Sub-diffusion processes in Hilbert space and their associated stochastic differential equations and Fokker-Planck-Kolmogorov equations
aa r X i v : . [ m a t h . P R ] O c t Sub-diffusion processes in Hilbert spaceand their associated stochastic differen-tial equations and Fokker-Planck-Kolmogorovequations
Lise Chlebak, Patricia Garmirian and Qiong Wu
Abstract.
This paper focuses on the time-changed Q-Wienerprocess, a Hilbert space-valued sub-diffusion. It is a martin-gale with respect to an appropriate filtration, hence a stochas-tic integral with respect to it is definable. For the resultingintegral, two change of variables formulas are derived. Via aduality theorem for integrals, existence and uniqueness theo-rems for stochastic differential equations (SDEs) driven by thetime-changed Q-Wiener process are discussed. Associated frac-tional Fokker-Planck-Kolmogorov equations are derived usingeither a time-changed Itˆo formula or duality. Connections areestablished between three integrals driven by time-changed ver-sions of the Q-Wiener process, cylindrical Wiener process, andmartingale measure.
Keywords. inverse stable subordinator, time-changed Q -Wienerprocesses, Hilbert space-valued sub-diffusion processes, time-changed stochastic differential equations (SDEs) in Hilbert space,fractional Fokker-Planck-Kolmogorov (FPK) equations.
1. Introduction
The Q -Wiener process is a stochastic process with values in a separa-ble Hilbert space which is usually infinite dimensional. Analogous toa classic Brownian motion in finite dimensional space, the Q -Wienerprocess is a Gaussian diffusion process which has independent andstationary increments and a covariance operator Q [1, 2, 3]. Stochas-tic integrals in Hilbert space with respect to the Q -Wiener processare constructed as an infinite sum of real-valued stochastic integralswith respect to infinitely many independent standard Brownian mo-tions [4, 5, 6, 7]. A Hilbert space of functions is a natural setting forthe semigroup approach to deal with stochastic partial differentialequations (SPDEs) since the solutions to these equations are typi-cally elements of such a space [8, 9, 10]. For motivation and details onthe semigroup approach, see [11, 12]. From the monographs [1, 4, 5],stochastic differential equations (SDEs) driven by the Q -Wiener pro-cess in Hilbert space are expressed as dX ( t ) = [ AX ( t ) + F ( t, X ( t ))]d t + C d W t X (0) = x ∈ H, (1.1)where the operator, A , is typically an elliptic differential operatorand W t is the Q -Wiener process. This type of SDE usually corre-sponds to stochastic partial differential equations (SPDEs) and thesolution, X ( t ), of the SDE (1.1) is also a diffusion process in Hilbertspace. Semigroups and distributions at a fixed time associated withthe diffusion processes in Hilbert spaces are studied in [13] and [14].Furthermore, deterministic Fokker-Planck-Kolmogorov (FPK) equa-tions corresponding to the diffusion process, X ( t ), in Hilbert spacetogether with the existence and uniqueness of their solutions havebeen investigated in [15, 16, 17]. Specifically, Bogachev and othershave started the study of FPK equations on Hilbert space in [15, 17]and [16] continues this study by proving existence and uniquenessresults for irregular, even non-continuous drift coefficients.Sub-diffusion processes in a finite dimensional space are inves-tigated by Meerschaert and Scheffler [18, 19] and Magdziarz [20]. Inparticular, the sub-diffusion processes arising as the scaling limit ofcontinuous-time random walks are considered in [18, 19]. The samplepath properties of this type of process are investigated by the mar-tingale approach in [20]. It is known that a Brownian motion withan embedded time-change which is the first hitting time process of astable subordinator of index between 0 and 1 is a sub-diffusion pro-cess [21]. Since the stable subordinator process is an increasing L´evyprocess with jumps, there are intervals for which the inverse process isconstant. Therefore, a time-changed Brownian motion process oftenmodels phenomena for which there are intervals where the process Lise Chlebak, Patricia Garmirian and Qiong Wudoes not change [22, 23, 24]. The transition probabilities, i.e., the den-sities of the time-changed Brownian motion, satisfy a time-fractionalFPK equation appearing in [25]. Furthermore, the stochastic calculusfor a time-changed semimartingale and the associated SDEs drivenby the time-changed semimartingale are investigated in [26]. See ref-erences [27, 28, 29] as well as the forthcoming book in [30] for furtherdiscussion and generalizations of the topics mentioned earlier in thisparagraph.Corresponding sub-diffusion processes in Hilbert space, specif-ically, the time-changed Q -Wiener processes have not been investi-gated. Also the SDEs driven by a time-changed Q -Wiener processand their associated fractional FPK equations have not yet beendiscussed. In this paper, we focus on the time-changed Q -Wienerprocess and its associated stochastic calculus. Also, SDEs driven bythe time-changed Q -Wiener process and fractional FPK equationsof the solutions to the time-changed SDEs are investigated. Specifi-cally, section 2 introduces the concept of a time-changed Q -Wienerprocess which is a sub-diffusion process. Similar to the time-changedBrownian motion, the time-changed Q -Wiener process is proven tobe a square-integrable martingale in Hilbert space with respect to anappropriate filtration. Furthermore, the increasing process and thequadratic variation process of the time-changed Q -Wiener processare explicitly derived.Section 3 develops the stochastic integral with respect to thetime-changed Q -Wiener process. First and second change of vari-able formulas for the time-changed stochastic integral are provided.Also the time-changed Itˆo formula for an Itˆo process driven by thetime-changed Q -Wiener process is developed. Moreover, SDEs drivenby the time-changed Q -Wiener process are introduced. Based on re-sults from Grecksch and Tudor [31] for general martingales in Hilbertspaces, the existence and uniqueness of a mild solution to a typeof time-changed SDE is provided. From the duality developed be-tween time-changed SDEs and their corresponding non-time-changedSDEs, the existence and uniqueness of strong solutions to the asso-ciated time-changed SDEs are established.Section 4 discusses a connection between three stochastic inte-grals, the one driven by a time-changed Q -Wiener process in Hilbertspace, the one driven by a time-changed cylindrical Wiener process inHilbert space and the one driven by the martingale measure, knownas Walsh’s integral [32, 7]. This connection allows one to potentiallyinterpret mild solutions to SDEs driven by time-changed Q -Wienerprocesses in Hilbert spaces as random field solutions from the Walshapproach to SPDEs by applying the results of Dalang and Quer-Sardanyons [33].Section 5 derives time-fractional FPK equations associated withthe solutions to SDEs driven by time-changed Q -Wiener processesin two ways. The first method uses the time-changed Itˆo formuladeveloped in Section 3. The same time-fractional FPK equationsare next derived by applying the duality theorem also developed inSection 3 and the FPK equations associated with the solution ofSDEs driven by the classic Q -Wiener process. Lise Chlebak, Patricia Garmirian and Qiong Wu
2. The time-changed Q -Wiener process After recalling the definition of the Q -Wiener process in Hilbertspace, we introduce the time-changed Q -Wiener process in Hilbertspace. The related martingale properties of the time-changed Q -Wiener process will also be investigated. Definition 2.1.
Following [5], Let Q be a nonnegative definite, sym-metric, trace-class operator on a separable Hilbert space K , let { f j } ∞ j =1 be an orthonormal basis in K diagonalizing Q , and let the corre-sponding eigenvalues be { λ j } ∞ j =1 . Let { w j ( t ) } t ≥ , j = 1 , , · · · , bea sequence of independent Brownian motions defined on a filteredprobability space (Ω , F , {F t } t ≥ , P ). Then the process W t := ∞ X j =1 λ / j w j ( t ) f j is called a Q -Wiener process in K .For more details of properties of Q -Wiener processes, see [4, 5].Before introducing the time-changed Q -Wiener processes, weneed to introduce the time-change process which is used throughoutthis paper. The time-change applied in this paper is the first hittingtime process of a β -stable subordinator defined as E t := E β ( t ) = inf { τ > U β ( τ ) > t } , (2.1)where U β ( t ) is the β -stable subordinator which has index β ∈ (0 , E ( e − uU β ( τ ) ) = e − τu β . (2.2)Note that E t is also called the inverse β -stable subordinator. Let B ( t ) denote a one-dimensional standard Brownian motion. Consider Z β ( t ) := B ( E t ), a subordinated Brownian motion which has beentime-changed by E t . The following result of Magdziarz [20] showsthat Z β ( t ) is a square integrable martingale with respect to the ap-propriate right-continuous filtration,¯ F t = \ u>t { σ [ B ( s ) : 0 ≤ s ≤ u ] ∨ σ [ E s : s ≥ } , (2.3)where ¯ F is assumed to be complete. Theorem 2.2.
Magdziarz, [20]
The time-changed Brownian motion Z β ( t ) is a mean zero and square integrable martingale with respectto the filtration { ¯ F E t } t ≥ . The quadratic variation process of Z β ( t ) is h Z β ( t ) , Z β ( t ) i = E t . The time-changed Brownian motion, Z β ( t ), is a sub-diffusionprocess. By incorporating the time-change, E t , into the independentBrownian motions in Definition 2.1, we define a Hilbert space-valuedtime-changed Q -Wiener process as follows: Definition 2.3.
With Q , { f j } ∞ j =1 , and { λ j } ∞ j =1 as defined in 2.1, let { w j ( t ) } t ≥ , j = 1 , , · · · , be a sequence of independent Brownianmotions defined on (Ω , F , {F t } t ≥ , P ) which are independent of E t .Then the process W E t := ∞ X j =1 λ / j w j ( E t ) f j (2.4)is called a time-changed Q -Wiener process in K . Lise Chlebak, Patricia Garmirian and Qiong WuThis time-changed Q -Wiener process can be considered as aninfinite-dimensional Hilbert space analog of a one-dimensional time-changed Brownian motion. It is a sub-diffusion process in Hilbertspace. Let µ t be the Borel probability measure induced by the time-changed Q -Wiener process, W E t on K , i.e., E ( W E t ) = R K xµ t (d x ).Then the time-fractional FPK equation corresponding to the time-changed Q -Wiener process in the following theorem can be consid-ered as a special case of Theorem 5.3 in Section 5. Theorem 2.4.
Suppose µ t is the probability measure induced by thetime-changed Q -Wiener process, W E , on K . Then µ t satisfies thefollowing time-fractional PDE D βt µ t = D x µ t , where D x denotes the second-order Fr´echet derivative in space and D βt is the Caputo time fractional derivative operator defined as D βt f ( t ) = 1Γ(1 − β ) Z t f ′ ( τ )( t − τ ) β d τ, where β is the index associated with the β -stable subordinator U β ( t ) and Γ( β ) is the gamma function. The time-changed Q -Wiener process is our main object of study.In order to define integrals with respect to this process, and, ulti-mately, to consider SDEs driven by this process, it is advantageousto view the time-changed Q -Wiener process as a martingale with re-spect to an appropriate filtration. To prove this, we begin with thedefinition of a martingale in a Hilbert space. Definition 2.5.
Following [5], let K be a separable Hilbert space en-dowed with its Borel σ -field B ( K ). Fix T > , F , {F t } t ≤ T , P )be a filtered probability space and { M t } t ≤ T be a K -valued processadapted to the filtration {F t } t ≤ T . Assume that M t is integrable, i.e., E k M t k < ∞ .a) If for any 0 ≤ s ≤ t , E ( M t |F s ) = M s , P -a.s., then M t is calledan F t -martingale.b) If E k M T k < ∞ , the martingale M t is called square integrableon 0 ≤ t ≤ T .The condition for { M t } t
The time-changed Q -Wiener process defined by Defi-nition 2.3 is a K -valued square integrable martingale with respect tothe filtration G t := ˜ F E t .Proof. From [20], all moments of the time change, E t , are finite, i.e., E ( E nt ) = t nβ n !Γ( nβ + 1) , for n = 1 , , · · · . Let f E t be the density function for E t . Then, thesecond moment for the time-changed Brownian motion w j ( E t ) isgiven by E ( w j ( E t )) = Z ∞ E ( w j ( τ )) f E t ( τ )d τ = Z ∞ τ f E t ( τ )d τ = t β Γ( β + 1) . Thus, E k W E t k K = E (cid:28) ∞ X j =1 λ / j w j ( E t ) f j , ∞ X i =1 λ / i w i ( E t ) f i (cid:29) K = E ∞ X j =1 λ j w j ( E t )= ∞ X j =1 λ j E ( w j ( E t )) = t β Γ( β + 1) ∞ X j =1 λ j < ∞ . The sum is finite since Q is a trace-class operator. Also, since the Q -Wiener process, W t , is a square integrable martingale in the Hilbertspace K , it follows that for any h ∈ K , the process X t defined by X t := h W t , h i K F t . This means that in order to prove the time-changed Q -Wiener process, W E t , is a square integrable martingale, it suffices toverify that the time-changed real-valued process, X E t , defined by X E t := h W E t , h i K , is a square integrable martingale with respect to the filtration G t =˜ F E t . Define the sequence of { ˜ F τ } -stopping times, T n , by T n = inf { τ > | X ( τ ) | ≥ n } . It is known that the stopped process X ( T n ∧ τ ) is a bounded martin-gale with respect to ˜ F τ . Thus, by Doob’s Optional Sampling Theo-rem, for s < t , E ( X ( T n ∧ E t ) | G s ) = X ( T n ∧ E s ) (2.6)The right hand side of (2.6) converges to X ( E s ) as n → ∞ . For theleft hand side, | X ( T n ∧ E t ) | ≤ sup ≤ s ≤ t | X ( E s ) | . E ( sup ≤ s ≤ t | X ( E s ) | ) = E ( sup ≤ s ≤ E t | X ( s ) | ) = Z ∞ E ( sup ≤ s ≤ τ | X ( s ) | | E t = τ ) f E t ( τ )d τ = Z ∞ E ( sup ≤ s ≤ τ | X ( s ) | ) f E t ( τ )d τ ≤ Z ∞ E ( | X ( τ ) | ) f E t ( τ )d τ = 4 Z ∞ E ( |h W τ , h i K | ) f E t ( τ )d τ ≤ Z ∞ E ( k W τ k K k h k K ) f E t ( τ )d τ = 4 k h k K Z ∞ E k W τ k f E t ( τ ) dτ = 4 k h k K E k W E t k K < ∞ . Also by Holder’s inequality, E ( sup ≤ s ≤ t | X E s | ) ≤ ( E ( sup ≤ s ≤ t | X E s | ) ) / . Thus, E ( sup ≤ s ≤ t | X E s | ) < ∞ . By the dominated convergence theorem, E ( X ( T n ∧ E t ) | G s ) −→ E ( X ( E t ) | G s ) , as n → ∞ . Therefore, from (2.6), E ( X ( E t ) | G s ) = X ( E s ) , which implies that X ( E t ) = h W E t , h i K is a martingale with respectto the filtration ˜ F E t . Therefore, W E t is a square integrable martin-gale in the Hilbert space K . (cid:3) Q -Wiener process W E t . Definition 2.7.
Following [5], let M t ∈ M T ( K ). Denote by h M i t the unique adapted continuous increasing process starting from 0such that k M t k K − h M i t is a continuous martingale. The quadraticvariation process hh M ii t of M t is an adapted continuous processstarting from 0, with values in the space of nonnegative definite trace-class operators on K , such that for all h, g ∈ K , h M t , h i K h M t , g i K − (cid:10) hh M ii t ( h ) , g (cid:11) K is a martingale. Lemma 2.8.
