there exists δ > δ δ we havelim sup γ →∞ P (cid:16) M ( R γ ; δ ) > ε (cid:17) P (cid:16) lim sup γ →∞ { M ( R γ ; δ ) > ε } (cid:17) = P (cid:16) lim sup γ →∞ M ( R γ ; δ ) > ε (cid:17) ε . Since inequality (2.4) implies M ( A γ ; δ ) = M ( R γ + S γ ; δ ) M ( R γ ; δ ) + sup s ,s ∈ [0 ,T ] | s − s | δ | S γ ( s ) − S γ ( s ) | M ( R γ ; δ ) + δ sup γ> kh v, F γ ( · ) b ik ∞ ,
27t follows that for all δ δ where δ := min { δ , ε (sup γ> kh v, F γ ( · ) b ik ∞ ) − } we havelim sup γ →∞ P (cid:16) M ( A γ ; δ ) > ε (cid:17) lim sup γ →∞ P (cid:16) M ( R γ ; δ ) > ε (cid:17) ε . (4.27)4) It follows from (4.21) and(4.26) that there exists δ > γ > E γ := ( sup | t − t | δ | C γ ( t ) − C γ ( t ) | ε ) ∩ ( sup t ∈ [0 ,T ] | B γ ( t ) | ε ) satisfies P ( E γ ) > − ε . It follows from (4.27) by the inequalities (2.3) and (2.4) for all δ min { δ , δ } thatlim sup γ →∞ P (cid:16) M ( I γ ; δ )) > ε (cid:17) lim sup γ →∞ P (cid:0) M ( A γ ; δ ) > ε (cid:1) + lim sup γ →∞ P ( E cγ ) ε , which shows (4.19) and thus, Condition (ii) in Theorem 3.2. In this section we consider the special case that the integrands can be diagonalised. Forthat purpose, assume that ( e k ) k ∈ N is an orthonormal basis of V . Corollary 4.5.
Let
F, F γ ∈ H ( U, V ) , γ > , be functions of the form F ∗ ( s ) e k = ϕ k ( s ) Ge k , F ∗ γ ( s ) e k = ϕ kγ ( s ) Ge k for all s ∈ [0 , T ] , k ∈ N , for c`adl`ag functions ϕ k , ϕ kγ : [0 , T ] → R and G ∈ L ( V, U ) . If (i) sup γ> (cid:13)(cid:13) ϕ kγ (cid:13)(cid:13) ∞ < ∞ for all k ∈ N ; (4.28)(ii) sup γ> (cid:13)(cid:13) ϕ kγ (cid:13)(cid:13) T V < ∞ for all k ∈ N ; (4.29)(iii) lim γ →∞ ϕ kγ = ϕ k in (cid:0) D ([0 , T ]; R ) , d M (cid:1) for all k ∈ N . (4.30) then it follows for each ε > and k ∈ N that lim γ →∞ (cid:0) F γ ∗ L ( t ) : t ∈ [0 , T ] (cid:1) = (cid:0) F ∗ L ( t ) : t ∈ [0 , T ] (cid:1) in probability in the product topology (cid:0) D ([0 , T ]; V ) , d eM (cid:1) .Proof. The proof is analogously to the proof of Theorem 4.4 for D = { e , e , . . . } butonly the estimate (4.26) is derived in the following way: for each k ∈ N and γ > E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t F ∗ γ ( t − s ) e k dX ( s ) (cid:12)(cid:12)(cid:12)(cid:12) κ √ T k ϕ k ( · ) k T V Z k u k α |h u, Ge k i| ν ( du ) ! / , where κ := 32 √ R p ln(1 /s ) ds and α > ε , ε > k ∈ N theconstant α > P sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t F ∗ γ ( t − s ) e k dX ( s ) (cid:12)(cid:12)(cid:12)(cid:12) > ε ! ε γ > , ϕ k , ϕ kγ and (4.28). Condition (4.17) is satisfied sinceCondition (4.30) implies by Theorem 12.7.2 in [28] for each k ∈ N and u ∈ U thatlim γ →∞ h F γ ( · ) u, e k i = lim γ →∞ ϕ kγ ( · ) h u, Ge k i = ϕ k ( · ) h u, Ge k i = h F ( · ) u, e k i , in (cid:0) D ([0 , T ]; R ) , d M (cid:1) . Corollary 4.6.