Following [5] , the quadratic variation process of a mar-tingale M t ∈ M T ( K ) exists and is unique. Moreover, h M i t = tr ( hh M ii t ) . Proposition 2.9.
The increasing process and quadratic variation pro-cess of the time-changed Q -Wiener process in Definition 2.3 are re-spectively h W E i t = tr ( Q ) E t and hh W E ii t = QE t . Proof.
Let Q be a nonnegative definite, symmetric, trace-class opera-tor on a separable Hilbert space K and let { f j } ∞ j =1 be an orthonormalbasis in K diagonalizing Q with corresponding eigenvalues { λ j } ∞ j =1 .4 Lise Chlebak, Patricia Garmirian and Qiong WuThen, k W E t k K = (cid:28) P ∞ j =1 λ / j w j ( E t ) f j , P ∞ i =1 λ / i w i ( E t ) f i (cid:29) K = P ∞ j =1 λ j w j ( E t ) . On the other hand, tr ( Q ) E t = E t ∞ X j =1 λ j . So define the process, N E t , as N E t = k W E t k K − tr ( Q ) E t = ∞ X j =1 λ j ( w j ( E t ) − E t ) , which can be considered as a time-change of N t where N t = k W t k K − tr ( Q ) t = ∞ X j =1 λ j ( w j ( t ) − t ) . From Definition 2.7, N t is a real-valued martingale since tr ( Q ) t is theunique increasing process of the Q -Wiener process. By an argumentsimilar to that of Theorem 2.6, the time-changed process, N E t , is amartingale. Further, W E t is a martingale and there is an increasingprocess, h W E i t , such that k W E t k K − h W E i t is a martingale. Finally,since k W E t k K is a real-valued submartingale, by the uniqueness ofthe Doob-Meyer decomposition [34], h W E i t = tr ( Q ) E t . (2.7)Again, by Theorem 2.6 and Lemma 2.8, the quadratic process, hh W E ii t ,of the time-changed Q -Wiener process, W E t , exists and is unique,5and satisfies tr (cid:0) hh W E ii t (cid:1) = h W E i t . (2.8)Therefore, from (2.7) and (2.8), hh W E ii t = QE t . (cid:3)
3. SDEs driven by the time-changed Q-Wiener process
In this section, we begin by developing the Itˆo stochastic integralwith respect to the time-changed Q -Wiener process in Hilbert space.Also a time-changed Itˆo formula for an Itˆo process driven by a time-changed Q -Wiener process is developed. Finally, the existence anduniqueness of solutions to the time-changed Hilbert space-valuedSDEs are investigated. Q -Wienerprocess In order to construct an Itˆo stochastic integral with respect to thetime-changed Q -Wiener process, we briefly recall Itˆo stochastic inte-grals with respect to a Q-Wiener process without a time change asin [4, 5].As in Section 2, let K and H be two separable Hilbert spaces,and Q be a symmetric, nonnegative definite trace-class operator on K . Let { f j } ∞ j =1 be an orthonormal basis (ONB) in K such that Qf j = λ j f j , where these eigenvalues λ j > j = 1 , , · · · . Then the6 Lise Chlebak, Patricia Garmirian and Qiong Wuseparable Hilbert space K Q = Q / K with an ONB { λ / j f j } ∞ j =1 isendowed with the following scalar product h u, v i K Q = ∞ X j =1 λ j h u, f j i K h v, f j i K . Let L ( K Q , H ) be the space of Hilbert-Schmidt operators from K Q to H . The Hilbert-Schmidt norm of an operator L ∈ L ( K Q , H ) isgiven by k L k L ( K Q ,H ) = k LQ / k L ( K,H ) = tr (( LQ / )( LQ / ) ∗ ) . The scalar product between two operators
L, M ∈ L ( K Q , H ) isdefined by h L, M i L ( K Q ,H ) = tr (( LQ / )( M Q / ) ∗ ) . Define Λ ( K Q , H ) as the class of L ( K Q , H )-valued processes whichare measurable mappings from([0 , T ] × Ω , B ([0 , T ]) × F )to ( L ( K Q , H ) , B ( L ( K Q , H ))) , adapted to the filtration {F t } t ≤ T , and satisfying the condition E Z T k Φ( t ) k L ( K Q ,H ) dt < ∞ . ( K Q , H ) is a Hilbert space if it is equipped with thenorm k Φ k Λ ( K Q ,H ) := E Z T k Φ( t ) k L ( K Q ,H ) dt ! / . The following lemma from [5] can be considered as a definition of astochastic integral with respect to the Q -Wiener process: Lemma 3.1.
Mandrekar, [5]
Let W t be a Q-Wiener process in a sep-arable Hilbert space K , Φ ∈ Λ ( K Q , H ) , and { f j } ∞ j =1 be an ONB in K consisting of eigenvectors of Q . Then, Z t Φ( s )d W s = ∞ X j =1 Z t (Φ( s ) λ / j f j )d h W s , λ / j f j i K Q . In order to incorporate the time-change, E t , into the Itˆo sto-chastic integral, the generalized ˜Λ ( K Q , H ) is also considered as theclass of L ( K Q , H )-valued processes which are measurable mappingsfrom ([0 , T ] × Ω , B ([0 , T ]) × ˜ F t )to ( L ( K Q , H ) , B ( L ( K Q , H ))) , adapted to the filtration { ˜ F E t } t ≤ T , and satisfying the condition E Z T k Φ( t ) k L ( K Q ,H ) dE t < ∞ . Similarly, ˜Λ ( K Q , H ) is a separable Hilbert space if it is equippedwith the norm k Φ k ˜Λ ( K Q ,H ) := E Z T k Φ( t ) k L ( K Q ,H ) dE t ! / . Q -Wiener process can be introduced. Definition 3.2.
Let W E t be a time-changed Q -Wiener process in aseparable Hilbert space K , Φ ∈ ˜Λ ( K Q , H ), and let { f j } ∞ j =1 be anONB in K consisting of eigenvectors of Q . Then, Z t Φ( s )d W E s = ∞ X j =1 Z t (Φ( s ) λ / j f j )d h W E s , λ / j f j i K Q . Now that the Itˆo integral with respect to the time-changed Q -Wiener process has been established, the next step is to derive theItˆo isometry first for elementary processes, and then by extension, forarbitrary processes in ˜Λ ( K Q , H ). Consider the class of {G t } -adaptedelementary processes of the formΦ( t, ω ) = φ ( ω )1 { } ( t ) + n − X j =0 φ j ( ω )1 ( t j ,t j +1 ] ( t ) , (3.1)where 0 ≤ t ≤ t ≤ · · · ≤ t n = T and φ, φ j , j = 0 , , · · · , n − G -measurable and G t j -measurable L ( K Q , H )-valuedrandom variables such that φ ( ω ) , φ j ( ω ) are linear, bounded oper-ators from K to H . Let E ( L ( K, H )) denote this class of elementaryprocesses. Proceeding to the Itˆo isometry for an elementary processΦ( t, ω ), we need the following useful lemma.
Lemma 3.3.
Let { f j } ∞ j =1 be an ONB in K consisting of eigenvectorsof Q and G s = ˜ F E s be the filtration. Then, for l = l ′ and t > s > , E (cid:18) E (cid:18)(cid:28) W E t − W E s , f l (cid:29) K (cid:28) W E t − W E s , f l ′ (cid:29) K (cid:12)(cid:12)(cid:12)(cid:12) G s (cid:19)(cid:19) = 0 . Proof.
From Definition 2.3 in Section 2, W E t = ∞ X j =1 λ / j w j ( E t ) f j . Using the definition of W E t , E (cid:18) E (cid:18)(cid:28) W E t − W E s , f l (cid:29) K (cid:28) W E t − W E s , f l ′ (cid:29) K (cid:12)(cid:12)(cid:12)(cid:12) G s (cid:19)(cid:19) = E (cid:26) E (cid:18)(cid:28) W E t , f l (cid:29) K (cid:28) W E t , f l ′ (cid:29) K − (cid:28) W E s , f l (cid:29) K (cid:28) W E t , f l ′ (cid:29) K − (cid:28) W E s , f l ′ (cid:29) K (cid:28) W E t , f l (cid:29) K + (cid:28) W E s , f l (cid:29) K (cid:28) W E s , f l ′ (cid:29) K (cid:12)(cid:12)(cid:12)(cid:12) G s (cid:19)(cid:27) = E (cid:18) E (cid:18) λ / l w l ( E t ) λ / l ′ w l ′ ( E t ) |G s (cid:19)(cid:19) − E (cid:18) E (cid:18) λ / l w l ( E s ) λ / l ′ w l ′ ( E t ) |G s (cid:19)(cid:19) − E (cid:18) E (cid:18) λ / l ′ w l ′ ( E s ) λ / l w l ( E t ) |G s (cid:19)(cid:19) + E (cid:18) λ / l w l ( E s ) λ / l ′ w l ′ ( E s ) (cid:19) := I − I − I + I . Since w l is independent of w l ′ , conditioning on E t to compute thefirst term yields I = E (cid:18) E (cid:18) λ / l w l ( E t ) λ / l ′ w l ′ ( E t ) |G s (cid:19)(cid:19) = E ( λ / l w l ( E t ) λ / l ′ w l ′ ( E t ))= λ / l λ / l ′ Z ∞ E ( w l ( τ ) w l ′ ( τ )) f E t ( τ )d τ = λ / l λ / l ′ Z ∞ · f E t ( τ )d τ = 0 . On the other hand, I = E (cid:18) E (cid:18) λ / l w l ( E s ) λ / l ′ w l ′ ( E t ) (cid:12)(cid:12)(cid:12)(cid:12) G s (cid:19)(cid:19) = λ / l λ / l ′ E (cid:18) E (cid:18) w l ( E s ) w l ′ ( E t ) (cid:12)(cid:12)(cid:12)(cid:12) G s (cid:19)(cid:19) = λ / l λ / l ′ E (cid:18) E (cid:18) w l ( E s ) (cid:18) w l ′ ( E t ) − w l ′ ( E s ) (cid:19) + w l ( E s ) w l ′ ( E s ) (cid:12)(cid:12)(cid:12)(cid:12) G s (cid:19)(cid:19) = λ / l λ / l ′ E (cid:18) w l ( E s ) E (cid:18) w l ′ ( E t ) − w l ′ ( E s ) (cid:12)(cid:12)(cid:12)(cid:12) G s (cid:19) + w l ′ ( E s ) w l ( E s ) (cid:19) = 0 , E ( w l ′ ( E t ) − w l ′ ( E s ) |G s ) = 0 by the martingale property of W E t and E ( w l ′ ( E s ) w l ( E s )) = 0 by the same conditioning argumentpreviously used in computing term I . Similarly, the third term, I ,and the fourth term, I , are also equal to 0. (cid:3) Theorem 3.4.
Let Φ ∈ E ( L ( K, H )) be a bounded elementary process.Then, for t ∈ [0 , T ] , E (cid:13)(cid:13)(cid:13)(cid:13) Z t Φ( s ) dW E s (cid:13)(cid:13)(cid:13)(cid:13) H = E Z t (cid:13)(cid:13) Φ( s ) (cid:13)(cid:13) L ( K Q ,H ) dE s < ∞ . Proof.
First, without loss of generality, assume that t = T . Then, forthe bounded elementary process, Φ, defined in (3.1), E (cid:13)(cid:13)(cid:13)(cid:13) Z T Φ( s ) dW E s (cid:13)(cid:13)(cid:13)(cid:13) H = E (cid:13)(cid:13)(cid:13)(cid:13) n − X j =0 φ j ( W E tj +1 − W E tj ) (cid:13)(cid:13)(cid:13)(cid:13) H = n − X j =0 E (cid:13)(cid:13)(cid:13)(cid:13) φ j ( W E tj +1 − W E tj ) (cid:13)(cid:13)(cid:13)(cid:13) H + n − X i = j =0 E (cid:28) φ j ( W E tj +1 − W E tj ) , φ i ( W E ti +1 − W E ti ) (cid:29) H := I + II. { e m } ∞ m =1 in H and { f l } ∞ l =1 in K be ONBs. For fixed j , I j isdenoted by I j = E (cid:13)(cid:13)(cid:13)(cid:13) φ j ( W E tj +1 − W E tj ) (cid:13)(cid:13)(cid:13)(cid:13) H = E ∞ X m =1 (cid:28) φ j ( W E tj +1 − W E tj ) , e m (cid:29) H = ∞ X m =1 E (cid:18) E (cid:0) h φ j ( W E tj +1 − W E tj ) , e m i H |G t j (cid:1)(cid:19) = ∞ X m =1 E (cid:18) E (cid:18)(cid:28) W E tj +1 − W E tj , φ ∗ j e m (cid:29) K (cid:12)(cid:12)(cid:12)(cid:12) G t j (cid:19)(cid:19) = ∞ X m =1 E (cid:18) E (cid:18)(cid:18) ∞ X l =1 (cid:28) W E tj +1 − W E tj , f l (cid:29) K (cid:28) φ ∗ j e m , f l (cid:29) K (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) G t j (cid:19)(cid:19) = ∞ X m =1 E (cid:18) E (cid:18)(cid:18) ∞ X l =1 (cid:28) W E tj +1 − W E tj , f l (cid:29) K (cid:28) φ ∗ j e m , f l (cid:29) K (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) G t j (cid:19)(cid:19) + ∞ X m =1 E (cid:18) E (cid:18)(cid:18) ∞ X l = l ′ =1 h W E tj +1 − W E tj , f l i K h φ ∗ j e m , f l i K × h W E tj +1 − W E tj , f l ′ i K h φ ∗ j e m , f l ′ i K (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) G t j (cid:19)(cid:19) := J + J . φ ∗ j is G t j -measurable and W E tj +1 is a discrete martingale withrespect to G t j , the first term, J , becomes J = ∞ X m =1 E (cid:18) ∞ X l =1 (cid:28) φ ∗ j e m , f l (cid:29) K E (cid:18)(cid:28) W E tj +1 − W E tj , f l (cid:29) K (cid:12)(cid:12)(cid:12)(cid:12) G t j (cid:19)(cid:19) = ∞ X m =1 E (cid:26) ∞ X l =1 (cid:28) φ ∗ j e m , f l (cid:29) K (cid:18) E (cid:18)(cid:28) W E tj +1 , f l (cid:29) K (cid:12)(cid:12)(cid:12)(cid:12) G t j (cid:19) − (cid:28) W E tj , f l (cid:29) K (cid:19)(cid:27) = ∞ X m =1 E (cid:26) ∞ X l =1 (cid:28) φ ∗ j e m , f l (cid:29) K (cid:18) E (cid:18)(cid:28) W E tj +1 , f l (cid:29) K (cid:12)(cid:12)(cid:12)(cid:12) G t j (cid:19)(cid:27) − ∞ X m =1 E (cid:26) ∞ X l =1 (cid:28) φ ∗ j e m , f l (cid:29) K (cid:28) W E tj , f l (cid:29) K (cid:19)(cid:27) = E (cid:26) ∞ X m =1 (cid:18) E t j +1 − E t j (cid:19) ∞ X l =1 λ l (cid:28) φ ∗ j e m , f l (cid:29) K (cid:27) = E (cid:26)(cid:18) E t j +1 − E t j (cid:19) ∞ X m,l =1 (cid:28) φ j ( λ / l f l ) , e m (cid:29) H (cid:27) = E (cid:26)(cid:18) E t j +1 − E t j (cid:19) k φ j k L ( K Q ,H ) (cid:27) . Also using the G t j -measurability of φ ∗ j and Lemma 3.3, the secondterm, J , becomes J = ∞ X m =1 E (cid:26) ∞ X l = l ′ =1 (cid:28) φ ∗ j e m , f l (cid:29) K (cid:28) φ ∗ j e m , f l ′ (cid:29) K × E (cid:18)(cid:28) W E tj +1 − W E tj , f l (cid:29) K (cid:28) W E tj +1 − W E tj , f l ′ (cid:29) K (cid:12)(cid:12)(cid:12)(cid:12) G t j (cid:19)(cid:27) = 0 . Thus, I = n − X j =0 I j = n − X j =0 E (cid:26)(cid:18) E t j +1 − E t j (cid:19) k φ j k L ( K Q ,H ) (cid:27) = E Z T (cid:13)(cid:13) Φ( s ) (cid:13)(cid:13) L ( K Q ,H ) dE s < ∞ . i < j .From Lemma 3.3, II = E ∞ X m =1 E (cid:18) ∞ X l,l ′ =1 (cid:28) W E tj +1 − W E tj , f l (cid:29) K (cid:28) φ ∗ j e m , f l (cid:29) K × (cid:28) W E ti +1 − W E ti , f l ′ (cid:29) K (cid:28) φ ∗ i e m , f l ′ (cid:29) K (cid:12)(cid:12)(cid:12)(cid:12) G t j (cid:19) = 0 . (cid:3) Theorem 3.5. (Time-changed Itˆo Isometry) For t ∈ [0 , T ] , the sto-chastic integral Φ R t Φ( s )d W E s with respect to a K -valued time-changed Q -Wiener process W E t is an isometry between ˜Λ ( K Q , H ) and the space of continuous square-integrable martingales M T ( H ) ,i.e., E (cid:13)(cid:13)(cid:13)(cid:13) Z t Φ( s )d W E s (cid:13)(cid:13)(cid:13)(cid:13) H = E Z t (cid:13)(cid:13) Φ( s ) (cid:13)(cid:13) L ( K Q ,H ) d E s < ∞ . (3.2) Proof.