Let
F, F γ ∈ H ( U, V ) , γ > , be functions of the form F ∗ ( s ) e k = ϕ k ( s ) Ge k , F ∗ γ ( s ) e k = ϕ kγ ( s ) Ge k for all s ∈ [0 , T ] , k ∈ N , for c`adl`ag functions ϕ k , ϕ kγ : [0 , T ] → R and G ∈ L ( V, U ) . If (i) F ∗ ( · ) v, F ∗ γ ( · ) v ∈ D ([0 , T ]; U ) for all v ∈ V and γ >
0; (4.31)(ii) sup γ> sup k ∈ N (cid:13)(cid:13) ϕ kγ (cid:13)(cid:13) ∞ < ∞ ; (4.32)(iii) sup γ> sup k ∈ N (cid:13)(cid:13) ϕ kγ (cid:13)(cid:13) T V < ∞ ; (4.33)(iv) for each n ∈ N we have lim γ →∞ (cid:0) ϕ γ , . . . , ϕ nγ (cid:1) = (cid:0) ϕ , . . . , ϕ n (cid:1) in (cid:0) D ([0 , T ]; R n ) , d M (cid:1) , (4.34) then it follows that lim γ →∞ (cid:0) F γ ∗ L ( t ) : t ∈ [0 , T ] (cid:1) = (cid:0) F ∗ L ( t ) : t ∈ [0 , T ] (cid:1) weakly in probability in D ([0 , T ]; V ) .Proof. The proof is analogous to the proof of Theorem 4.4 for D = V but only theestimate (4.26) is derived in the following way: for each v ∈ V and γ > E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t F ∗ γ ( t − s ) v dX ( s ) (cid:12)(cid:12)(cid:12)(cid:12) κ √ T ∞ X k =1 |h v, e k i| (cid:13)(cid:13) ϕ kγ ( · ) (cid:13)(cid:13) T V Z k u k α |h u, Ge k i| ν ( du ) ! / κ √ T sup k ∈ N (cid:13)(cid:13) ϕ kγ ( · ) (cid:13)(cid:13) T V ∞ X k =1 h v, e k i ! / ∞ X k =1 Z k u k α h u, Ge k i ν ( du ) ! / = κ √ T sup k ∈ N (cid:13)(cid:13) ϕ kγ ( · ) (cid:13)(cid:13) T V k v k Z k u k α k G ∗ u k ν ( du ) ! / , where κ := 32 √ R p ln(1 /s ) ds and α > ε , ε > α > P sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t F ∗ γ ( t − s ) v dX ( s ) (cid:12)(cid:12)(cid:12)(cid:12) > ε ! ε γ > , u ∈ U and v ∈ V sup γ> kh F γ ( · ) u, v ik ∞ k G ∗ u k k v k sup γ> sup k ∈ N (cid:13)(cid:13) ϕ kγ (cid:13)(cid:13) ∞ < ∞ , which establishes Condition (4.15). For the last part, fix u ∈ U and v ∈ V and define foreach γ > n ∈ N the functions f := h F ( · ) u, v i , f γ := h F γ ( · ) u, v i ,f n := n X k =1 h u, Ge k ih v, e k i ϕ k , f γn := n X k =1 h u, Ge k ih v, e k i ϕ kγ . Cauchy–Schwarz inequality impliessup γ> k f γn − f γ k ∞ = sup γ> (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k = n +1 ϕ kγ ( · ) h u, Ge k ih v, e k i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ sup γ> sup k ∈ N (cid:13)(cid:13) ϕ kγ (cid:13)(cid:13) ∞ ∞ X k = n +1 h G ∗ u, e k i ! ∞ X k = n +1 h v, e k i ! → , n → ∞ , and analogously we obtain k f n − f k ∞ → n → ∞ . Since Condition (4.34) guaranteesby Theorem 12.7.2 in [28] that f γn → f n as γ → ∞ in (cid:0) D ([0 , T ]; R ) , d M ) for all n ∈ N ,Lemma 2.3 implieslim γ →∞ h F γ ( · ) u, v i = h F ( · ) u, v i in (cid:0) D ([0 , T ]; R ) , d M ),which shows Condition (4.17) and completes the proof. Let V be a separable Hilbert space and let A be the generator of a strongly continuoussemigroup ( S ( t )) t > in V . For γ > dY γ ( t ) = γAY γ ( t ) dt + G dL ( t ) for t ∈ [0 , T ] ,Y γ (0) = 0 , (5.1)where L denotes a L´evy process in U and G ∈ L ( U, V ). A progressively measurablestochastic process ( Y γ ( t ) : t ∈ [0 , T ]) is called a weak solution of (5.1) if it satisfies forevery v ∈ D ( A ∗ ) and t ∈ [0 , T ] P -a.s. the equation h Y γ ( t ) , v i = Z t h Y γ ( s ) , A ∗ v i ds + h L ( t ) , G ∗ v i . Since A γ := γA : D ( A ) ⊆ V → V is the generator of the C -semigroup ( S γ ( t )) t > where S γ ( t ) := S ( γt ) for all t >
0, Theorem 2.3 in [6] implies that there exists a uniqueweak solution Y γ := ( Y γ ( t ) : t ∈ [0 , T ]) of (5.1), and which can be represented by Y γ ( t ) = Z t S γ ( t − s ) G dL ( s ) for all t ∈ [0 , T ] .