For elementary processes Φ ∈ E ( L ( K, H ), Theorem 3.4 es-tablishes the desired equality (3.2) and consequently the square-integrability of the integral Z t Φ( s )d W E s . Furthermore, since thetime-changed Q-Wiener process, W E t , is a K-valued martingale, forany h ∈ H and s < t , E (cid:18)(cid:28) Z t Φ( r ) dW E r , h (cid:29) H (cid:12)(cid:12)(cid:12)(cid:12) G s (cid:19) = E (cid:18)(cid:28) n − X j =0 φ j ( W E tj +1 ∧ t − W E tj ∧ t ) , h (cid:29) H (cid:12)(cid:12)(cid:12)(cid:12) G s (cid:19) = n − X j =0 E (cid:18)(cid:28) W E tj +1 ∧ t − W E tj ∧ t , φ ∗ ( h ) (cid:29) K (cid:12)(cid:12)(cid:12)(cid:12) G s (cid:19) = n − X j =0 (cid:28) W E tj +1 ∧ s − W E tj ∧ s , φ ∗ ( h ) (cid:29) K = (cid:28) n − X j =0 φ ( W E tj +1 ∧ s − W E tj ∧ s ) , h (cid:29) H = (cid:28) Z s Φ( s ) dW E s , h (cid:29) H , R t Φ( s )d W E s is a square-integrable martingale. Therefore, the desired result holds when Φ( s )is an elementary process.Now, let { Φ n } ∞ n =1 be a sequence of elementary processes ap-proximating Φ ∈ ˜Λ ( K Q , H ). Assume that Φ = 0 and || Φ n +1 − Φ n || ˜Λ ( K Q ,H ) < n . Then, by Doob’s Maximal Inequality, ∞ X n =1 P (cid:18) sup t ≤ T (cid:13)(cid:13)(cid:13)(cid:13)Z t Φ n +1 ( s ) dW E s − Z t Φ n ( s ) dW E s (cid:13)(cid:13)(cid:13)(cid:13) H > n (cid:19) ≤ ∞ X n =1 n E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T (Φ n +1 ( s ) − Φ n ( s )) dW E s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H = ∞ X n =1 n E Z T || Φ n +1 ( s ) − Φ n ( s ) || L ( K Q ,H ) dE s ≤ T β Γ( β + 1) ∞ X n =1 n n < ∞ . By Borel-Cantelli, it follows that for some k ( ω ) > t ≤ T (cid:13)(cid:13)(cid:13)(cid:13)Z t Φ n +1 ( s ) dW E s − Z t Φ n ( s ) dW E s (cid:13)(cid:13)(cid:13)(cid:13) H ≤ n , n > k ( ω ) , holds P -almost surely. Therefore, for every t ≤ T , ∞ X n =1 (cid:18)Z t Φ n +1 ( s ) dW E s − Z t Φ n ( s ) dW E s (cid:19) → Z t Φ( s ) dW E s in L (Ω , H ) , which also converges P -a.s. to a continuous version of the integral.Thus, the map Φ Z t Φ( s ) dW E s , viewed as an isometry fromelementary processes to the space of continuous square-integrablemartingales, has an extension to Φ ∈ ˜Λ ( K Q , H ) by the completenessproperty of H . (cid:3) The following two change of variable formulas concern the Itˆostochastic integral related to the time-change E t . They are needed5later and can be considered as the Hilbert space extensions of for-mulas in [26]. Theorem 3.6. ( ) Let W t be a Q -Wiener process in a separable Hilbert space K , Φ ∈ ˜Λ ( K Q , H ) , and E t be the inverse of a β -stable subordinator. Then,with probability one, for all t ≥ , Z E t Φ( s )d W s = Z t Φ( E s )d W E s . Proof.
Let { f j } ∞ j =1 be an ONB in the separable Hilbert space K consisting of eigenvectors of Q . Then, it follows from Lemma 3.1that Z E t Φ( s )d W s = ∞ X j =1 Z E t (Φ( s ) λ / j f j )d h W s , λ / j f j i K Q . For any h ∈ H , (cid:28) Z E t Φ( s )d W s , h (cid:29) H = (cid:28) ∞ X j =1 Z E t (Φ( s ) λ / j f j )d h W s , λ / j f j i K Q , h (cid:29) H = ∞ X j =1 Z E t h (Φ( s ) λ / j f j ) , h i H d h W s , λ / j f j i K Q = ∞ X j =1 Z t h (Φ( E s ) λ / j f j ) , h i H d h W E s , λ / j f j i K Q = (cid:28) ∞ X j =1 Z t (Φ( E s ) λ / j f j )d h W E s , λ / j f j i K Q , h (cid:29) H = (cid:28) Z t Φ( E s )d W E s , h (cid:29) H . The third equality follows from the first change of variable formulaof the real-valued stochastic integral from [26]. (cid:3)
Theorem 3.7. ( )Let W t be a Q-Wiener process in a separable Hilbert space K and Φ ∈ ˜Λ ( K Q , H ) . Let U t be a β -stable subordinator with β ∈ (0 , and E t be its inverse stable subordinator. Then, with probability one,for all t ≥ , Z t Φ( s )d W E s = Z E t Φ( U s − )d W s . Proof.
Let { f j } ∞ j =1 be an ONB in the separable Hilbert space K consisting of eigenvectors of Q . Applying Definition 3.2 yields, Z t Φ( s )d W E s = ∞ X j =1 Z t (Φ( s ) λ / j f j )d h W E s , λ / j f j i K Q . For any h ∈ H , (cid:28) Z t Φ( s )d W E s , h (cid:29) H = (cid:28) ∞ X j =1 Z t (Φ( s ) λ / j f j )d h W E s , λ / j f j i K Q , h (cid:29) H = ∞ X j =1 Z t h (Φ( s ) λ / j f j ) , h i H d h W E s , λ / j f j i K Q = ∞ X j =1 Z E t h (Φ( U s − ) λ / j f j ) , h i H d h W s , λ / j f j i K Q = (cid:28) ∞ X j =1 Z E t (Φ( U s − ) λ / j f j )d h W s , λ / j f j i K Q , h (cid:29) H = (cid:28) Z E t Φ( U s − )d W s , h (cid:29) H . The first equality follows from Definition 3.2 and the third equalityfollows from the second change of variable formula for real-valuedstochastic integrals from [26]. (cid:3) The technique used to develop the time-changed Itˆo formula in thissection is inspired by the proof of the standard Itˆo formula of Theo-rem 2.9 in [5].
Theorem 3.8. (Time-changed Itˆo formula) Let Q be a symmetric,nonnegative definite trace-class operator on a separable Hilbert space K , and let { W E t } ≤ t ≤ T be a time-changed Q-Wiener process on a fil-tered probability space (Ω , G , {G t } ≤ t ≤ T , P ) . Assume that a stochasticprocess X ( t ) is given by X ( t ) = X (0) + Z t ψ ( s )d s + Z t γ ( s )d E s + Z t φ ( s )d W E s , where X (0) is a G -measurable H -valued random variable, ψ ( s ) and γ ( s ) are H-valued G s -measurable P -a.s. integrable processes on [0 , T ] such that Z T k ψ ( s ) k H d s < ∞ and Z T k γ ( s ) k H d E s < ∞ , and φ ∈ ˜Λ ( K Q , H ) . Also assume that F : H → R is continuousand its Fr´echet derivatives F x : H → L ( H, R ) and F xx : H →L ( H, L ( H, R )) are continuous and bounded on bounded subsets of H . Then, F ( X ( t )) − F ( X (0)) = Z t (cid:10) F x ( X ( s )) , ψ ( s ) (cid:11) H d s + Z E t (cid:10) F x ( X ( U ( s − ))) , γ ( U ( s − )) (cid:11) H d s + Z E t (cid:10) F x ( X ( U ( s − ))) , φ ( U ( s − ))d W s (cid:11) H + 12 Z E t tr ( F xx ( X ( U ( s − )))( φ ( U ( s − )) Q / )( φ ( U ( s − )) Q / ) ∗ )d s, P -a.s. for all t ∈ [0 , T ] .Proof. First, the desired Itˆo formula is reduced to the case where ψ ( s ) = ψ , γ ( s ) = γ , and φ ( s ) = φ are constant processes for s ∈ [0 , T ]. Let C > τ C = inf (cid:26) t ∈ [0 , T ] : max (cid:18) || X ( t ) || H , Z t || φ ( s ) || H d s, Z t || γ ( s ) || H d E s , Z t || φ ( s ) || L ( K Q ,H ) d E s (cid:19) ≥ C (cid:27) . Then, define X C ( t ) as X C ( t ) = X C (0) + Z t ψ C ( s )d s + Z t γ C ( s )d E s + Z t φ C ( s )d W E s , t ∈ [0 , T ] , where X C ( t ) = X ( t ∧ τ C ) , ψ C ( t ) = ψ ( t )1 [0 ,τ C ] ( t ) , γ C ( t ) = γ ( t )1 [0 ,τ C ] ( t ),and φ C ( t ) = φ ( t )1 [0 ,τ C ] ( t ). It is enough to prove the Itˆo formula forthe processes stopped at τ C . Since P (cid:18) Z T k ψ C ( s ) k H d s < ∞ (cid:19) = 1 , P (cid:18) Z T || γ C ( s ) || H d E s < ∞ (cid:19) = 1 , and E Z T || φ C ( s ) || L ( K Q ,H ) d E s < ∞ , ψ C , γ C , and φ C can be approximated respectivelyby sequences of bounded elementary processes ψ C,n , γ C,n , and φ C,n such that as n → ∞ Z t k ψ C,n ( s ) − ψ C ( s ) k H d s → , Z t k γ C,n ( s ) − γ C ( s ) k H d E s → , and (cid:13)(cid:13)(cid:13)(cid:13)Z t φ C,n ( s ) dW E s − Z t φ C ( s )d W E s (cid:13)(cid:13)(cid:13)(cid:13) H → , P − a.s. (3.3)uniformly in t ∈ [0 , T ]. Let X C,n ( t ) = X (0) + Z t ψ C,n ( s )d s + Z t γ C,n ( s )d E s + Z t φ C,n ( s )d W E s . Then, as n → ∞ sup t ≤ T k X C,n ( t ) − X C ( t ) k H → , P − a.s. . (3.4)Assume the Itˆo formula for X C,n ( t ) holds P -a.s. for all t ∈ [0 , T ], i.e., F ( X C,n ( t )) − F ( X (0)) = Z t h F x ( X C,n ( s )) , φ C,n ( s )d W E s i H + Z t h F x ( X C,n ( s )) , ψ C,n ( s ) i H d s + Z t h F x ( X C,n ( s )) , γ C,n ( s ) i H d E s + Z t tr [ F xx ( X C,n ( s ))( φ C,n ( s ) Q / )( φ C,n ( s ) Q / ) ∗ ]d E s := I C,n + I C,n + I C,n + I C,n . (3.5)By using the continuity of F and the continuity and local bounded-ness of F x and F xx , it will suffice to show that the following holds0 Lise Chlebak, Patricia Garmirian and Qiong Wu P -a.s. for all t ∈ [0 , T ]: F ( X C ( t )) − F ( X (0)) = Z t h F x ( X C ( s )) , φ C ( s )d W E s i H + Z t h F x ( X C ( s )) , ψ C ( s ) i H d s + Z t h F x ( X C ( s )) , γ C ( s ) i H d E s + 12 Z t tr [ F xx ( X C ( s ))( φ C ( s ) Q / )( φ C ( s ) Q / ) ∗ ]d E s := I C + I C + I C + I C . (3.6)Consider, term by term, the difference between both sides of (3.5)and (3.6). Due to the continuity of F and almost sure convergencein (3.4), the left hand side of (3.5) converges to the left hand sideof (3.6) P -a.s for all t ≤ T , i.e., F ( X C,n ( t )) → F ( X C ( t )) , P − a.s. as n → ∞ . (3.7)Turn to the first terms in both right hand sides of (3.5) and (3.6), E | I C,n − I C | = E (cid:12)(cid:12)(cid:12)(cid:12) Z t (cid:18) φ ∗ C,n ( s ) F x ( X C,n ( s )) − φ ∗ C ( s ) F x ( X C ( s )) (cid:19) d W E s (cid:12)(cid:12)(cid:12)(cid:12) ≤ E Z t k ( φ ∗ C,n ( s ) − φ ∗ C ( s )) F x ( X C,n ( s )) k L ( K Q ,R ) d E s + 2 E Z t k φ ∗ C ( s )( F x ( X C,n ( s )) − F x ( X C ( s ))) k L ( K Q ,R ) d E s ≤ E Z t (cid:18) k φ ∗ C ( s ) − φ ∗ C,n k L ( K Q ,H ) k F x ( X C,n ( s ) k H (cid:19) d E s + 2 E Z t (cid:18) k φ ∗ C ( s ) k L ( K Q ,H ) k F x ( X C,n ( s )) − F x ( X C ( s )) k H (cid:19) d E s := J + J , where R t β ∗ ( s ) α ( s )d W E s := R t h α ( s ) , β ( s )d W E s i H and β ∗ ( s ) is theadjoint operator of β ( s ). Since F x is bounded on bounded subsets of1 H , there exists an M > J ≤ M E t Z k φ ∗ C ( s ) − φ ∗ C,n k L ( K Q ,H ) d E s → , as n → ∞ . Since φ C ( s ) is square integrable in the space ˜Λ ( K Q , H ) and F x isbounded in H , J → I C,n converges to I C in mean square, i.e., E | I C,n − I C | → , (3.8)and thus converges in probability. For the second terms, I C,n and I C , the RHSs of (3.5) and (3.6), applying the conditions of (3.3)and (3.4) leads to I C,n − I C = Z t (cid:18)(cid:28) F x ( X C,n ( s )) − F x ( X C ( s )) , ψ C,n ( s ) (cid:29) H + (cid:28) F x ( X C ( s )) , ψ C,n ( s ) − ψ C ( s ) (cid:29) H (cid:19) d s → , P − a.s. . (3.9)Similarly, for the third terms, I C,n and I C , the RHSs of (3.5) and (3.6), I C,n − I C = Z t (cid:18)(cid:28) F x ( X C,n ( s )) − F x ( X C ( s )) , γ C,n ( s ) (cid:29) H + (cid:28) F x ( X C ( s )) , γ C,n ( s ) − γ C ( s ) (cid:29) H (cid:19) d E s → , P − a.s. . (3.10)Before proceeding to the fourth terms, I C,n and I C , of RHSs of (3.5)and (3.6), note that k φ C,n ( s ) − φ C ( s ) k ˜Λ ( K Q ,H ) → , n k such that for all s ≤ T k φ C,n k ( s ) − φ C ( s ) || L ( K Q ,H ) → , P − a.s. . Thus, for the eigenvectors { f j } ∞ j =1 of Q and all t ≤ T , || φ C,n k ( s ) f j − φ C ( s ) f j || H → , P − a.s. (3.11)On the other hand, for the ONB, { f j } ∞ j =1 , in the Hilbert space K , tr ( F xx ( X C,n k ( s )) φ C,n k ( s ) Qφ ∗ C,n k ( s )) = tr ( φ ∗ C,n k ( s ) F xx ( X C,n k ( s )) φ C,n k ( s ) Q )= ∞ X j =1 λ j h F xx ( X C,n k ( s )) φ C,n k ( s ) f j , φ C,n k ( s ) f j i H , where λ j is the eigenvalue associated with eigenvector f j of Q . Since X C,n k ( s ) is bounded and F xx is continuous, (3.11) implies that for s ≤ T h F xx ( X C,n k ( s )) φ C,n k ( s ) f j , φ C,n k ( s ) f j i H → h F xx ( X C ( s )) φ C ( s ) f j , φ C ( s ) f j i H , P − a.s.. By the Lebesgue Dominated Convergence Theorem (DCT) (with re-spect to the counting measure), it holds a.e. on [0 , T ] × Ω that tr ( F xx ( X C,n k ( s )) φ C,n k ( s ) Qφ ∗ C,n k ( s )) → tr ( F xx ( X C ( s )) φ C ( s ) Qφ ∗ C ( s )) . (3.12)Moreover, the left hand side of (3.12) is bounded above by η n ( s ) := k F xx ( X C,n k ( s )) k L ( H ) k φ C,n k k ( K Q ,H ) , , T ] × Ω η n ( s ) → η ( s ) = k F xx ( X C ( s )) k L ( H ) k φ C k ( K Q ,H ) . So, by the boundedness of F xx , R t η n ( s ) dE s → R t η ( s ) dE s and bythe general Lebesgue DCT, it holds P -a.s. that for t ≤ TI C,n k − I C = Z t tr [ F xx ( X C,n k ( s ))( φ C,n k ( s ) Q / )( φ C,n k ( s ) Q / ) ∗ ]d E s − Z t tr [ F xx ( X C ( s ))( φ C ( s ) Q / )( φ C ( s ) Q / ) ∗ ]d E s → . (3.13)Therefore, from (3.7), (3.8), (3.9), (3.10) and (3.13), the Itˆo formulafor the process X C,n ( t ), (3.5), converges in probability to the Itˆoformula for the process X C ( t ), (3.6), and possibly for a subsequence, n k , converges P -a.s.Second, the proof can be reduced to the case where X ( t ) = X (0) + ψt + γE t + φW E t where ψ , γ , and φ are G -measurable random variables independentof t . Define the function u ( t , t , x ) : R + × R + × H → R as u ( t, E t , W E t ) = F ( X (0) + ψt + γE t + φW E t ) = F ( X ( t )) . Now, we prove that the Itˆo formula holds for the function u ( t , t , x ).First, let 0 = t < t < ... < t n = t ≤ T be a partition of an interval4 Lise Chlebak, Patricia Garmirian and Qiong Wu[0 , t ], then u ( t, E t , W E t ) − u (0 , ,