30e require that Y γ has c`adl`ag trajectories, which is satisfied for example if the semigroup( S ( t )) t > is analytic or contractive; see [18]. Then the integrated Ornstein-Uhlenbeckprocess ( X γ ( t ) : t ∈ [0 , T ]) is defined by X γ ( t ) := γ Z t Y γ ( s ) ds for all t ∈ [0 , T ] . Corollary 5.1.
Assume that the semigroup ( S ( t )) t > is diagonalisable, i.e. there existsan orthonormal basis e := ( e k ) k ∈ N of V such that S ( t ) e k = e − λ k t e k for all t > , k ∈ N , for some λ k > with λ k → ∞ for k → ∞ . Then the integrated Ornstein–Uhlenbeckprocess X γ satisfies lim γ →∞ (cid:0) AX γ ( t ) : t ∈ [0 , T ] (cid:1) = (cid:0) − GL ( t ) : t ∈ [0 , T ] (cid:1) in probability in the product topology (cid:0) D ([0 , T ]; V ) , d eM (cid:1) .Proof. For every t ∈ [0 , T ] and v ∈ D ( A ∗ ) Fubini’s theorem implies by (4.3) h AX γ ( t ) , v i = γ Z t h Y γ ( s ) , A ∗ v i ds = γ Z t (cid:18)Z s G ∗ S ∗ γ ( s − r ) A ∗ v dL ( r ) (cid:19) ds = γ Z t (cid:18)Z tr G ∗ S ∗ γ ( s − r ) A ∗ v ds (cid:19) dL ( r ) . (5.2)It follows from properties of the adjoint semigroup (see [25, Proposition 1.2.2]) that forevery r ∈ [0 , t ] we have Z tr G ∗ S ∗ γ ( s − r ) A ∗ v ds = 1 γ G ∗ A ∗ γ Z t − r S ∗ γ ( s ) v ds = 1 γ (cid:0) G ∗ S ∗ γ ( t − r ) v − G ∗ v (cid:1) . (5.3)Define a L´evy process K in V by K ( t ) := GL ( t ) for all t >
0. By combining (5.3) with(5.2) we obtain from (4.3) and (4.5) that h AX γ ( t ) , v i = Z t (cid:0) G ∗ S ∗ γ ( t − r ) v − G ∗ v (cid:1) dL ( r ) = h Z t (cid:0) S γ ( t − r ) − Id (cid:1) dK ( r ) , v i . Consider the functions F : [ − , T ] → L ( V, V ) , F ( t ) = ( − Id , if t ∈ [0 , T ] , , if t ∈ [ − , ,F γ : [ − , T ] → L ( V, V ) , F γ ( t ) = ( S ( γ t ) − Id , if t ∈ [0 , T ] , , if t ∈ [ − , . Defining the functions above on [ − , T ] and not only on [0 , T ] with a jump at 0 enablesus to consider c`adl`ag functions. By means of Corollary 4.5 (with an obvious adaptionfor considering the interval [ − , T ]) we show thatlim γ →∞ (cid:18)Z t (cid:0) S γ ( t − r ) − Id (cid:1) dK ( r ) : t ∈ [ − , T ] (cid:19) = (cid:18)Z t F ( t − s ) dK ( s ) : t ∈ [ − , T ] (cid:19) (5.4)31n probability in the product topology (cid:0) D ([ − , T ]; V ) , d eM (cid:1) . The functions F γ and F areof the form F ∗ ( t ) e k = ϕ k ( t ) e k , F ∗ γ ( t ) e k = ϕ kγ ( t ) e k for all t ∈ [ − , T ] , k ∈ N , where the real-valued functions ϕ k , ϕ kγ : [ − , T ] → R are defined by ϕ k ( t ) = − [0 ,T ] ( t ) , ϕ kγ ( t ) = [0 ,T ] ( t )( e − λ k γt − . For every k ∈ N the sequence ( ϕ kγ ) γ> meets Condition (4.28), assup γ> (cid:13)(cid:13) ϕ kγ (cid:13)(cid:13) ∞ = sup γ> sup s ∈ [0 ,T ] (cid:12)(cid:12) e − λ k γs − (cid:12)(cid:12) . Since ϕ kγ is a decreasing function it has finite variation and thus (cid:13)(cid:13) ϕ kγ (cid:13)(cid:13) T V = 0 whichverifies Condition (4.29). Since ϕ kγ is monotone for each γ > k ∈ N and satisfies foreach k ∈ N lim γ →∞ ϕ kγ ( s ) = ϕ k ( s ) for all s ∈ [ − , T ] \{ } , it follows from Corollary 12.5.1 in [28], that ϕ kγ → ϕ k as γ → ∞ in (cid:0) D ([0 , T ]; R ) , d M (cid:1) ,which is Condition (4.30). Thus, we can apply Corollary 4.5 to conclude (5.4). References [1] R. J. Adler.
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