0) = n − X j =1 [ u ( t j +1 , E t j +1 , W E tj +1 ) − u ( t j , E t j +1 , W E tj +1 )]+ n − X j =1 [ u ( t j , E t j +1 , W E tj +1 ) − u ( t j , E t j , W E tj +1 )]+ n − X j =1 [ u ( t j , E t j , W E tj +1 ) − u ( t j , E t j , W E tj )] . Also, let ∆ t j = t j +1 − t j , ∆ E j = E t j +1 − E t j and ∆ W j = W E tj +1 − W E tj . Let θ j ∈ [0 ,
1] be a random variable, and ¯ t j = t j + θ j ( t j +1 − t j ),¯ E j = E t j + θ j ( E t j +1 − E t j ) and ¯ W j = W E tj + θ j ( W E tj +1 − W E tj ).5Using Taylor’s formula, u ( t, E t , W E t ) − u (0 , , n − X j =1 u t (¯ t j , E t j +1 , W E tj +1 )∆ t j + n − X j =1 u t ( t j , ¯ E j , W E tj +1 )∆ E j + n − X j =1 [ h u x ( t j , E t j , W E tj ) , ∆ W j i K + 12 h u xx ( t j , E t j , ¯ W j )(∆ W j ) , ∆ W j i K ]= n − X j =1 u t ( t j , E t j +1 , W E tj +1 )∆ t j + n − X j =1 u t ( t j , E t j , W E tj +1 )∆ E j + n − X j =1 h u x ( t j , E t j , W E tj ) , ∆ W j i K + 12 n − X j =1 h u xx ( t j , E t j , W E tj )(∆ W j ) , ∆ W j i K + n − X j =1 [ u t (¯ t j , E t j +1 , W E tj +1 ) − u t ( t j , E t j +1 , W E tj +1 )]∆ t j + n − X j =1 [ u t ( t j , ¯ E j +1 , W E tj +1 ) − u t ( t j , E t j , W E tj +1 )]∆ E j + 12 n − X j =1 h [ u xx ( t j , E t j , ¯ W j )(∆ W j ) − u xx ( t j , E t j , W E tj )(∆ W j )] , ∆ W j i K := I + I + I + I + I + I + I . (3.14)By the uniform continuity of the mappings:[0 , T ] × [0 , T ] × [0 , T ] ∋ ( t, s, r ) → u t ( t, E s , W E r ) ∈ R [0 , T ] × [0 , T ] × [0 , T ] ∋ ( t, s, r ) → u t ( t, E s , W E r ) ∈ R , T ] ∋ t → u x ( t, E t , W E t ) ∈ K ∗ , thefollowing holds P -a.s. I = n − X j =1 u t ( t j , E j +1 , W E tj +1 )∆ t j → Z t u t ( s, E s , W E s )d s,I = n − X j =1 u t ( t j , E j , W E tj +1 )∆ E t j → Z t u t ( s, E s , W E s )d E s ,I = n − X j =1 h u x ( s, E s , W E tj ) , ∆ W j i K → Z t h u x ( s, E s , W E s ) , d W E s i K . (3.15)Also since the time change E t has bounded variation, | I | = (cid:12)(cid:12)(cid:12)(cid:12) n − X j =1 [ u t (¯ t j , E t j +1 , W E tj +1 ) − u t ( t j , E t j +1 , W E tj +1 )]∆ t j (cid:12)(cid:12)(cid:12)(cid:12) ≤ T sup j ≤ n − (cid:12)(cid:12)(cid:12)(cid:12) u t (¯ t j , E t j +1 , W E tj +1 ) − u t ( t j , E t j +1 , W E tj +1 ) (cid:12)(cid:12)(cid:12)(cid:12) → , | I | = (cid:12)(cid:12)(cid:12)(cid:12) n − X j =1 [ u t ( t j , ¯ E j +1 , W E tj +1 ) − u t ( t j , E t j , W E tj +1 )]∆ E j (cid:12)(cid:12)(cid:12)(cid:12) ≤ n − X j =1 | ∆ E t j | sup j ≤ n − (cid:12)(cid:12)(cid:12)(cid:12) u t ( t j , ¯ E j +1 , W E tj +1 ) − u t ( t j , E t j , W E tj +1 ) (cid:12)(cid:12)(cid:12)(cid:12) → . (3.16)7Similarly, by the continuity of the map K ∋ x → u xx ( t , t , x ) ∈L ( K, K ), | I | = 12 (cid:12)(cid:12)(cid:12)(cid:12) n − X j =1 h [ u xx ( t j , E t j , ¯ W j )(∆ W j ) − u xx ( t j , E t j , W E tj )(∆ W j )] , ∆ W j i K (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup j ≤ n − || u xx ( t j , E t j , ¯ W j )(∆ W j ) − u xx ( t j , E t j , W E tj )(∆ W j ) || L ( K,K ) × n − X j =1 || ∆ W j || K → n → ∞ since the function u has the samesmoothness as F and ¯ W j → W E tj as the increments t j +1 − t j getsmaller. It remains to deal with the fourth term, I . Let 1 Nj =1 { max {k W Eti k K ≤ N, i ≤ j }} which is G t j -measurable. To handle I , thefollowing computations are helpful. First, E (cid:18) h Nj u xx ( t j , E t j , W E tj )(∆ W j ) , ∆ W j i K (cid:12)(cid:12)(cid:12)(cid:12) G t j (cid:19) = E (cid:18)(cid:28) Nj u xx ( t j , E t j , W E tj ) ∞ X k =1 λ / k ( w k ( E t j +1 ) − w k ( E t j )) f k , ∞ X l =1 λ / l ( w l ( E t j +1 ) − w l ( E t j )) f l (cid:29) K (cid:12)(cid:12)(cid:12)(cid:12) G t j (cid:19) = ∞ X k =1 E (cid:18) λ k h Nj u xx ( t j , E t j , W E tj ) f k , f k i K ( w k ( E t j +1 ) − w k ( E t j )) (cid:12)(cid:12)(cid:12)(cid:12) G t j (cid:19) = tr (1 Nj u xx ( t j , E t j , W E tj ) Q )∆ E j . (3.18)8 Lise Chlebak, Patricia Garmirian and Qiong WuSecond, for the cross term arising in the computation below, withoutloss of generality, assume i < j . Then, I N := E (cid:26)(cid:18)(cid:10) Ni u xx ( t j , E t j , W E ti )(∆ W i ) , ∆ W i (cid:11) K − tr (cid:0) Ni u xx ( t j , E t j , W E ti ) Q (cid:1) ∆ E i (cid:19) × (cid:18)(cid:10) Nj u xx ( t j , E t j , W E tj )(∆ W j ) , ∆ W j (cid:11) K − tr (cid:0) Nj u xx ( t j , E t j , W E tj ) Q (cid:1) ∆ E j (cid:19)(cid:27) = E (cid:26)(cid:18)(cid:10) Ni u xx ( t j , E t j , W E ti )(∆ W i ) , ∆ W i (cid:11) K − tr (cid:0) Ni u xx ( t j , E t j , W E ti ) Q (cid:1) ∆ E i (cid:19) × E (cid:18)(cid:10) Nj u xx ( t j , E t j , W E tj )(∆ W j ) , ∆ W j (cid:11) K − tr (cid:0) Nj u xx ( t j , E t j , W E tj ) Q (cid:1) ∆ E j (cid:12)(cid:12)(cid:12)(cid:12) G t ( i +1) (cid:19)(cid:27) . I N = E (cid:26)(cid:18)(cid:10) Ni u xx ( t j , E t j , W E ti )(∆ W i ) , ∆ W i (cid:11) K − tr (cid:0) Ni u xx ( t j , E t j , W E ti ) Q (cid:1) ∆ E i (cid:19) × E (cid:18) E (cid:0)(cid:10) Nj u xx ( t j , E t j , W E tj )(∆ W j ) , ∆ W j (cid:11) K |G t j (cid:1) − tr (cid:0) Nj u xx ( t j , E t j , W E tj ) Q (cid:1) ∆ E j (cid:12)(cid:12)(cid:12)(cid:12) G t ( i +1) (cid:19)(cid:27) = E (cid:26)(cid:18)(cid:10) Ni u xx ( t j , E t j , W E ti )(∆ W i ) , ∆ W i (cid:11) K − tr (cid:0) Ni u xx ( t j , E t j , W E ti ) Q (cid:1) ∆ E i (cid:19) × E (cid:18) E (cid:0) tr (cid:0) Nj u xx ( t j , E t j , W E tj ) Q (cid:1) ∆ E j − tr (cid:0) Nj u xx ( t j , E t j , W E tj ) Q (cid:1) ∆ E j (cid:12)(cid:12)(cid:12)(cid:12) G t ( i +1) (cid:19)(cid:27) = 0 . (3.19)Third, let f E tj +1 ,E tj ( τ , τ ) be the joint density function of randomvariables, E t j +1 and E t j . Then, for t j +1 > t j , letting D = { ( τ , τ ) ∈ R + × R + : 0 ≤ τ ≤ τ } , E (cid:18)(cid:10) Nj u xx ( t j , E t j , W E tj )(∆ W j ) , ∆ W j (cid:11) K × tr (cid:0) Nj u xx ( t j , E t j , W E tj ) Q (cid:1) ∆ E j (cid:19) = Z Z D E (cid:18)(cid:10) Nj u xx ( t j , E t j , W E tj )(∆ W j ) , ∆ W j (cid:11) K × tr (cid:0) Nj u xx ( t j , E t j , W E tj ) Q (cid:1) ∆ E j (cid:12)(cid:12)(cid:12)(cid:12) E t j +1 = τ , E t j = τ (cid:19) f E tj +1 ,E tj ( τ , τ )d( τ , τ )= Z Z D E (cid:18) tr (cid:0) Nj u xx ( t j , τ , W τ ) Q (cid:1) ( τ − τ ) (cid:19) f E tj +1 ,E tj ( τ , τ )d( τ , τ )= E (cid:18) tr (cid:0) Nj u xx ( t j , E t j , W E tj ) Q (cid:1) (∆ E t j ) (cid:19) , (3.20)Finally, still for t j +1 > t j : E k ∆ W j k K = E k W E tj +1 − W E tj k K = Z Z D E (cid:18) k W τ − W τ k K (cid:12)(cid:12)(cid:12)(cid:12) E t j +1 = τ , E t j = τ (cid:19) × f E tj +1 ,E tj ( τ , τ )d( τ , τ )= Z Z D trQ ) ( τ − τ ) f E tj +1 ,E tj ( τ , τ )d( τ , τ )= 3( trQ ) E ( E t j +1 − E t j ) = 3( trQ ) E (∆ E j ) . (3.21)1Noting that u xx is bounded on bounded subsets of H , apply (3.18), (3.19),(3.20) and (3.21) to yield E (cid:18) n − X j =1 (cid:0)(cid:10) Nj u xx ( t j , E t j , W E tj )(∆ W j ) , ∆ W j (cid:11) K − tr (cid:0) Nj u xx ( t j , E t j , W E tj ) Q (cid:1) ∆ E j (cid:1)(cid:19) = n − X j =1 E (cid:18)(cid:10) Nj u xx ( t j , E t j , W E tj )(∆ W j ) , ∆ W j (cid:11) K − tr (cid:0) Nj u xx ( t j , E t j , W E tj ) Q (cid:1) ∆ E j (cid:19) + n − X i = j =1 E (cid:26)(cid:18)(cid:10) Ni u xx ( t i , E t i , W E ti )(∆ W i ) , ∆ W i (cid:11) K − tr (cid:0) Ni u xx ( t i , E t i , W E ti ) Q (cid:1) ∆ E i (cid:19) × (cid:18)(cid:10) Nj u xx ( t j , E t j , W E tj )(∆ W j ) , ∆ W j (cid:11) K − tr (cid:0) Nj u xx ( t j , E t j , W E tj ) Q (cid:1) ∆ E j (cid:19)(cid:27) = n − X j =1 (cid:26) E (cid:10) Nj u xx ( t j , E t j , W E tj )(∆ W j ) , ∆ W j (cid:11) K − E (cid:18) tr (1 Nj u xx ( t j , E t j , W E tj ) Q ) (∆ E j ) (cid:19)(cid:27) ≤ sup s ≤ t, k h k H ≤ N | u xx ( s, E s , h ) | L ( H ) n − X j =1 (cid:0) E || ∆ W j || K − ( trQ ) E (∆ E j ) (cid:1) = 2 sup s ≤ t, k h k H ≤ N | u xx ( s, E s , h ) | L ( H ) ( trQ ) E n − X j =1 (∆ E j ) → , E t has finite bounded variation. Addi-tionally, P (cid:18) n − X j =1 (1 − Nj ) (cid:8)(cid:10) u xx ( t j , E t j , W E tj )(∆ W j ) , ∆ W j (cid:11) K − tr (cid:0) Nj u xx ( t j , E t j , W E tj ) Q (cid:1) ∆ E j (cid:9) = 0 (cid:19) ≤ P (cid:18) sup s ≤ t {|| W E s || > N } (cid:19) → N → ∞ . Thus, it follows that I = 12 n − X j =1 h u xx ( t j , E t j , W E tj )(∆ W j ) , ∆ W j i K → Z t tr [ u xx ( s, E s , W E s ) Q ]d E s (3.22)in probability. Combining (3.15), (3.17) and (3.22), and taking thelimit on the right hand side of (3.14) yields the Itˆo formula for thefunction u ( t, E t , W E t ): u ( t, E t , W E t ) = u (0 , ,
0) + Z t u t ( s, E s , W E s )d s + Z t u t ( s, E s , W E s )d E s + 12 Z t tr [ u xx ( s, E s , W E s ) Q ]d E s + Z t h u x ( s, E s , W E s ) , dW E s i K , (3.23)3in probability. Also note that u t ( t, E t , k ) = h F x ( X (0) + ψt + γE t + φk ) , ψ i H u t ( t, E t , k ) = h F x ( X (0) + ψt + γE t + φk ) , γ i H u x ( t, E t , k ) = φ ∗ F x ( X (0) + ψt + γE t + φk ) ,u xx ( t, E t , k ) = φ ∗ F xx ( X (0) + ψt + γE t + φk ) φ, (3.24)and tr [ F xx ( X ( s ))( φQ / )( φQ / ) ∗ ] = tr [ φ ∗ F xx ( X ( s )) φQ ] . (3.25)Substituting (3.24) and (3.25) into (3.23) yields F ( X ( t )) = F ( X (0)) + Z t h F x ( X ( s )) , ψ ( s ) i H d s + Z t h F x ( X ( s )) , γ ( s ) i H d E s + Z t h F x ( X ( s )) , φ ( s )d W E s i H + 12 Z t tr ( F xx ( X ( s ))( φ ( s ) Q / )( φ ( s ) Q / ) ∗ )d E s (3.26)in probability. Consequently, there is a subsequence such that theequality (3.26) holds almost surely. Therefore applying the secondchange of variable formula yields the desired result. (cid:3) Q -Wiener process Let K and H be real separable Hilbert spaces, and let M ( t ) := W E t be a time-changed K -valued Q -Wiener process on a completefiltered probability space (Ω , G , {G} t ≤ T , P ) with the filtration G t =˜ F E t satisfying the usual conditions. Consider the following type of4 Lise Chlebak, Patricia Garmirian and Qiong Wusemilinear SDE driven by the time-changed Q -Wiener process on[0 , T ] in H : du ( t ) = Au ( t )d t + B ( u, t )d M ( t ) (3.27)with initial condition u (0) = u , where A is the generator of a C -semigroup of operators { S ( t ) , t ≥ } . Theorem 3.9.
Assume that the following hypotheses are satisfied: S ( t ) is a contraction C -semigroup generated by A . B ( t, u ) : D ( R + , H ) → ˜Λ ( K Q , H ) is non-anticipating where D ( R + , H ) denotes the H -valued cadlag adapted processes with R + as the time interval. (Local Lipschitz property) For every r > , there exists a con-stant K r > such that for every x, y ∈ D ( R + , H ) and t ≥ k B ( t, x ) − B ( t, y ) || L ( K Q ,H ) ≤ K r sup s Assume that the following hypotheses are satisfied: W t is a K -valued Q -Wiener process on a complete filtered prob-ability space (Ω , F , {F t } t ≤ T , P ) with the filtration F t satisfyingthe usual conditions and E t is the inverse β -stable subordinatorwhich is independent of W t . A is a linear bounded operator. The coefficients F : Ω × [0 , T ] × C ([0 , T ] , H ) → H and B :Ω × [0 , T ] × C ([0 , T ] , H ) → L ( K Q , H ) , where C ([0 , T ] , H ) isthe Banach space of H -valued continuous functions on [0 , T ] ,satisfy the following conditions (a) F and B are jointly measurable, and for every ≤ t ≤ T , they are measurable with respect to the product σ -field F t × C t on Ω × C ([0 , T ] , H ) , where C t is a σ -field generatedby cylinders with bases over [0 , t ] . (b) There exists a constant L such that for all x ∈ C ([0 , T ] , H ) , k F ( ω, t, x ) k H + k B ( ω, t, x ) k L ( K Q ,H ) ≤ L (1 + sup ≤ s ≤ T k x ( s ) k H ) for ω ∈ Ω and ≤ t ≤ T . (c) For all x, y ∈ C ([0 , T ] , H ) , ω ∈ Ω , ≤ t ≤ T , there exists K > such that k F ( ω, t, x ) − F ( ω, t, y ) k H + k B ( ω, t, x ) − B ( ω, t, y ) k L ( K Q ,H ) ≤ K sup ≤ s ≤ T || x ( s ) − y ( s ) || H . E R T k B ( t, Y ( t )) k L ( K Q ,H ) dt < ∞ . Y ( t ) is in the domain of A d P × d t -almost everywhere. x is an F -measurable H -valued random variable.Then, the time-changed SDE (3.30) has a unique strong solution, X ( t ) , satisfying X ( t ) = x + Z t ( AX ( s ) + F ( E s , X ( s )))d E s + Z t B ( E s , X ( s ))d W E s . (3.32) Proof. From [5], based on conditions in Theorem 3.12, there is aunique solution Y ( t ) that satisfies the SDE (3.29). Moreover, it fol-lows from Theorem 3.11 that X ( t ) := Y ( E t ) satisfies the SDE (3.30).Therefore, there exists a solution to the time-changed SDE (3.30).Now suppose there exists another solution to the SDE (3.30).Call this solution ˆ X ( t ). Then, by Theorem 3.11, the process ˆ Y ( t ) :=ˆ X ( U t − ) is a solution to the SDE (3.29). Since the solution to theSDE (3.29) is unique from [5], it must be that ˆ Y ( t ) = Y ( t ). Thusˆ X ( U t − ) = Y ( t ) which implies that ˆ X ( t ) = Y ( E t ) = X ( t ). Therefore,the solution X ( t ) of the SDE (3.30) is unique and satisfies the desiredintegral equation (3.32). (cid:3) 4. Connections between Hilbert space-valued integralsdriven by time-changed Q -Wiener processes andtime-changed cylindrical Wiener processes,respectively, and Walsh-type integrals The objective of this section is to establish the equality of the threeintegrals given below for appropriate integrands: Z T Z R N g ( t, x ) M E (d t, d x ) = Z T g ( t )d ˜ W E t = Z T Φ gt ◦ J − d W E t , where M E is a time-changed version of a worthy martingale measure,˜ W E t is a time-changed version of a cylindrical Wiener process and W E t is a time-changed Q-Wiener process. Section 4.1 focuses on thecase of no time change. Section 4.2 provides the extension to thetime-changed case. The equality of these integrals in the case where there is no timechange are made in [7, 33]. The general idea is to first define a specificrandom field F and an associated Hilbert space K . After identifyingthese objects, the integral with respect to the resulting cylindricalWiener process and the integral with respect to the developed mar-tingale measure are shown to be the same for a particular class ofintegrands. Also a connection between the cylindrical Wiener processand a Q-Wiener process is made that leads to an equality of theirrespective integrals. Thus, a combination of those results proves thatall three integrals are equal.0 Lise Chlebak, Patricia Garmirian and Qiong WuLet { F ( φ ) | φ ∈ C ∞ ( R + × R N ) } be a family of mean zero Gauss-ian random variables, called a Gaussian random field. The covarianceof F is given by E ( F ( φ ) F ( ψ )) = Z R + Z R N Z R N φ ( s, x ) f ( x − y ) ψ ( s, y )d y d x d s, (4.1)where f satisfies the following two conditions:1) f is a non-negative, non-negative definite continuous functionon R N \{ } , which is integrable in a neighborhood of 0;2) for all γ ∈ S ( R N ), the space of C ∞ functions which are rapidlydecreasing along with all their derivatives, the following condi-tions hold:a) there exists a tempered measure µ on R N such that forany m ∈ N + Z R N (1 + | ξ | ) − m µ (d ξ ) < ∞ , andb) Z R N f ( x ) γ ( x )d x = Z R N F γ ( ξ ) µ (d ξ ), where F γ is the Fouriertransform of γ .For the remainder of Section 4, fix the specific random field F chosenin (4.1). We now define a Hilbert space associated with the fixedrandom field F . Let K be the completion of the Schwartz space S ( R N ) with the semi-inner product h γ, ψ i K := Z R N Z R N γ ( x ) f ( x − y ) ψ ( y )d y d x = Z R N µ (d ξ ) F γ ( ξ ) F ψ ( ξ ) , (4.2)1where γ, ψ ∈ K , and associated semi-norm ||·|| K . Then K is a Hilbertspace, see [36]. In the remaining of this section, the Hilbert space K always refers to (4.2).Moreover, after fixing a time interval [0 , T ], it is possible toconsider the set K T := L ([0 , T ] , ; K ) with the norm k g k K T = Z T k g ( s ) k K d s. Note that C ∞ ([0 , T ] × R N ) is dense in K T . It should also be notedthat although F was originally defined on smooth, compactly sup-ported functions, it is possible to extend F to functions of the form1 [0 ,T ] ( · ) ϕ ( ∗ ) where ϕ ∈ S ( R N ). This follows since such an F is arandom linear functional such that γ F ( γ ) is an isometry from( C ∞ ([0 , T ] × R N ) , k · k K T ) into L (Ω , F , P ), i.e., a family of meanzero Gaussian random variables characterized by (4.1). For furtherdetails, see [33]. Definition 4.1. A cylindrical Wiener process on a Hilbert space K as defined in (4.2) is a family of random variables { ˜ W t , t ≥ } suchthat:1. for each h ∈ K , { ˜ W t ( h ) , t ≥ } defines a Brownian motion withmean 0 and variance t h h, h i K ;2. for all s, t ∈ R + and h, g ∈ K , E ( ˜ W s ( h ) ˜ W t ( g )) = ( s ∧ t ) h h, g i K where s ∧ t := min( s, t ).From [33], the stochastic process { ˜ W t , t ≥ } defined in termsof the fixed random field F with covariance chosen in (4.1) is given2 Lise Chlebak, Patricia Garmirian and Qiong Wuby ˜ W t ( ϕ ) := F (1 [0 ,t ] ( · ) ϕ ( ∗ )) , for ϕ ∈ K, (4.3)is a cylindrical Wiener process on the Hilbert space K . A completeorthonormal basis { f j } can be chosen such that { f j } ⊂ S ( R N ) since S ( R N ) is a dense subspace of K . Consider the space L (Ω × [0 , T ]; K )of predictable processes g such that E Z T || g ( s ) || K d s ! < ∞ . For g ∈ L (Ω × [0 , T ]; K ), the stochastic integral in Hilbert space H with respect to the cylindrical Wiener process ˜ W t is defined as Z T g ( s )d ˜ W s := ∞ X j =1 Z T h g ( s ) , f j i K d ˜ W s ( f j )where particularly H = R and { f j } is an orthonormal basis of K in (4.2). The series is convergent in L (Ω , F , P ), and the sum doesnot depend on the choice of orthonormal basis. Additionally, thefollowing isometry holds: E Z T g ( s )d ˜ W s ! = E Z T || g ( s ) || K d s ! . On the other hand, consider M t defined by M t ( A ) := F (1 [0 ,t ] ( · )1 A ( ∗ )) , t ∈ [0 , T ] , A ∈ B b ( R N ) , (4.4)where F is the specific random field chosen in (4.1) and B b ( R N ) de-notes the set of bounded Borel sets of R N . The covariance of { M t ( A ) } E ( M t ( A ) M t ( B )) = Z R + Z R N Z R N [0 ,t ] ( s )1 A ( x ) f ( x − y )1 B ( y )d y d x d s = t Z R N Z R N A ( x ) f ( x − y )1 B ( y )d y d x. Therefore, M t is a martingale measure in the following sense. Definition 4.2. ([33] ) A process { M t ( A ) } t ≥ ,A ∈B ( R N ) is a martingalemeasure with respect to {F t } t ≥ if:1. for all A ∈ B ( R N ), M ( A ) = 0 a.s.;2. for t > M t is a sigma-finite L ( P )-valued signed measure;and3. for all A ∈ B ( R N ), { M t ( A ) } t ≥ is a mean-zero martingale withrespect to the filtration {F t } t ≥ .Further, in order to define stochastic integrals with respect toa martingale measure, the martingale measure needs to be worthy. Definition 4.3. ([33]) A martingale measure M is worthy if thereexists a random sigma-finite measure K ( A × B × C, ω ), where A, B ∈B ( R N ), C ∈ B ( R + ), and ω ∈ Ω, such that:1. A × B × C K ( A × B × C, ω ) is nonnegative definite andsymmetric;2. { K ( A × B × (0 , t ]) } t ≥ is a predictable process for all A, B ∈B ( R N );3. for all compact sets A, B ∈ B ( R N ) and t > E ( K ( A × B × (0 , t ])) < ∞ ;4 Lise Chlebak, Patricia Garmirian and Qiong Wu4. for all A, B ∈ B ( R N ) and t > | E ( M t ( A ) M t ( B )) | ≤ K ( A × B × (0 , t ]) a.s.In particular, the stochastic integral with respect to a worthymartingale measure is defined in such a way that it is itself a mar-tingale measure. First, consider elementary processes g of the form g ( s, x, ω ) = 1 ( a,b ] ( s )1 A ( x ) X ( ω ) (4.5)where 0 ≤ a < b ≤ T , X is bounded and F a -measurable, and A ∈B ( R N ). If g is an elementary process as in (4.5), then define g · M by g · M t ( B ) := Z t Z B g ( s, x ) M (d s, d x )= X ( ω ) ( M t ∧ b ( A ∩ B ) − M t ∧ a ( A ∩ B )) . The definition of g · M can be extended by linearity to simple pro-cesses, which are finite sums of elementary processes. Let P + denotethe set of predictable processes ( ω, t, x ) g ( t, x ; ω ) such that k g k := E Z T Z R N Z R N | g ( t, x ) | f ( x − y ) | g ( t, y ) | d y d x d t ! < ∞ . Taking limits of simple processes, the definition of g · M extendsto all g ∈ P + . From [7], g · M is a worthy martingale measure if g ∈ P + . Therefore, it makes sense to define the stochastic integralwith respect to M as a martingale measure in the following way: Z t Z A g ( s, x ) M (d s, d x ) =: g · M t ( A ) . Proposition 4.4. ( [33] ) Suppose g ∈ P + . Then g ∈ L (Ω × [0 , T ]; K ) and Z T Z R N g ( t, x ) M (d t, d x ) = Z T g ( t )d ˜ W t , where M is the worthy martingale measure defined in (4.4) and ˜ W t is the cylindrical Wiener process defined in (4.3) . Finally, Proposition 4.6 below provides conditions under whichthese integrals coincide with the integral with respect to a Q-Wienerprocess. Define the operator J : K → K by J ( h ) := ∞ X j =1 λ / j h h, f j i K f j , h ∈ K, (4.6)where { f j } is an orthonormal basis in K and λ j ≥ ∞ X j =1 λ j < ∞ . Lemma 4.5. Let Q = JJ ∗ : K → K . Then Q has eigenvalues λ j corresponding to the eigenvectors f j , i.e. Qf j = λ j f j .Proof. Let f j be the orthonormal basis as in (4.6), then Qf j = ( JJ ∗ )( f j ) = ∞ X l =1 λ / l h J ∗ f j , f l i K f l = ∞ X l =1 λ / l h f l , Jf j i K f l = ∞ X l =1 λ / l h f j , ∞ X i =1 λ / i h f l , f i i K f i i K f l = ∞ X l =1 λ l h f j , f l i K f l = λ j f j . (cid:3) Additionally, Q is symmetric (self-adjoint), non-negative defi-nite, and trQ = P ∞ j =1 λ j < ∞ . The operator J : K → K Q is anisometry since k h k K = k Q − / J ( h ) k K = k J ( h ) k K Q , h ∈ K, where Q − / denotes the pseudo-inverse of Q / . Further, the inverseoperator J − : K Q = Q / K → K is also an isometry. Therefore,taking { w j ( t ) } ∞ j =1 to be the family of independent Brownian motionprocesses defined by 4.3, a Q-Wiener process in K as defined inSection 2 can be constructed as W t := ∞ X j =1 w j ( t ) J ( f j ) = ∞ X j =1 w j ( t ) λ / j f j . (4.7)As seen in Section 3, a predictable process φ will be integrablewith respect to the Q -Wiener process W t if E Z T || φ ( t ) || L ( K Q ,H ) d t ! < ∞ , (4.8)Consider g ∈ L (Ω × [0 , T ]; K ), and define the operator, Φ gs : K → R ,by Φ gs ( η ) = h g ( s ) , η i K , η ∈ K. (4.9)The following proposition is from Dalang and Quer-Sardanyons [33];the proof is included here since it contains information that will beused later.7 Proposition 4.6. If g ∈ P + and Φ gt as defined in (4.9) , then Φ gt ◦ J − satisfies the condition (4.8) and Z T Φ gt ◦ J − d W t = Z T g ( t )d ˜ W t , where W t is the Q -Wiener process defined in (4.7) , J is the operatordefined in (4.6) and ˜ W t is the cylindrical Wiener process definedin (4.3) .Proof. First, to show that Φ gt ◦ J − ∈ L ( K Q , R ), note that || Φ gt ◦ J − || L ( K Q , R ) = ∞ X j =1 [(Φ gt ◦ J − )( λ / j f j )] = ∞ X j =1 [Φ gt ( J − λ / j f j )] = ∞ X j =1 h g ( t ) , J − Q / f j i K = ∞ X j =1 h g ( t ) , f j i K = || g ( t ) || K . Therefore, E Z T || Φ gt ◦ J − || L ( K Q , R ) d t ! = E Z T || g ( t ) || K d t ! < ∞ , since g ∈ P + implies that g ∈ L (Ω × [0 , T ]; K ). Thus, Φ gt ◦ J − satisfies condition (4.8). Also, from Lemma 3.1, Z T Φ gt ◦ J − dW t = ∞ X j =1 Z T (Φ gt ◦ J − )( λ / j f j )d h W t , λ / j f j i K Q = ∞ X j =1 Z T h g ( t ) , f j i K d w j ( t )= ∞ X j =1 Z T h g ( t ) , f j i K d ˜ W t ( f j )= Z T g ( t )d ˜ W t . (cid:3) Therefore, combining Propositions 4.4 and 4.6 yields the desiredintegral connections. Corollary 4.7. For g ∈ P + and Φ gt as defined in (4.9) , Z T Z R N g ( t, x ) M (d t, d x ) = Z T g ( t )d ˜ W t = Z T Φ gt ◦ J − d W t , where M is the martingale measure defined in (4.4) , ˜ W t is the cylin-drical Wiener process defined in (4.3) , J is the operator definedin (4.6) and W t is the Q -Wiener process defined in (4.7) . The previous section summarized results from [7, 33, 35] establish-ing the equality of integrals with respect to a martingale measure,a cylindrical Wiener process, and a Q-Wiener process. This sectionwill extend those results to the time-changed case. The procedure forshowing the equivalence of the integrals will be very similar to thatused in the previous section. The same random field F and Hilbertspace K are used to define time-changed versions of a cylindricalWiener process and a martingale measure. Their associated inte-grals are then shown to be equal. Finally, a connection between thegiven time-changed cylindrical Wiener process and a time-changedQ-Wiener process leads to the equality of all three integrals.First, recall that the time change, E t , is the inverse of a β -stablesubordinator. Then, the time-changed cylindrical Wiener process isdefined as follows.9 Definition 4.8. Let K be a separable Hilbert space. A family of ran-dom variables { ˜ W E t , t ≥ } is a time-changed cylindrical Wienerprocess on K if the following conditions hold:1. for any k ∈ K , { ˜ W E t ( k ) , t ≥ } defines a time-changed Brown-ian motion with mean 0 and variance E ( E t ) h k, k i K ; and2. for all s, t ∈ R + and k, h ∈ K , E ( ˜ W E s ( k ) ˜ W E t ( h )) = E ( E s ∧ t ) h k, h i K . Let g : R + × Ω → K be any predictable process such that E Z T k g ( s ) k K d E s ! < ∞ . (4.10)Consider the series ∞ X j =1 Z T h g s , f j i K d ˜ W E s ( f j ) . Convergence of this series in L (Ω , G , P ) is established in Proposition4.9. This justifies defining the stochastic integral of g with respect toa time-changed cylindrical Wiener process as follows: Z T g ( s )d ˜ W E s := ∞ X j =1 Z T h g s , f j i K d ˜ W E s ( f j ) . Proposition 4.9. Let g : R + × Ω → K in (4.2) be any predictableprocess satisfying condition (4.10) . Then, the series ∞ X j =1 Z T h g s , f j i K d ˜ W E s ( f j ) (4.11) converges in L (Ω , G , P ) . Proof. Let Y n := n X j =1 Z T h g s , f j i K d ˜ W E s ( f j ) . In order to show the convergence of the series defined in (4.11), itis sufficient to show { Y n } is a Cauchy sequence in L (Ω , G , P ). For n > m , || Y n − Y m || = E (cid:18) n X j = m +1 Z T h g s , f j i K d ˜ W E s ( f j ) (cid:19) = E (cid:20)(cid:18) n X j = m +1 Z T h g s , f j i K d ˜ W E s ( f j ) (cid:19)(cid:18) n X i = m +1 Z T h g s , f i i K d ˜ W E s ( f i ) (cid:19)(cid:21) = E (cid:18) n X j = i = m +1 (cid:20) Z T h g s , f j i K d ˜ W E s ( f j ) (cid:21) (cid:19) + E (cid:18) n X j = i = m +1 (cid:20) Z T h g s , f j i K d ˜ W E s ( f j ) (cid:21)(cid:20) Z T h g s , f i i K d ˜ W E s ( f i ) (cid:21)(cid:19) =: I + II. Since the time-changed Q -Wiener process W E t defined in (2.4)is a square integrable martingale with respect to the filtration G t =˜ F E t defined in (2.5), for h = λ − / j f j ∈ K, j = 1 , , · · · , h W E t , h i K isalso a square integrable martingale with respect to the filtration G t ,i.e., for 0 < s < t , E ( h W E t , h i K |G s ) = h W E s , h i k . This implies that each projection, which is a time-changed Brown-ian motion, w j ( E t ) , j = 1 , , · · · , is also a square integrable mar-tingale with respect to the same filtration G t , i.e., for 0 < s < t ,1 E ( w j ( E t ) |G s ) = w j ( E s ). Therefore, the integral, Z T h g s , f j i K d w j ( E s ) , is also a square integral martingale with respect to the filtration G t ,and it follows from the Itˆo isometry that E (cid:20) Z T h g s , f j i K d w j ( E s ) (cid:21) = E (cid:20) Z T h g s , f j i K d E s (cid:21) . Further, by a proof similar to that of Theorem 2.6, the product w i ( E t ) w j ( E t ) is a square integrable martingale for i = j . This meansthe quadratic covariation process of martingales w i ( E t ) and w j ( E t )is zero, i.e., [ w i ( E t ) , w j ( E t )] = 0. Thus, using the martingale prop-erty of w j ( E s ) and its associated integral, along with the Cauchy-Schwartz inequality I := E (cid:18) n X j = i = m +1 (cid:20) Z T h g s , f j i K d ˜ W E s ( f j ) (cid:21) (cid:19) = n X j = i = m +1 E (cid:20) Z T h g s , f j i K d w j ( E s ) (cid:21) = n X j = i = m +1 E (cid:20) Z T h g s , f j i K d E s (cid:21) ≤ ∞ X j = i = m +1 E (cid:20) Z T h g s , f j i K d E s (cid:21) . (4.12)By assumption (4.10), ∞ X j =1 E (cid:20) Z T h g s , f j i K d E s (cid:21) = E (cid:20) Z T k g s k K d E s (cid:21) < ∞ , m (hence n ) → ∞ . Meanwhile, II := E (cid:18) n X j = i = m +1 (cid:20) Z T h g s , f j i K d ˜ W E s ( f j ) (cid:21)(cid:20) Z T h g s , f i i K d ˜ W E s ( f i ) (cid:21)(cid:19) = n X j = i = m +1 E (cid:20)(cid:18) Z T h g s , f j i K d ˜ W E s ( f j ) (cid:19)(cid:18) Z T h g s , f i i K d ˜ W E s ( f i ) (cid:19)(cid:21) = n X j = i = m +1 E (cid:20)(cid:18) Z T h g s , f j i K d w j ( E s ) (cid:19)(cid:18) Z T h g s , f i i K d w i ( E s ) (cid:19)(cid:21) = n X j = i = m +1 E (cid:18) Z T h g s , f j i K h g s , f i i K d[ w j ( E s ) , w i ( E s )] (cid:19) = 0 . (4.13)Combining (4.12) and (4.13) yields || Y n − Y m || = I + II → n, m → ∞ . Thus, { Y n } is a Cauchy sequence in L (Ω , G , P ), com-pleting the proof. (cid:3) The next proposition shows how to define a cylindrical processfrom the fixed random field F chosen in (4.1) . Proposition 4.10. For t ≥ and φ ∈ K , set ˜ W E t ( φ ) := F (1 [0 ,E t ] ( · ) φ ( ∗ )) . (4.14) Then, the process ˜ W E t is a time-changed cylindrical Wiener process.Proof. Consider a fixed φ ∈ K . ˜ W E t ( φ ) is a time-changed Brownianmotion by construction. Additionally, E [ ˜ W E t ( φ )] = E [ F (1 [0 ,E t ] ( · ) φ ( ∗ ))] = Z ∞ E [ F (1 [0 ,τ ] ( · ) φ ( ∗ ))] f E t ( τ )d τ = Z ∞ · f E t ( τ )d τ = 0 , E [ ˜ W E t ( φ ) ˜ W E t ( φ )] = E [ F (1 [0 ,E t ] ( · ) φ ( ∗ )) F (1 [0 ,E t ] ( · ) φ ( ∗ ))]= E (cid:20)Z R + [0 ,E t ] ( s )1 [0 ,E t ] ( s ) Z R N Z R N φ ( x ) f ( x − y ) φ ( y )d y d x d s (cid:21) = E (cid:20) h φ, φ i K Z R + [0 ,E t ] ( s )d s (cid:21) = E ( E t ) h φ, φ i K . Further, for fixed s, t ∈ R + and φ, ψ ∈ K , E [ ˜ W E t ( φ ) ˜ W E s ( ψ )] = E [ F (1 [0 ,E t ] ( · ) φ ( ∗ )) F (1 [0 ,E s ] ( · ) ψ ( ∗ ))]= E Z R + [0 ,E t ] ( r )1 [0 ,E s ] ( r ) Z R N Z R N φ ( x ) f ( x − y ) ψ ( y ) dydxdr = E Z R + h φ, ψ i K [0 ,E t ∧ E s ] ( r ) dr = E ( E t ∧ s ) h φ, ψ i K . Therefore, according to the Definition 4.8, ˜ W E t is a time-changedcylindrical Wiener process. (cid:3) Moreover, it follows from Proposition 4.9 that the integral with re-spect to the process ˜ W E t defined in (4.14) is well-defined.On the other hand, E t is independent of the martingale measure M t ( A ). So, define a time-changed version of { M t ( A ) } by M E t ( A ) = M ( A × [0 , E t ]) := F (1 [0 ,E t ] ( · )1 A ( ∗ )) . (4.15)By conditioning on the time change, the covariance for M E t ( A ) is E ( M E t ( A ) M E t ( B )) = Z ∞ E ( M τ ( A ) M τ ( B )) f E t ( τ )d τ = Z ∞ τ (cid:18)Z R N Z R N A ( x ) f ( x − y )1 B ( y )d y d x (cid:19) f E t ( τ )d τ = E ( E t ) Z R N Z R N A ( x ) f ( x − y )1 B ( y )d y d x, f E t is the density function of E t . Also E [ M E t ( A )] = E ( E t ) Z R N Z R N A ( x ) f ( x − y )1 A ( y )d y d x < ∞ for all A ∈ B b ( R N ). Thus, M E t ( A ) has a finite second moment forall A ∈ B b ( R N ). Then the following theorem shows that { M E t ( A ) } is also a martingale measure. Theorem 4.11. { M E t ( A ) } t ≥ ,A ∈B ( R N ) is a martingale measure withrespect to the filtration { ˜ F E t } t ≥ , where ˜ F t in (2.5) is generated bythe time change E t and independent Brownian motions ˜ W t ( f j ) , j =1 , , · · · defined by 4.3.Proof. It suffices to check the conditions in Definition 4.2. First, since E = 0 a.s., M E ( A ) = M ( A ) = 0 a.s. because M t ( A ) is a martin-gale measure.Second, let A , B ∈ B ( R N ) be disjoint. Then, for fixed τ , M τ ( A ∪ B ) and M τ ( A ) + M τ ( B ) are mean zero Gaussian random variablesandVar( M τ ( A ∪ B )) = Z Z ( A ∪ B ) × ( A ∪ B ) f ( x − y )d y d x = Z Z A × A f ( x − y ) dydx + Z Z B × B f ( x − y ) dydx + 2 Z Z A × B f ( x − y )d y d x = Var( M τ ( A )) + Var( M τ ( B )) + 2 E ( M τ ( A ) M τ ( B ))= Var (cid:18) M τ ( A ) + M τ ( B ) (cid:19) . Also note that M τ ( A ∪ B ) = M τ ( A ) + M τ ( B ) a.s.5Thus, conditioning on the time change yields P ( M E t ( A ∪ B ) = M E t ( A ) + M E t ( B ))= Z ∞ P ( M τ ( A ∪ B ) = M τ ( A ) + M τ ( B )) f E t ( τ )d τ = Z ∞ f E t ( τ )d τ = 1 , which means M E t ( · ) is additive a.s. Furthermore, assume A ⊃ A ⊃ ... such that ∩ n A n = ∅ , E [ M E t ( A n )] = E ( E t ) Z Z A n × A n f ( x − y )d y d x → n → ∞ , and so M E t ( A n ) → L ( P ) . This proves the countableadditivity of M E t ( · ).Finally, since { M E t ( A ) } has a finite second moment, a simi-lar argument to the proof of Theorem 2.6 shows that { M E t ( A ) } isa martingale for all A ∈ B ( R N ). Thus, { M E t ( A ) } is a martingalemeasure with respect to the filtration G t = ˜ F E t . (cid:3) Define the dominating measure K by K ( A × B × C ):= E ( λ ( { E s ( ω ) : s ∈ C } )) Z R N Z R N A ( x ) f ( x − y )1 B ( y )d y d x, (4.16)where A, B ∈ B ( R N ) and λ is the Lebesgue measure on C ∈ B ( R + ). Then, the following theorem shows that the martingale measure { M E t ( A ) } is worthy.6 Lise Chlebak, Patricia Garmirian and Qiong Wu Theorem 4.12. The martingale measure { M E t ( A ) } t ≥ ,A ∈B ( R N ) is wor-thy with respect to the filtration { ˜ F E t } t ≥ , i.e. the same one as de-fined in Theorem 4.11.Proof. To show that the martingale measure { M E t ( A ) } is worthy,it suffices to show that the dominating measure K defined in (4.16)satisfies the conditions of Definition 4.3.1. For all C ∈ B ( R + ), K ( A × B × C ) = E ( λ ( { E s ( ω ) : s ∈ C } )) Z R N Z R N A ( x ) f ( x − y )1 B ( y )d y d x = E ( λ ( { E s ( ω ) : s ∈ C } )) Z R N Z R N B ( x ) f ( x − y )1 A ( y )d y d x = K ( B × A × C ) . Additionally, for all A ∈ B ( R N ), C ∈ B ( R + ), since f is non-negative definite, K ( A × A × C ) = E ( λ ( { E s ( ω ) : s ∈ C } )) Z R N Z R N A ( x ) f ( x − y )1 A ( y )d y d x ≥ . 2. For all A, B ∈ B ( R N ), t > K ( A × B × (0 , t ]) = E ( E t ) Z R N Z R N A ( x ) f ( x − y )1 B ( y )d y d x is ˜ F E t -measurable.3. For all compact sets A, B ∈ B ( R N ) and t > E | K ( A × B × (0 , t ]) | = E ( E t ) Z R N Z R N A ( x ) f ( x − y )1 B ( y )d y d x < ∞ . 74. For all A, B ∈ B ( R N ) and t > | E ( M E t ( A ) M E t ( B )) | = E ( E t ) Z R N Z R N A ( x ) f ( x − y )1 B ( y )d y d x = K ( A × B × (0 , t ]) . Thus, the martingale measure { M E t ( A ) } is worthy. (cid:3) As shown in the non-time-changed case, the stochastic integralwith respect to a worthy time-changed martingale measure is also aworthy martingale measure. First, consider elementary processes g of the form g ( s, x, ω ) = 1 ( a,b ] ( s )1 A ( x ) X ( ω ) (4.17)where 0 ≤ a < b ≤ T , A ∈ B ( R N ) and X is both bounded and G a := ˜ F E a -measurable. For t, r ∈ [0 , T ], let M E t ∧ E r := M E t ∧ r , and let M E (d s, d x ) denote integration with respect to the martingalemeasure in the both s and x . Then, define g · M E by g · M E t ( B ) := X ( ω )( M E t ∧ E b ( A ∩ B ) − M E t ∧ E a ( A ∩ B ))= Z t Z R N g ( s, x ) M E (d s, d x ) . As usual, this definition of g · M E can be extended to finite sumsof elementary processes and finally to predictable processes g such8 Lise Chlebak, Patricia Garmirian and Qiong Wuthat || g || † := E Z T Z R N Z R N | g ( t, x ) | f ( x − y ) | g ( t, y ) | d y d x d E t ! < ∞ . (4.18)Let P † denote the set of predictable processes ( ω, t, x ) g ( t, x ; ω )such that (4.18) holds. For g ∈ P † , g · M E is a worthy martingalemeasure and the stochastic integral with respect to M E is defined by Z t Z A g ( s, x ) M E ( ds, dx ) =: g · M E t ( A ) . The following theorem connects an integral with respect to a time-changed martingale measure with an integral with respect to a time-changed cylindrical Wiener process. Theorem 4.13. Let M E t be the time-changed martingale measure de-fined in (4.15) and ˜ W E t be the time-changed cylindrical process de-fined in (4.14) . For g ∈ P † , then, Z T Z R N g ( t, x ) M E (d t, d x ) = Z T g ( t )d ˜ W E t . Proof. First notice that since g ∈ P † , E Z T || g ( s ) || K dE s ! = E Z T Z R N Z R N g ( s, x ) f ( x − y ) g ( s, y )d y d x d E s ! ≤ || g || † < ∞ , which means g satisfies the condition (4.10). Further, since the setof elementary processes is dense in P † , it suffices to check that the9integrals coincide for elementary processes of the form g ( s, x, ω ) = 1 ( a,b ] ( s )1 A ( x ) X ( ω )where 0 ≤ a < b ≤ T , A ∈ B ( R N ) and X ( ω ) is both bounded and˜ F E a -measurable. For this, note that Z T Z R N g ( t, x ) M E (d t, d x ) = X ( M E T ∧ E b ( A ) − M E T ∧ E a ( A ))= X ( M E b ( A ) − M E a ( A ))= X ( F (1 [0 ,E b ] ( · )1 A ( ∗ )) − F (1 [0 ,E a ] ( · )1 A ( ∗ )))= X ( F (1 ( E a ,E b ] ( · )1 A ( ∗ ))) . On the other hand, using the linearity of F Z T g ( t ) d ˜ W E t = ∞ X j =1 Z ba X h A , f j i K d ˜ W E t ( f j )= X ∞ X j =1 h A , f j i K ( ˜ W E b ( f j ) − ˜ W E a ( f j ))= X ∞ X j =1 h A , f j i K [ F (1 [0 ,E b ] ( · ) f j ) − F (1 [0 ,E a ] ( · ) f j )]= X ∞ X j =1 h A , f j i K [ F (1 ( E a ,E b ] ( · ) f j )]= X [ F (1 ( E a ,E b ] ( · ) ∞ X j =1 h A , f j i K f j ]= X [ F (1 ( E a ,E b ] ( · )1 A ( ∗ ))] . (cid:3) Next, a connection between the time-changed cylindrical Wienerprocess and the time-changed Q -Wiener process will be established.0 Lise Chlebak, Patricia Garmirian and Qiong WuDefine W E t := ∞ X j =1 w j ( E t ) J ( f j ) (4.19)where w j ( E t ) := ˜ W E t ( f j ) are the time-changed Brownian motionsdefined in Proposition 4.5. Also for g ∈ L (Ω × [0 , T ] , K ), define theoperator Φ gs by Φ gs ( η ) := h g ( s ) , η i K η ∈ K. (4.20)A predictable process φ will be integrable with respect to W E t if E Z T || φ ( t ) || L ( K Q ,H ) dE t ! < ∞ . (4.21)The next result provides the connection between the integral withrespect to the time-changed cylindrical Wiener process and the time-changed Q -Wiener process. Theorem 4.14. Let ˜ W E t be the time-changed cylindrical Wiener pro-cess as in (4.14) and let W E t be the time-changed Q -Wiener processas in (4.19) . Let g ∈ P † . Then, Φ gs ◦ J − satisfies condition (4.21) and Z T Φ gs ◦ J − dW E s = Z T g ( s ) d ˜ W E s . Proof. First, from the proof of Proposition 4.6, k Φ gs ◦ J − k L ( K Q , R ) = k g ( s ) k K . Thus, E Z T || Φ gt ◦ J − || L ( K Q , R ) d E t ! = E Z T || g ( t ) || K d E t ! < ∞ , g ∈ P † implies that con-dition (4.10) holds. So, Φ gt ◦ J − satisfies the condition (4.21). Alsofrom Definition 3.2, Z T Φ gs ◦ J − dW E s = ∞ X j =1 Z T Φ gs ◦ J − ( λ / j f j )d h W s , λ / j f j i K Q = ∞ X j =1 Z T h g ( s ) , f j i K d w j ( E s )= ∞ X j =1 Z T h g ( s ) , f j i K d ˜ W E s ( f j )= Z T g ( s )d ˜ W E s . (cid:3) Finally, combining the results of Theorems 4.13 and 4.14 yieldsthe desired correspondence of integrals with respect to time-changedprocesses. Corollary 4.15. For g ∈ P † and Φ gt as defined in (4.20) , Z T Z R N g ( t, x ) M E ( dt, dx ) = Z T g ( t )d ˜ W E t = Z T Φ gt ◦ J − dW E t , where M E t is the time-changed martingale measure as in (4.15) , ˜ W E t is the time-changed cylindrical process as in (4.14) , J is the operatordefined in (4.6) and W E t is the time-changed Q -Wiener process asin (4.19) . 5. Fokker-Planck-Kolmogorov equations associated withthe time-changed stochastic differential equations In this section, the Fokker-Plank-Kolmogorov (FPK) equations cor-responding to sub-diffusion processes in Hilbert space are introduced.A FPK equation is a deterministic differential equation whose so-lution is the probability density function for a stochastic process.A fundamental example of such an equation in finite dimensions isthe heat equation, whose solution is the density for Brownian mo-tion. These equations are important for several reasons. As in theheat equation, these equations are often helpful in understanding ascientific phenomenon. The connection of these equations to a sto-chastic process allows one to use information about the stochasticprocess to study these phenomena. Conversely, knowing FPK equa-tions corresponding to a particular stochastic process is helpful in thesimulation of this stochastic process, see [25, 26, 29]. More recently,the FPK equations corresponding to diffusion processes in infinitedimensional Hilbert spaces have been analyzed, see [15, 16, 17].An introduction to the sub-diffusion processes on Hilbert spacesconsidered here and their corresponding FPK equations requires apreliminary discussion of diffusion processes on Hilbert space andtheir FPK equations. Consider the following classic SDE driven bythe Q -Wiener process dY ( t ) = [ AY ( t ) + F ( t, Y ( t ))]d t + C d W t Y (0) = x ∈ H (5.1)3where A : D ( A ) ⊂ H → H is the infinitesimal generator of a C -semigroup S ( t ) = e tA , t ≥ 0, in H , and W t is a K -valued Q -Wienerprocess on a complete filtered probability space (Ω , F , {F t } t ≤ T , P )with the filtration F t satisfying the usual conditions. Suppose that F : Ω × [0 , T ] × H → H , C : Ω × K → H and C ∈ Λ ( K Q , H ). Alsoassume the initial value x is an F -measurable, H -valued randomvariable. Let Y ( t ) be a strong solution of the SDE (5.1) so that Y ( t )will satisfy the following integral equation: Y ( t ) = x + Z t [ AY ( s ) + F ( s, Y ( s ))] ds + CW t . The Kolmogorov operator L corresponding to the classic SDE (5.1)is L φ ( x ) = h x, A ∗ D x φ ( x ) i H + h F ( t, x ) , D x φ ( x ) i H + 12 tr [( CQ / )( CQ / ) ∗ D x φ ( x )] , (5.2)where x ∈ H , t ∈ [0 , T ], and D x , D x denote the first- and second-order Fr´echet derivatives in space, respectively. D ( L ) denotes thedomain of the operator L and A ∗ denotes the adjoint of the operator A . More details on the domain D ( L ) are given in [15, 16, 17].Let µ (d t, d x ) be a product measure on [0 , T ] × H of the type µ (d t, d x ) = µ t (d x )d t, where µ t ∈ P ( H ) is a Borel probability measure on the Hilbert space H for all t ∈ [0 , T ]. Let P Yt be the transition evolution operatoron B b ( H ), the space of bounded, Borel-measurable functions on H ,4 Lise Chlebak, Patricia Garmirian and Qiong Wudefined by P Yt φ ( x ) = E ( φ ( Y ( t )) | Y (0) = x ) , ≤ t ≤ T, φ ∈ B b ( H ) , (5.3)and let ( P Yt ) ∗ be its adjoint operator. Note that ( P Yt ) is a semi-group generated by the Markov process Y ( t ). According to [17], it ispossible to define the measure µ Yt induced by the solution Y ( t ) as µ Yt ( dy ) := ( P Yt ) ∗ ξ ( dy ) , (5.4)where ξ ∈ P ( H ) is the measure associated with the initial value x .The induced measure, µ Yt ( dy ), is defined as Z H φ ( y ) µ Yt ( dy ) = Z H P Yt φ ( y ) ξ ( dy ) , for all φ ∈ B b ( H ) . (5.5)Under the assumption Z [0 ,T ] × H (cid:18) |h y, A ∗ h i H | + k F ( t, y ) k H (cid:19) µ (d t, d y ) < ∞ , where h ∈ D ( A ∗ ), the induced measure µ Yt satisfies the followingFPK equationdd t Z H φ ( y ) µ Yt ( dy ) = Z H L φ ( y ) µ Yt ( dy ) , for dt -a.e., t ∈ [0 , T ] , (5.6)where the initial condition islim t → Z H φ ( y ) µ Yt (d y ) = Z H φ ( y ) ξ ( dy ) . (5.7)5Further, if the domain of the Kolmogorov operator L is comprisedof test functions, integration by parts yields ∂∂t µ Yt = L ∗ µ Yt , µ Y = ξ. (5.8)Further details are given in [17].The following lemma is needed to extend the FPK equations (5.6)and (5.7) or (5.8) associated to the solution of the classic SDE (5.1)to the case of the solution to an SDE driven by a time-changed Q -Wiener process. Lemma 5.1. Let U β ( t ) be a β -stable subordinator with the cumulativedistribution function F τ ( t ) = P ( U β ( τ ) ≤ t ) and density function f τ ( t ) . Suppose the inverse of U β ( t ) is E t with the density function f E t ( τ ) . Then, for any integrable function h ( τ ) on (0 , ∞ ) , the function q ( t ) defined by the following integral q ( t ) := Z ∞ f E t ( τ ) h ( τ ) dτ has Laplace transform L t → s { q ( t ) } = s β − [ g h ( τ )]( s β ) , where g h ( τ )( s ) = L τ → s { h ( τ ) } .Proof. Using the self-similarity property of the β -stable subordina-tor U β ( t ), the distribution function F E t ( τ ) associated with the time6 Lise Chlebak, Patricia Garmirian and Qiong Wuchange E t is F E t ( τ ) = P ( E t ≤ τ ) = P ( U β ( τ ) > t ) = 1 − P ( τ /β U β (1) ≤ t )= 1 − P (cid:18) U β (1) ≤ tτ /β (cid:19) = 1 − F (cid:18) tτ /β (cid:19) . (5.9)Differentiating both sides of (5.9), f E t ( τ ) = − ∂∂τ (cid:26) τ /β ( Jf ) (cid:18) tτ /β (cid:19)(cid:27) , where J is an integral operator defined by( Jf ) (cid:18) ta (cid:19) = Z t f (cid:18) sa (cid:19) d s for all a > . (5.10)Therefore, the Laplace transform L t → s (cid:20) a ( Jf ) (cid:18) ta (cid:19)(cid:21) ( s ) = 1 a Z ∞ ( Jf ) (cid:18) ta (cid:19) e − st dt = 1 as Z ∞ f (cid:18) ta (cid:19) e − st dt = 1 s Z ∞ f (ˆ t ) e − as ˆ t d ˆ t = 1 s ˜ f ( as ) , (5.11)where e f ( s ) is the Laplace transform of the function f ( t ). Now con-sider q ( t ) := Z ∞ f E t ( τ ) h ( τ ) dτ = − Z ∞ ∂∂τ (cid:26) τ /β ( Jf ) (cid:18) tτ /β (cid:19)(cid:27) h ( τ ) dτ. Applying the Laplace transform of the β -stable subordinator, U β ( t ),in (2.2) and using (5.11) yields L t → s { q ( t ) } = − Z ∞ ∂∂τ (cid:26) s e − τs β (cid:27) h ( τ ) dτ = s β − [ g h ( τ )]( s β ) . (cid:3) Let W E t be a time-changed Q -Wiener process on a completefiltered probability space (Ω , G , {G t } t ≤ T , P ) with the filtration G t :=7˜ F E t satisfying the usual conditions. Suppose that x is an F -measurableand H -valued random variable. Let the coefficients A, F, and C bethe same as in the classic SDE (5.1). Consider the following au-tonomous SDE on H driven by W E t and on the time interval [0 , T ]: d X ( t ) = ( AX ( t ) + F ( X ( t )))d E t + C d W E t ,X (0) = x ∈ H, t ≥ . (5.12)We derive the time-fractional FPK equation associated with thetime-changed SDE (5.12) via two different methods: first by apply-ing the time-changed Itˆo formula and second by using the duality inTheorem 3.11. The advantage of the first approach is that it directlyreveals the connection between the time-fractional FPK equations inTheorems 5.2 and 5.3, and the time-changed SDE (5.12). The advan-tage of the second approach is that it reveals the connection betweenthe time-fractional FPK equations in Theorems 5.2 and 5.3, and theclassic FPK equations in (5.6) and (5.7) or (5.8). Theorem 5.2. Suppose the coefficients A, F, and C of the time-changedSDE (5.12) satisfy the conditions in Theorem 3.12. Let X ( t ) be thesolution to (5.12) . Also suppose that X ( U β ( t )) is independent of E t .Then the probability kernel µ Xt (d x ) induced by the solution X ( t ) sat-isfies the following fractional integral equation D βt Z H φ ( x ) µ Xt (d x ) = Z H L φ ( x ) µ Xt (d x ) , (5.13)8 Lise Chlebak, Patricia Garmirian and Qiong Wu with initial condition µ X (d x ) = ξ (d x ) , where φ ∈ D ( L ) , L is theKolmogorov operator defined in (5.2) and D βt denotes the Caputofractional derivative operator as defined in Theorem 2.4.Proof. ( Method via the time-changed Itˆo formula ) Let Y ( t ) := X ( U t ).Since E t and W E t are both constant on [ U t − , U t ], the integrals Z t ( AX ( s ) + F ( X ( s )))d E s and Z t B ( X ( s ))d W E s are also constant on [ U t − , U t ]. Therefore, since X ( t ) is the solutionto the SDE (5.12), X ( t ) is also constant on [ U t − , U t ] and satisfies Y ( E t ) = X ( U E t ) = X ( t ). So, for a fixed time t ∈ [0 , T ], let µ Xt and µ Yt denote the probability measures induced on H by the stochasticprocesses X ( t ) and Y ( t ), respectively. Thus, for φ ∈ D( L ), E ( φ ( X ( t ))) = Z H φ ( x ) µ Xt (d x ) . (5.14)Since X ( t ) = Y ( E t ), taking the expectation of X ( t ) conditioned on E t , E ( φ ( X ( t ))) = E ( φ ( Y ( E t ))) = Z ∞ E ( φ ( Y τ ) | E t = τ ) f E t ( τ )d τ, where f E t ( τ ) is the density function of E t . By the assumption that Y ( t ) = X ( U t ) is independent of E t , E ( φ ( X ( t ))) = Z ∞ Z H φ ( x ) µ Yτ (d x ) f E t ( τ )d τ = Z H φ ( x ) Z ∞ µ Yτ (d x ) f E t ( τ )d τ. (5.15)9Since φ ∈ D( L ) is arbitrary, combining (5.14) and (5.15) yields µ Xt (d x ) = Z ∞ µ Yτ (d x ) f E t ( τ )d τ. (5.16)Since X ( t ) is constant on every interval [ U r − , U r ], X ( U r − ) = X ( U r ) = Y ( r ). Thus, by the time-changed Itˆo formula, φ ( X ( t )) − φ ( x ) = Z E t L φ ( X ( U r − ))d r + Z E t h φ x ( X ( U r − )) , C d W r i = Z E t L φ ( Y ( r ))d r + Z E t h φ x ( Y ( r )) , C d W r i . (5.17)Since φ ∈ D( L ), the integral M ( τ ) = Z τ h φ x ( Y ( r )) , C d W r i H is a square integrable F τ -martingale. Taking expectations on bothsides of (5.17) and conditioning on E t gives E [ φ ( X ( t )) | X (0) = x ] − φ ( x )= Z ∞ E [ Z τ L φ ( Y ( r ))d r + M ( τ ) | E t = τ, X (0) = x ] f E t ( τ )d τ = Z ∞ Z τ E [ L φ ( Y ( r )) | X (0) = x ]d rf E t ( τ )d τ = Z ∞ Z τ Z H L φ ( x ) µ Yr (d x )d rf E t ( τ )d τ = Z ∞ (cid:18) JP Y ( τ ) (cid:19) f E t ( τ )d τ, (5.18)where J is the integral operator as defined in (5.10) and P Y ( r ) isdefined by P Y ( r ) = Z H L φ ( x ) µ Yr (d x ) . E [ φ ( X ( t )) | X (0) = x ] − E [ φ ( x ) | X (0) = x ]= Z H φ ( x ) µ Xt (d x ) − Z H φ ( x ) ξ (d x ) , (5.19)which also implies the initial condition µ X (d x ) = ξ (d x ). Combin-ing (5.18), (5.19), Lemma 5.1 yields L t → s (cid:26) Z H φ ( x ) µ Xt (d x ) (cid:27) − s Z H φ ( x ) ξ (d x )= s β − [ ^ J t P Y ( τ )]( s β ) = s β − s β [ ^ P Y ( τ )]( s β ) , which, in turn, implies s β L t → s (cid:26) Z H φ ( x ) µ Xt (d x ) (cid:27) − s β − Z H φ ( x ) ξ (d x ) = s β − [ ^ P Y ( τ )]( s β ) . (5.20)Combining (5.16) and Fubini’s Theorem yields Z H L φ ( x ) µ Xt (d x ) = Z H L φ ( x ) Z ∞ µ Yτ (d x ) f E t ( τ )d τ = Z ∞ Z H L φ ( x ) µ Yτ (d x ) f E t ( τ )d τ = Z ∞ P Y ( τ ) f E t ( τ )d τ. (5.21)Taking Laplace transforms of both sides of (5.21) gives L t → s (cid:26) Z H L φ ( x ) µ Xt (d x ) (cid:27) = L t → s (cid:26) Z ∞ P Y ( τ ) f E t ( τ )d τ (cid:27) = s β − [ ^ P Y ( τ )]( s β ) . (5.22)Recall that the following equality holds: L t → s (cid:8) D βt f ( t ) (cid:9) = s β L t → s (cid:8) f ( t ) (cid:9) − s β − f (0) , (5.23)1where f ( t ) is a real-valued function on t ≥ 0. Therefore, combin-ing (5.20) and (5.22) yields s β L t → s (cid:26) Z H φ ( x ) µ Xt (d x ) (cid:27) − s β − Z H φ ( x ) ξ (d x ) = L t → s (cid:26) Z H L φ ( x ) µ Xt (d x ) (cid:27) , which, together with (5.23), implies that D βt Z H φ ( x ) µ Xt (d x ) = Z H L φ ( x ) µ Xt (d x ) . (cid:3) The next theorem gives the familiar differential form of the FPKequation for the solution to the time-changed SDE (5.12). Theorem 5.3. Suppose the conditions in Theorem 5.2 hold. If thedomain of the operator L defined in (5.2) is a set of test functions,then the probability measure µ Xt induced by the solution X ( t ) satisfiesthe following time-fractional PDE D βt µ Xt = L ∗ µ Xt , (5.24) with initial condition µ X (d x ) = ξ (d x ) , where L ∗ is the adjoint of theoperator L and D βt denotes the Caputo fractional derivative opera-tor.Proof. The proof of Theorem 5.2 gives Z H φ ( x ) µ Xt (d x ) − Z H φ ( x ) ξ (d x ) = Z ∞ Z τ Z H L φ ( x ) µ Yr (d x )d rf E t ( τ )d τ. φ ∈ D ( L ) is a test function, applying the integration by partsoperator yields Z H φ ( x ) µ Xt (d x ) − Z H φ ( x ) ξ (d x ) = Z H φ ( x ) Z ∞ Z τ L ∗ µ Yr (d x )d rf E t ( τ )d τ, which means µ Xt (d x ) − ξ (d x ) = Z ∞ Z τ L ∗ µ Yr (d x )d rf E t ( τ )d τ. (5.25)Additional information on the adjoint operator, L ∗ , is given in [37].The proof of Theorem 5.2 also gives µ Xt (d x ) = Z ∞ µ Yτ (d x ) f E t ( τ )d τ. (5.26)Taking the Laplace transforms in (5.25) and (5.26) gives s β L t → s (cid:26) µ Xt (d x ) (cid:27) − s β − ξ (d x ) = L t → s (cid:26) L ∗ µ Xt (d x ) (cid:27) , which implies D βt µ Xt = L ∗ µ Xt , with initial condition µ X (d x ) = ξ (d x ). (cid:3) We now use the second approach based on duality to derive theFPK equation for the solution to the time-changed SDEs (5.12). Theorem 5.4. Under the assumptions of Theorem 5.2, the time-fractional FPK equation associated with the time-changed SDE (5.12) follows from (5.13) . Also if the domain of the operator L definedin (5.2) is a set of test functions, then the time-fractional FPK equa-tion has the form (5.24) . Proof. ( Via duality ) From the duality theorem, Theorem 3.11, thesolution of the time-changed SDE (5.12) is X ( t ) = Y ( E t ) , (5.27)where Y ( t ) is the solution to the classic SDE (5.1). As in (5.3) and(5.4), define the transition evolution operator, P Xt , induced by thesolution, X ( t ), as follows: P Xt φ ( x ) = E ( φ ( X ( t )) | X (0) = x ) = E ( φ ( Y ( E t )) | X (0) = x )= Z ∞ P Yτ φ ( x ) f E t ( τ )d τ, ≤ t ≤ T, φ ∈ B b ( H ) . (5.28)The probability measure, µ Xt (d x ), induced by X ( t ) is µ Xt (d x ) := ( P Xt ) ∗ ξ (d x ) , (5.29)which means for all φ ∈ B b ( H ), Z H φ ( x ) µ Xt ( dx ) := Z H P Xt φ ( x ) ξ ( dx ) = Z H Z ∞ P Yτ φ ( x ) ξ (d x ) f E t ( τ ) dτ. (5.30)Therefore, the connection between the probability measures, µ Xt (d x )and µ Yt (d y ), is obtained from (5.30) by applying Fubini’s theorem Z H φ ( x ) µ Xt ( dx ) = Z ∞ Z H φ ( x ) µ Yτ (d x ) f E t ( τ ) dτ, for all φ ∈ B b ( H ) , (5.31)i.e., µ Xt ( dx ) = Z ∞ µ Yτ (d x ) f E t ( τ ) dτ, for all φ ∈ B b ( H ) . (5.32)4 Lise Chlebak, Patricia Garmirian and Qiong WuAppealing to Lemma 5.1, and taking Laplace transforms of bothsides of (5.31) leads to L t → s (cid:26) Z H φ ( x ) µ Xt ( dx ) (cid:27) = s β − L τ → s (cid:26) Z H φ ( x ) µ Yτ ( dx ) (cid:27) ( s β ) . (5.33)On the other hand, taking Laplace transforms on both sides of (5.6)leads to s L t → s (cid:26) Z H φ ( y ) µ Yt ( dy ) (cid:27) − Z H φ ( y ) ξ (d y ) = L t → s (cid:26) Z H L φ ( y ) µ Yt ( dy ) (cid:27) . (5.34)Replacing s by s β in (5.34) yields s β L t → s (cid:26) Z H φ ( y ) µ Yt ( dy ) (cid:27) ( s β ) − Z H φ ( y ) ξ (d y )= L t → s (cid:26) Z H L φ ( y ) µ Yt ( dy ) (cid:27) ( s β ) . (5.35)Thus, combining (5.33) and (5.35) gives s β L t → s (cid:26) Z H φ ( x ) µ Xt ( dx ) (cid:27) − s β − Z H φ ( y ) ξ (d y )= s β − L t → s (cid:26) Z H L φ ( y ) µ Yt ( dy ) (cid:27) ( s β ) , which implies D βt Z H φ ( x ) µ Xt ( dx ) = Z ∞ Z H L φ ( y ) µ Yτ ( dy ) f E t ( τ )d τ = Z H L φ ( y ) Z ∞ µ Yτ ( dy ) f E t ( τ )d τ = Z H L φ ( y ) µ Xt ( dy ) , (5.36)as required. Further, if the domain of the Kolmogorov operator, L ,is comprised of test functions, taking Laplace transforms on both5sides of (5.8) yields s L t → s (cid:26) µ Yt (cid:27) − µ Y = L t → s (cid:26) L ∗ µ Yt (cid:27) , (5.37)while taking Laplace transforms of both sides of (5.32) and applyingLemma 5.1 yields L t → s (cid:26) µ Xt (cid:27) = s β − L τ → s (cid:26) µ Yτ (cid:27) ( s β ) . (5.38)Finally, combining (5.37) and (5.38) yields D βt µ Xt = L ∗ µ Xt . 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Expositiones Mathematicae Stochastic integration and differential equations , Vol.21, Springer.[35] Robert Dalang, R.C., Khoshnevisan, D., Mueller C., Nualart, D, andXiao, Y. 2009. A minicourse on stochastic partial differential equations ,Vol. 1962, Springer, Berlin, Germany.[36] Dalang, R.C. 1999. Extending the martingale measure stochastic in-tegral with applications to spatially homogeneous spde’s. Electron. J.Probab K. D. Elworthy. Gaussian measures on Banach spaces and manifolds.Global analysis and its applications. 1974. Lise ChlebakTufts University, Department of Mathematics,503 Boston Avenue, Medford, MA 02155, USA.Patricia GarmirianTufts University, Department of Mathematics,503 Boston Avenue, Medford, MA 02155, USA.e-mail: