Optimal control of stochastic Volterra integral equations with completely monotone kernels and stochastic differential equations on Hilbert spaces with unbounded control and diffusion operators

Optimal con trol of sto c hastic V olterra in tegral equations with completely monotone k ernels and sto c hastic diﬀerential equations on Hilb ert spaces with unbounded con trol and diﬀusion op erators Gabriele Bolli ∗ Filipp o de F eo † F ebruary 20, 2026 Abstract The dynamic programming approac h is one of the most p o werful ones in optimal con trol. Ho wev er, when dealing with optimal control problems of sto c hastic V olterra in tegral equations (SVIEs) with completely monotone kernels, deep mathematical diﬃculties arise and it is still not understo od. These very classical problems ha ve applications in most ﬁelds and hav e now b ecome even more p opular due to their applications in mathematical ﬁnance under rough v olatility . In this article, w e consider a class of optimal con trol problems of SVIEs with completely monotone kernels. Via a recen t Marko vian lift [ 27 ], the problem can b e reform ulated as an optimal control problem of sto chastic diﬀeren tial equations (SDEs) on suitable Hilb ert spaces, which due to the roughness of the kernel, presents a generator of an analytic semigroup and un b ounded control and diﬀusion op erators. This analysis leads us to study a general class of optimal control problems of abstract SDEs on Hilb ert spaces with unbounded control and diﬀusion op erators. This class includes optimal control problems of SVIEs with completely monotone kernels, but it is also motiv ated b y other mo dels. W e analyze the regularity of the asso ciated Ornstein-Uhlen b ec k transition semigroup. W e pro v e that the semigroup exhibits a new smoothing prop ert y in con trol directions through a general observ ation operator Γ , whic h we call Γ -smo othing. This allows us to establish existence and uniqueness of mild solutions of the Hamilton-Jacobi-Bellman equation, establish a veriﬁcation theorem, and construct optimal feedback controls. W e apply these results to optimal control problems of SVIEs with completely monotone k ernels. T o the b est of our knowledge these are the ﬁrst results of this kind for this abstract class of inﬁnite dimensional problems and for the optimal control of SVIEs with completely monotone k ernels. MSC Classiﬁcation : 93E20, 60H15,45D05, 49L20, 35R15 Key words : Sto c hastic optimal con trol; sto c hastic V olterra integral equations; completely monotone kernels; path- dep enden t control; abstract SDEs; Γ -smo othing; HJB equation Con ten ts 1 In tro duction 2 2 The abstract sto c hastic optimal control problem 4 3 Γ -Smo othing for the Ornstein-Uhlen b ec k Semigroup 5 4 Minim um Energy Analysis via Virtual Control 7 5 Γ –Smo othing prop erties of the conv olution 8 6 The Abstract HJB Equation 9 7 V eriﬁcation theorem 12 8 Optimal F eedback Con trols 16 9 Optimal Control of Sto c hastic V olterra Integral Equations 17 A Completely monotone kernels 22 ∗ Dipartimento di Matematica “Guido Casteln uov o”, Sapienza Università di Roma, Roma, Italy . Email: gabriele.b olli@uniroma1.it. † Institut für Mathematik, T echnisc he Universität Berlin, Berlin, Germany , Email: defeo@math.tu-berlin.de. Filippo de F eo ackno wledges funding by the Deutsche F orsch ungsgemeinschaft (DFG, German Research F oundation) – CRC/TRR 388 "Rough Analysis, Sto chastic Dynamics and Related Fields" – Pro ject ID 516748464 and by INdAM (Instituto Nazionale di Alta Matematica F. Severi) - GNAMP A (Gruppo Nazionale p er l’Analisi Matematica, la Probabilità e le loro Applicazioni) 1 1 In tro duction 1.1 Mo del problems Optimal control of SVIEs with completely monotone k ernels. Consider a controlled sto c hastic V olterra in tegral equation (SVIE) of the form y ( s ) = z ( s ) + Z s 0 K ( s − r )[ cy ( r ) + bu ( r )] d r + Z s 0 K ( s − r ) g d W ( r ) , (1.1) where K : (0 , ∞ ) → R is a completely monotone kernel, z is an initial curve, and u ( · ) is a control process. The goal here is to minimize, ov er all admissible controls u ( · ) , a cost functional of the form ˜ J ( t, z ; u ) = E " Z T t  1 ( u ( s )) d s + ˜ φ ( y ( T )) # . (1.2) The state equation ( 1.1 ) is non-Marko vian and cannot b e treated using the standard dynamic programming approac h in ﬁnite dimensions. Ho w ever, Marko vianity can b e regained by lifting the problem onto suitable inﬁnite dimensional spaces and many possible c hoices hav e b een inv estigated in the literature. Here we will use the Marko vian lift recently prop osed in [ 27 ] (see also [ 7 , 52 ]). With this pro cedure, the state ( 1.1 ) is rewritten as an abstract sto c hastic evolution equation of the form ( 1.3 ) for the state v ariable X ( s ) ov er the Hilb ert space H := H η : = L 2 ( R + , (1 + x ) η ¯ µ (d x )) for suitable parameter η ∈ R and where ¯ µ : = δ 0 + µ with µ is characterized by the Laplace-Bernstein representation of K . Here, A, B , G are unbounded op erators, due to the roughness of the kernel. While A is an unbounded op erator generating an analytic semigroup on H , the op erators B and G b ecome b ounded ov er the weak er spaces H := H η ′ ←  H η , when η ′ < η is small enough. Moreo ver, A extends to an analytic semigroup onto the w eaker space H . The state y ( s ) is recov ered through a suitable reconstruction op erator Γ ∈ L ( H , R n ) via y ( s ) = Γ X ( s ) . On the other hand, the lift φ ( x ) = ˜ φ (Γ x ) of the ﬁnal cost is only contin uous in H η . In Section 9 we will detail this lifting pro cedure. Optimal control of SDEs on Hilbert spaces with un b ounded control and diﬀusion op erators. The previous analysis leads to the study of the following class of optimal con trol problems on Hilb ert spaces. Let H , K , Ξ b e arbitrary separable Hilb ert spaces, resp ectiv ely represen ting, the state, noise the and control spaces, and consider the follo wing controlled sto c hastic diﬀeren tial equation (SDEs) on H of the form d X ( s ) = ( AX ( s ) + B u ( s )) d s + G d W ( s ) , s ∈ [ t, T ] , X ( t ) = x ∈ H , (1.3) where A : D ( A ) ⊆ H → H is the inﬁnitesimal generator of a strongly con tinuous semigroup S ( t ) , B : K → H, G : Ξ → H are linear unbounded op erators, W is a cylindrical Wiener pro cess on Ξ , and u ( · ) is a control pro cess with v alues in K . The goal is to minimize, ov er all admissible controls u ( · ) , a cost functional of the form J ( t, x, u ) = E " Z T t  1 ( u ( s )) d s + φ ( X ( T )) # , (1.4) where φ = ¯ φ ◦ Γ for a certain observ ation op erator Γ ∈ L ( H , Y ) , b eing Y another separable Hilb ert space. T o handle un b ounded control and diﬀusion op erators, we introduce a larger separable Hilb ert space H with contin uous and dense em b edding H  → H and w e assume that B , G are b ounded op erators on H and that S ( t ) can b e extended to a strongly con tinuous semigroup S ( t ) on H and satisﬁes the following analytic smo othing prop ert y: for an y t > 0 , S ( t )( H ) ⊆ D ( A ) ⊂ H . On the other hand, φ is contin uous only in H . 1.2 State of the art Optimal control of SDEs in inﬁnite dimensions. The optimal con trol of SDEs on Hilb ert spaces is a very active area of research, where deep mathematical diﬃculties usually arise, due to the lac k of lo cal compactness of the state spaces and the presence of un b ounded operators. These c hallenging mo dels are motiv ated by the optimal control of some of the most prominen t families of sto c hastic partial diﬀerential equations (SPDEs) [ 19 , 28 , 47 , 48 ], SDEs with dela ys in the state and/or in the control [ 15 , 14 , 16 , 18 , 31 , 32 , 34 , 38 , 42 ], SVIEs [ 39 , 45 ], partially observed sto c hastic systems [ 36 ], particle systems in Hilb ert spaces [ 17 ], and many others. A classical monograph on the sub ject is [ 26 ]. While techniques to treat linear unbounded terms of the form AX t in the drift are now standard, e.g., via the theory of C 0 -semigroups, the presence of un b ounded op erators in the con trol v ariable and/or in the diﬀusion is still not 2 w ell understo od with few results a v ailable in the literature only in sp eciﬁc settings, although these mo dels arise in man y real-world applications, such as path-dependent problems, b oundary con trol, or partial observ ation. W e refer to [ 8 , 26 , 33 , 35 , 37 ] for some key con tributions that, how ever, do not cov er our setting. Among these, the closest one to the setting of our control problem ( 1.3 ) - ( 1.4 ) is [ 33 ]. In this pap er, motiv ated b y applications in problems with delays in the control and boundary con trol, they consider a particular case of the abstract state equation ( 1.3 ) , where the control op erator B is an unbounded linear op erator like in our setting, while G is a b ounded op erator from H to H . As they cannot exp ect smo othing prop erties for the underlying Marko v transition semigroup 1 , they introduce a sp eciﬁc concept of partial deriv ative, designed for this situation, and develop the so-called partial smo othing metho d to pro ve that the asso ciated HJB equation has a solution with enough regularity to ﬁnd optimal contr ols in feedbac k form. Optimal control of SVIEs. Optimal control of SVIEs is a rapidly expanding area of research, with applications spanning a wide range of disciplines. In terest in the topic has b een further stim ulated by the adven t of rough volatilit y mo dels in mathematical ﬁnance [ 6 , 24 , 23 , 30 ]. SVIEs are not Marko v or not even a semi-martingale in general, so the ﬂow prop erty do es not apply . Similarly to other path-dep enden t mo dels reviewed ab o ve, this raises deep mathematical c hallenges and Mark ovianit y can typically b e regained only by lifting the state equation onto a suitable inﬁnite dimensional space, typically Banach or Hilb ert spaces, i.e. the so called Marko vian lifts. W e refer to [ 1 , 2 , 3 , 4 , 9 , 10 , 20 , 21 , 40 , 41 , 39 , 45 , 51 ] for several key works on optimal control of SVIEs. Among these contributions, how ever, the case of completely monotone kernels, fundamen tal in applications in mathematical ﬁnance under rough v olatility , is only co v ered via Riccati equations in [ 1 , 2 , 40 , 41 ] in the linear quadratic case, via Peng Maxim um Principle in [ 39 ], and via forward-bac kward systems in [ 9 ] in the case where the noise term is driven b y a pure jump Lévy noise and the control acts on the in tensity of the jumps (no Wiener pro cess is considered) and in [ 10 ] for particular sto chasic V olterra equations of the form d dt R t −∞ a ( t − s ) u ( s )d s = Au ( t ) + g ( t, u ( t ))[ r ( t, u ( t ) , γ ( t )) + ˙ W ( t )] . Finally , for pioneering works on Kolmogoro v PDEs (no control problems are considered) related to SVIEs, we refer to [ 5 , 50 ] for the case of regular kernels and, for rough kernels, to [ 11 , 43 , 49 ] for gaussian pro cesses and [ 29 ] for the general case. 1.3 Our results F rom the literature review ab o v e, it is eviden t that • the dynamic programming approac h for the optimal control problem of SVIEs ( 1.1 ) - ( 1.2 ) in the case of completely monotone kernels is not understoo d. In particular, the corresp onding Hamilton–Jacobi–Bellm an (HJB) equation has not b een solved, and there are currently no av ailable results establishing veriﬁcation theorems or providing a construction of optimal feedback controls in this setting. • The optimal con trol problem of SDEs on Hilb ert spaces ( 1.3 ) - ( 1.4 ) with unbounded con trol and diﬀusion op erators has not b een studied in the literature, although these are an important class of problems motiv ated not only by con trol problems of SVIEs but also, e.g., by SPDEs with b oundary noise and b oundary con trol [ 26 , 37 ]. The goal of this pap er is then to ﬁll these gaps in the literature, b y solving the HJB equations corresp onding to b oth problems, establishing veriﬁcation theorems, and constructing optimal feedbac k controls. Since optimal control problems of SVIEs of the form ( 1.1 ) - ( 1.2 ) can b e rewritten as optimal control problems of the form ( 1.3 ) - ( 1.4 ) , w e start attac king the latter ones ﬁrst. Optimal con trol of SDEs on Hilb ert spaces with unbounded control and diﬀusion op erators. T o attac k optimal control problems of the form ( 1.3 ) - ( 1.4 ) , in tro duced in Section 2 , w e generalize the approach of [ 33 ] to this new setup where not only B but also G is also unbounded on H . T o solve optimal control problems a standard strategy is the follo wing: pro ve existence and uniqueness of mild solutions of the HJB equation directly and use this solution to prov e v eriﬁcation theorems and construct optimal feedbac k control. T o follow this strategy , how ever, we need to sho w some smo othing prop erties of the asso ciated Ornstein-Uhlen b ec k (OU) semigroup. In the literature, tw o smo othing prop erties hav e b een studied, i.e. the standard (full) smoothing condition [ 12 , 26 ] or a weak er v ersion, the so called partial smo othing in control directions [ 33 ]. Ho wev er, in optimal control problems of SVIEs ( 1.1 ) - ( 1.2 ) , we cannot exp ect either one. Nevertheless, we are able to show that the OU semigroup asso ciated to SVIEs ( 1.1 ) satisﬁes a new smo othing prop ert y in control directions through the reconstruction op erator Γ ∈ L ( H , R ) . Therefore, in Section 3 , in the general setting ( 1.3 ) - ( 1.4 ) , we derive a more general notion of smo othing, whic h we call Γ -smo othing in control directions, or Γ -smo othing for brevit y , for a 1 see [ 12 , 26 ] for these kind of prop erties 3 general observ ation op erator Γ ∈ L ( H , Y ) , where the observ ation space Y is another separable Hilb ert space. In this setting, we prov e existence of B -directional deriv atives for functions deﬁned on H , see Theorem 3.4 . In Section 4 , inspired by the classical w ork [ 53 ], we establish a rigorous control-theoretic interpretation of the Γ -smo othing through a minimum energy analysis via virtual control, see Theorem 4.1 . This analysis is also new in the case of partial smo othing [ 33 ]. In order to solv e the HJB equation on the extended space H , it is necessary to preserve regularity through the time con volution R t 0 P t − s [ · ]d s . Therefore, in Section 5 , w e prov e Γ –smo othing prop erties of the conv olution, extending the partial smo othing ones in [ 31 , Section 4] and [ 33 , Section5]. In Section 6 , we pro ve existence, uniqueness, and suitable regularity prop erties of mild solutions of the HJB equation, see Theorem 6.2 . Our ﬁnal goal is to completely solv e the con trol problem b y establishing veriﬁcation theorems and constructing optimal feedback controls. Therefore, in Section 7 , we prov e that our mild solution can b e approximated via K -strong solutions, i.e. classical solutions of p erturb ed HJB equations, also con verging in a suitable sense to the mild solution. This allows us to pro ve a v eriﬁcation theorem through a suitable approximation argumen t and to characterize the v alue function of the problem as the mild solution of the HJB equation, see Theorem 7.8 . In Section 8 , we construct feedback con trols by completely solving the optimal con trol problem ( 1.3 )-( 1.4 ). T o the b est of our knowledge these are the ﬁrst results of this kind for the class of control problems on Hilbert spaces with unbounded control and diﬀusion op erators ( 1.3 )-( 1.4 ). Optimal con trol of SVIEs with completely monotone kernels. In Section 9 we tackle to optimal control problems of SVIEs ( 1.1 ) - ( 1.2 ) . Using the Mark ovian lift recently prop osed in [ 27 ] (see also [ 7 , 52 ]), we lift the problem to Hilb ert spaces. W e verify that the assumptions of our theory , so that we can apply the theory developed to completely solve the control problem, i.e. solving the corresp onding HJB equation, establishing v eriﬁcation theorems, and constructing optimal feedbac k con trols. T o the b est of our kno wledge these are the ﬁrst results of this kind for the optimal con trol problems of SVIEs with completely monotone kernels ( 1.1 ) - ( 1.2 ) . Finally , for the ﬁnancially oriented reader, we recall examples of completely monotone kernels, particularly used in mathematical ﬁnance under rough v olatility , as well as other applications. 2 The abstract sto chastic optimal con trol problem Notations. Let H , K , Z b e real separable Hilb ert spaces and C ∈ L ( K, H ) . A function f : H → Z is C -dir e ctional ly diﬀer entiable at x ∈ H in the direction k ∈ K if the limit ∇ C f ( x ; k ) : = lim s → 0 f ( x + sC k ) − f ( x ) s exists in Z . The function is C -Gâte aux diﬀer entiable if k 7→ ∇ C f ( x ; k ) deﬁnes an e lemen t of L ( K, Z ) , and C -F r é chet diﬀer entiable if the conv ergence is uniform for k ∈ B K (0 , 1) . F or Z = R and α ∈ [0 , 1) , we deﬁne the follo wing Banach spaces: C 1 ,C b ( H ) is the space of f ∈ C b ( H ) with a con tinuous and bounded C -F réchet deriv ative ∇ C f : H → K . The space C 0 , 1 ,C α ([0 , T ] × H ) contains functions f ∈ C b ([0 , T ] × H ) such that f ( t, · ) ∈ C 1 ,C b ( H ) for t ∈ (0 , T ] , with ( t, x ) 7→ t α ∇ C f ( t, x ) contin uous and bounded from (0 , T ] × H to K , and norm ∥ f ∥ C 0 , 1 ,C α : = sup [0 ,T ] × H | f | + sup (0 ,T ] × H t α ∥∇ C f ∥ K . Finally , C 0 , 2 ,C α ([0 , T ] × H ) comprises f ∈ C 0 , 1 b ([0 , T ] × H ) such that ∇ C ∇ f ∈ L ( H , K ) exists for t ∈ (0 , T ] with ( t, x ) 7→ t α ∇ C ∇ f ( t, x ) con tinuous and b ounded from (0 , T ] × H to L ( H , K ) , with norm ∥ f ∥ C 0 , 2 ,C α : = sup [0 ,T ] × H | f | + sup [0 ,T ] × H ∥∇ f ∥ H + sup (0 ,T ] × H t α ∥∇ C ∇ f ∥ L ( H,K ) . Abstract framew ork. W e now delineate the functional analytic and probabilistic foundations necessary for the analysis of the inﬁnite-dimensional control problem. The mathematical setting is grounded on real, separable Hilbert spaces ( H , ⟨· , ·⟩ H ) , ( ˜ U , ⟨· , ·⟩ ˜ U ) , and (Ξ , ⟨· , ·⟩ Ξ ) , which resp ectiv ely denote the state space, the control space, and the noise space. T o rigorously handle singular control op erators that do not take v alues directly in the state space H w e in tro duce a larger separable Hilb ert space H with con tinuous and dense em b edding H  → H . (2.1) W e denote b y ∥·∥ H and ∥·∥ H the norms in H and H , resp ectiv ely . State equation. The sto c hastic analysis is conducted on a complete probabilit y space (Ω , F , P ) , endow ed with a ﬁltration F = ( F t ) t ≥ 0 satisfying the standard assumptions of right-con tinuit y and completeness. Within this en vironment, w e consider the state dynamics go verned by the linear sto c hastic ev olution equation on the extended space H : d X ( s ) = ( AX ( s ) + B u ( s )) d s + G d W ( s ) , s ∈ [ t, T ] , X ( t ) = x ∈ H . (2.2) Since B maps in to the larger space H deﬁned in ( 2.1 ) , this equation is understoo d strictly in the mild sense deﬁned b elo w in Deﬁnition 2.2 , consistent with the approac h for unbounded control operators [ 8 , 33 , 32 ]. 4 Hyp othesis 2.1 (Structural Assumptions on System Data) . The op erators app earing in the state equation ( 2.2 ) satisfy the following structural conditions, consistent with the framew ork of C 0 -semigroups [ 44 , 25 ]: (i) The op erator A : D ( A ) ⊆ H → H is the inﬁnitesimal generator of a strongly contin uous semigroup S ( t ) : = e tA on H . W e assume that S ( t ) can b e extended to a strongly contin uous semigroup S ( t ) : = e tA on the larger space H with v alues on H . Both semigroups satisfy the standard exp onen tial gro wth b ound: ∥ S ( t ) ∥ L ( H ) ≤ M e ω t ,   S ( t )   L ( H ) ≤ M e ω t , ∀ t ≥ 0 . (ii) The control op erator B ∈ L ( K, H ) . That is, the con trol acts eﬀectively in the larger space H but implies a singularit y in the basic state space H . Ho wev er, thanks to the smo othing condition in (i) , for an y t > 0 , the op erator S ( t ) B maps K into D ( A ) ⊂ H . (iii) The diﬀusion op erator G b elongs to L (Ξ , H ) . That is, the noise acts in the extended space H , allowing for singular diﬀusion kernels. How ever, thanks to the smo othing condition in (i) , for any t > 0 , the op erator S ( t ) G maps Ξ into D ( A ) ⊂ H . (iv) The pro cess W is an ( F t ) -adapted cylindrical Wiener pro cess with v alues in the Hilb ert space Ξ . (v) T o ensure the well-posedness of the sto chastic conv olution in H , the linear co v ariance op erator Q t ∈ L ( H ) , deﬁned formally by the integral in H : Q t : = Z t 0 S ( s ) GG ∗ S ( s ) ∗ d s, (2.3) is assumed to b e a trace class op erator on H for all t > 0 , i.e., T r( Q t ) < ∞ . (vi) The set of control v alues U is a b ounded and closed subset of K . The space of admissible controls U consists of all ( F s ) -progressiv ely measurable pro cesses u : [ t, ∞ ) × Ω → U . The concept of solution is the mild form on the extended space. Deﬁnition 2.2 (Mild Solution) . Given an initial time t ∈ [0 , T ] and an initial state x ∈ H , an H -v alued (for s > t ) adapted pro cess X ( · ; t, x, u ) is deﬁned as a mild solution to the sto c hastic diﬀerential equation ( 2.2 ) if it satisﬁes the follo wing integral equation in the space H , P -almost surely , for all s ∈ [ t, T ] : X ( s ) = S ( s − t ) x + Z s t S ( s − r ) B u ( r ) d r + Z s t S ( s − r ) G d W ( r ) . (2.4) Note that for s > t , eac h term on the right hand side b elongs to H due to the analytic smo othing assumptions. Clearly , uniqueness of mild solutions is immediate. Cost functional. Given t ∈ [0 , T ] , x ∈ H the goal is to minimize, ov er all admissible con trols u ( · ) ∈ U , the following cost functional J ( t, x, u ) = E " Z T t  1 ( u ( s )) d s + φ ( X ( T )) # , (2.5) where l 1 : U → R , φ : H → R are the running cost and terminal cost, resp ectiv ely . 3 Γ -Smo othing for the Ornstein-Uhlen b ec k Semigroup In applications with singular control op erators, the standard global smoothing on H [ 12 ] typically fails. Moreov er, in control problems of SVIEs (see Section 9 ), w e cannot use the so called partial smo othing approach [ 31 , 32 , 33 ]. Ho wev er, motiv ated by these applications, w e can obtain diﬀerentiabilit y in the directions aﬀected by the control op erator B ∈ L ( K, H ) through a suitable observ ation op erator Γ , which we call Γ -smo othing in control directions or Γ -Smo othing for brevity . W e analyze the regularity of the transition semigroup asso ciated with the uncontrolled pro cess. Let Z ( · ) be the solution to the linear sto c hastic equation evolv ed starting from H : d Z ( t ) = AZ ( t ) d t + G d W ( t ) , t ≥ 0 , Z (0) = x ∈ H . (3.1) The transition semigroup P t acting on b ounded Borel functions φ ∈ B b ( H ) is deﬁned by: P t [ φ ]( x ) := E [ φ ( Z ( t ; x ))] = Z H φ  S ( t ) x + y  N (0 , Q t )(d y ) , x ∈ H , t ≥ 0 . (3.2) Here, N (0 , Q t ) denotes the Gaussian measure on H with cov ariance op erator Q t . 5 Deﬁnition 3.1 ( Γ -Cylindrical F unctions) . Let H b e the regular state space and let Y b e another separable Hilb ert space. Consider a b ounded linear op erator Γ ∈ L ( H , Y ) . The set B Γ b ( H ) is deﬁned as the class of functions φ : H → R satisfying the following factorization prop erty: there exists a b ounded Borel measurable function ¯ φ ∈ B b ( Y ) such that φ ( x ) = ¯ φ (Γ x ) for all x ∈ H. (3.3) R emark 3.2 (W ell-p osedness of the ev aluation on H ) . The ev aluation P t [ φ ]( x ) for x ∈ H and t > 0 is well-deﬁned. Under Hyp othesis 2.1 , for t > 0 , S ( t ) x ∈ H , thus Γ can b e applied to the deterministic part. The noise term also liv es in H almost surely . Hyp othesis 3.3 ( Γ -Smo othing Condition) . W e assume that the noise regularizes the system in the sp eciﬁc directions driv en by the con trol op erator B ∈ L ( K, H ) through the observ ation op erator Γ . (i) F or every t > 0 , we assume the following inclusion of ranges in the observ ation space Y : Im  Γ S ( t ) B  ⊆ Im  ( Q Γ t ) 1 / 2  , wher e Q Γ t := Γ Q t Γ ∗ = Z t 0 Γ S ( s ) GG ∗ S ( s ) ∗ Γ ∗ d s. (3.4) By Hyp othesis 2.1 , the op erator S ( t ) B maps K in to H . Since Γ ∈ L ( H , Y ) , the composition Γ S ( t ) B is a w ell-deﬁned b ounded linear op erator from K to Y . Consequently , the regularization op erator Λ Γ ,B ( t ) : K → Y deﬁned by Λ Γ ,B ( t ) k := (Γ Q t Γ ∗ ) − 1 / 2 Γ S ( t ) B k (3.5) is well-deﬁned and b ounded. (ii) There exist κ 0 > 0 and γ ∈ (0 , 1) such that: ∥ Λ Γ ,B ( t ) ∥ L ( K, Y ) ≤ κ 0  t − γ ∨ 1  , ∀ t > 0 . (3.6) W e can no w state the main result on the existence of B -directional deriv atives for functions deﬁned on H . Theorem 3.4 ( B -Diﬀeren tiability via Γ -regularization) . L et φ ∈ B Γ b ( H ) with Γ ∈ L ( H , Y ) and assume Hyp othesis 3.3 holds. Then, for any t > 0 , P t [ φ ] is diﬀer entiable in the gener alize d dir e ctions of Im ( B ) . Sp e ciﬁc al ly, for any x ∈ H , ther e exists a ve ctor ∇ B P t [ φ ]( x ) ∈ K such that the derivative in dir e ction k satisﬁes the Bismut-Elworthy-Li typ e formula: ⟨∇ B P t [ φ ]( x ) , k ⟩ K = Z H φ  S ( t ) x + y  ⟨ Λ Γ ,B ( t ) k , (Γ Q t Γ ∗ ) − 1 / 2 Γ y ⟩ Y N (0 , Q t )(d y ) . (3.7) Her e, the term Γ y is understo o d in the sense of the pr oje cte d Gaussian me asur e on Y . Mor e over, we have the smo othing estimate: ∥∇ B P t [ φ ]( x ) ∥ K ≤ ∥ φ ∥ ∞ ∥ Λ Γ ,B ( t ) ∥ L ( K, Y ) ≤ κ 0 t − γ ∥ φ ∥ ∞ . (3.8) Pr o of. The pro of pro ceeds by reduction to the pro jected Gaussian measure on Y . Let φ ∈ B Γ b ( H ) , so φ ( z ) = ¯ φ (Γ z ) for z ∈ H . F or t > 0 and x ∈ H , the term S ( t ) x b elongs to H . Let ν t := N (0 , Q Γ t ) b e the Gaussian measure on Y deﬁned b y the pro jection of N (0 , Q t ) via Γ . The transition semigroup can b e written as P t [ φ ]( x ) = R Y ¯ φ (Γ S ( t ) x + ξ ) ν t (d ξ ) . Fix x ∈ H , a direction k ∈ K and α > 0 . W e consider the p erturbation of the initial condition in the direction B . The p erturbed tra jectory at time t is eﬀectively S ( t ) x + αS ( t ) B k (view ed in H ). Since S ( t ) B maps K in to H (Hyp othesis 2.1 ), the vector S ( t ) B k lies in the domain of Γ . Thus, by linearity , Γ  S ( t ) x + αS ( t ) B k  = Γ S ( t ) x + α Γ S ( t ) B k . Let h := Γ S ( t ) B k ∈ Y . Note that h is w ell-deﬁned precisely b ecause of the smo othing action of S ( t ) on the image of B . The diﬀerence quotient b ecomes: ∆ α [ φ ]( x ) := 1 α [ P t [ φ ]( x + αB k ) − P t [ φ ]( x )] = 1 α  Z Y ¯ φ (Γ S ( t ) x + αh + ξ ) ν t (d ξ ) − Z Y ¯ φ (Γ S ( t ) x + ξ ) ν t (d ξ )  . (3.9) W e now in vok e the Cameron-Martin Theorem on the space Y (see [ 13 , Prop osition 2.26]). By Hyp othesis 3.3 (i), the shift h satisﬁes h ∈ Im (Γ S ( t ) B ) ⊆ Im (( Q Γ t ) 1 / 2 ) . This inclusion guaran tees that the shifted measure ν t ( · − αh ) is absolutely con- tin uous with resp ect to ν t . The Radon-Nikodym deriv ative is ρ α ( ξ ) = exp  α ⟨ ( Q Γ t ) − 1 / 2 h, ( Q Γ t ) − 1 / 2 ξ ⟩ Y − α 2 2 ∥ ( Q Γ t ) − 1 / 2 h ∥ 2 Y  . Iden tifying ( Q Γ t ) − 1 / 2 h = Λ Γ ,B ( t ) k , and pro ceeding similarly to the standard case (taking the limit α → 0 ), we obtain: ⟨∇ B P t [ φ ]( x ) , k ⟩ K = Z Y ¯ φ (Γ S ( t ) x + ξ ) ⟨ Λ Γ ,B ( t ) k , ( Q Γ t ) − 1 / 2 ξ ⟩ Y ν t (d ξ ) . (3.10) Rewriting the integral ov er H (where ξ corresp onds to the pro jection Γ y of the noise y ∼ N (0 , Q t ) ), we recov er formula ( 3.7 ). The estimate ( 3.8 ) follo ws immediately from the Hölder inequalit y on Y : |⟨∇ B P t [ φ ]( x ) , k ⟩ K | ≤ ∥ ¯ φ ∥ ∞  E ν t     ⟨ Λ Γ ,B ( t ) k , ( Q Γ t ) − 1 / 2 ξ ⟩ Y    2  1 / 2 = ∥ φ ∥ ∞ ∥ Λ Γ ,B ( t ) k ∥ Y . (3.11) 6 4 Minim um Energy Analysis via Virtual Control In this section, we establish a rigorous control-theoretic interpretation of the singularit y op erator Λ Γ ,B ( t ) in tro duced in Hyp othesis 3.3 . W e demonstrate that its op erator norm quantiﬁes the minim um energy required to replicate the state displacemen t induced b y the control operator B using the system’s inherent noise channels. This analysis bridges the analytic prop erties of the singular k ernel with the controllabilit y of the pro jected system. W e generalize the approach of [ 53 , Section 15.1] to the case of partial controllabilit y where observ ations are restricted to the regular space H . Consider the linear stochastic system ( 2.2 ) with zero initial condition and zero drift. The diﬀusion term generates a reachable set in the extended space H whic h is instantaneously smo othed into H . T o analyze its geometric relation with the control operator B , w e formulate an auxiliary deterministic control problem. In this setting, the noise op erator G acts as the con trol op erator. Let v ∈ L 2 (0 , t ; Ξ) denote a virtual c ont r ol acting through G . The state tra jectory X v ( · ) (view ed in H ) is gov erned by the evolution equation: d X v ( s ) = AX v ( s ) d s + Gv ( s ) d s, s ∈ [0 , t ] , X v (0) = 0 . (4.1) Its mild solution is given b y X v ( s ) = R s 0 S ( s − r ) Gv ( r ) d r , s ∈ [0 , t ] . By the analytic smo othing condition (Hyp othesis 2.1 ), for an y r < s , the op erator S ( s − r ) G maps Ξ in to the regular space H . Consequently , the integral tak es v alues in H . This ensures that the application of the observ ation op erator Γ ∈ L ( H , Y ) is well-deﬁned. W e deﬁne the input-to-pr oje cte d-state operator L Γ t : L 2 (0 , t ; Ξ) → Y as the map taking a con trol history to the ﬁnal pro jected state: L Γ t v : = Γ X v ( t ) = Z t 0 Γ S ( t − r ) Gv ( r ) d r. (4.2) By duality , the adjoint op erator ( L Γ t ) ∗ : Y → L 2 (0 , t ; Ξ) is uniquely deﬁned. F or any z ∈ Y and v ∈ L 2 (0 , t ; Ξ) , the iden tity ⟨L Γ t v , z ⟩ Y = ⟨ v , ( L Γ t ) ∗ z ⟩ L 2 yields  ( L Γ t ) ∗ z  ( r ) = G ∗ S ( t − r ) ∗ Γ ∗ z , r ∈ [0 , t ] . A crucial observ ation is that the pro jected controllabilit y Gramian Q Γ t , deﬁned in ( 3.4 ), admits the factorization: Q Γ t = L Γ t ( L Γ t ) ∗ . (4.3) Fix a control direction k ∈ K and consider the displacement caused b y the ph ysical control term at time t . W e deﬁne the target vector in the observ ation space Y as: η k : = − Γ S ( t ) B k . (4.4) Note that η k is well-deﬁned b ecause S ( t ) B maps K in to H (Hyp othesis 2.1 ). The condition that the virtual system can n ullify the total pro jected state is equiv alent to the existence of a solution v to the linear op erator equation L Γ t v = η k . Theorem 4.1 (Minimum Energy Characterization) . L et t > 0 and k ∈ K . (i) The tar get displac ement η k lies in the r e achable set of the virtual system, i.e., η k ∈ Im ( L Γ t ) , if and only if the fol lowing r ange inclusion holds: η k ∈ Im  ( Q Γ t ) 1 / 2  . (4.5) Ther efor e, η k ∈ Im( L Γ t ) , for al l k ∈ K if and only if ( 3.4 ) holds. (ii) Assume that c ondition ( 4.5 ) holds. L et ˆ v b e the unique minimum-ener gy c ontr ol solving L Γ t v = η k , deﬁne d by ˆ v : = arg min  ∥ v ∥ L 2 (0 ,t ;Ξ) : L Γ t v = η k  . Then, the fol lowing isometric r elation holds: ∥ ˆ v ∥ L 2 (0 ,t ;Ξ) =    ( Q Γ t ) − 1 / 2 η k    Y = ∥ Λ Γ ,B ( t ) k ∥ Y . (4.6) Conse quently, ∥ Λ Γ ,B ( t ) ∥ L ( K, Y ) r epr esents the worst-c ase ener gy r e quir e d to ste er a unit ve ctor fr om K using the diﬀusion channel. (iii) If the str onger c ondition η k ∈ Im( Q Γ t ) holds, the optimal virtual c ontr ol is given explicitly by: ˆ v ( s ) = − G ∗ S ( t − s ) ∗ Γ ∗ ( Q Γ t ) † Γ S ( t ) B k , s ∈ [0 , t ] , (4.7) wher e ( Q Γ t ) † denotes the Mo or e-Penr ose pseudoinverse. Pr o of. W e employ the Douglas Lemma [ 22 ] (see also [ 13 , App endix B]), which characterizes range inclusions and factorizations of op erators on Hilbert spaces. Pr o of of (i) . Iden tifying the op erators via the factorization ( 4.3 ) , the Douglas Lemma implies the identit y of ranges: Im ( L Γ t ) = Im  ( L Γ t ( L Γ t ) ∗ ) 1 / 2  = Im  ( Q Γ t ) 1 / 2  . Th us, solv abilit y is equiv alent to η k b elonging to the range of the square ro ot of the Gramian. 7 Pr o of of (ii) . The set of admissible controls is the aﬃne subspace ˆ v + k er ( L Γ t ) . The minimum norm solution ˆ v is the unique elemen t in k er ( L Γ t ) ⊥ = Im(( L Γ t ) ∗ ) . The Douglas Lemma guarantees that the map L Γ t restricts to an isometry b et w een k er ( L Γ t ) ⊥ and the range space Im (( Q Γ t ) 1 / 2 ) equipp ed with the graph norm induced b y the inv erse op erator, i.e. ∥ ˆ v ∥ L 2 (0 ,t ;Ξ) = inf {∥ v ∥ : L Γ t v = η k } =    ( Q Γ t ) − 1 / 2 η k    Y =    ( Q Γ t ) − 1 / 2 ( − Γ S ( t ) B k )    Y = ∥ Λ Γ ,B ( t ) k ∥ Y . where to obtain the last tw o equalities we hav e used the deﬁnition of the regularization op erator in Hyp othesis 3.3 , Λ Γ ,B ( t ) k = ( Q Γ t ) − 1 / 2 Γ S ( t ) B k . and the deﬁnition of η k from ( 4.4 ). Pr o of of (iii) . Assume η k ∈ Im ( Q Γ t ) . Then, there exists a v ector z ∈ Y (formally z = ( Q Γ t ) † η k ) suc h that η k = L Γ t ( L Γ t ) ∗ z . The minim um norm solution is reco vered by lifting the m ultiplier z via the adjoin t op erator: ˆ v = ( L Γ t ) ∗ z . Using the explicit adjoint form ab ov e and substituting z = − ( Q Γ t ) † Γ S ( t ) B k , we obtain ( 4.7 ). 5 Γ –Smo othing prop erties of the con v olution In order to solve the HJB equation in Section 6 on the extended space H , it is necessary to preserve regularity through the time conv olution R t 0 P t − s [ · ]d s . Therefore, we pro ve Γ –smo othing prop erties of the con volution. Deﬁnition 5.1 (Smoothing Spaces on H ) . Let T > 0 and α ∈ (0 , 1) . W e deﬁne the space Σ 1 T ,α as the set of functions g ∈ C b ([0 , T ] × H ) p ossessing the follo wing structure: 1. There exists a function f ∈ C 0 , 1 α ([0 , T ] × Y ) ; 2. F or any t ∈ (0 , T ] and x ∈ H , the ev aluation is given b y g ( t, x ) = f ( t, Γ S ( t ) x ) . R emark 5.2 . F or an y g ∈ Σ 1 T , 1 / 2 , the chain rule yields ∇ B g ( t, x ) = (Γ S ( t ) B ) ∗ ∇ Y f ( t, Γ S ( t ) x ) , t ∈ (0 , T ] , x ∈ H . (5.1) This represen tation shows that ∇ B g inherits the factorization of g . Speciﬁcally , there exists a b ounded contin uous map ¯ f : (0 , T ] × Y → K such that t 1 / 2 ∇ B g ( t, x ) = ¯ f  t, Γ S ( t ) x  . (5.2) Lemma 5.3 (Preserv ation of Regularity on H ) . Assume Hyp othesis 3.3 with γ = 1 / 2 . L et ψ : K → R b e a Lipschitz c ontinuous function. (i) If g ∈ Σ 1 T , 1 / 2 , then the c onvolution ˆ g ( t, x ) := R t 0 P t − s [ ψ ( ∇ B g ( s, · ))]( x ) d s is wel l-deﬁne d for x ∈ H and b elongs to Σ 1 T , 1 / 2 . (ii) Ther e exists a c onstant C T > 0 such that: sup x ∈ H   ∇ B ( ˆ g ( t, · ))( x )   K ≤ C T  t 1 / 2 + ∥ g ∥ C 0 , 1 ,B 1 / 2  . (5.3) This lemma ensur es that the ﬁxe d-p oint map of the HJB e quation maps the solution sp ac e into itself. Pr o of. W e start b y proving that ˆ g deﬁned in (i) is B -F réchet diﬀerentiable on H and w e exhibit its B -F réchet deriv ative. Recalling the deﬁnition of the semigroup P t , we extend it to act on H via S ( t ) and write ˆ g as ˆ g ( t, x ) = Z t 0 P t − s h ψ  ∇ B ( g ( s, · )) i ( x ) d s = Z t 0 Z H ψ  ∇ B g ( s, S ( t − s ) x + y )  N (0 , Q t − s )(d y ) d s = Z t 0 Z H ψ  s − 1 / 2 ¯ f ( s, Γ S ( s ) S ( t − s ) x + Γ S ( s ) y )  N (0 , Q t − s )(d y ) d s = Z t 0 Z H ψ  s − 1 / 2 ¯ f ( s, Γ S ( t ) x + Γ S ( s ) y )  N (0 , Q t − s )(d y ) d s. where we hav e used the deﬁnition of the space Σ 1 T , 1 / 2 (Deﬁnition 5.1 ), i.e. w e know that g ( s, z ) = f ( s, Γ S ( s ) z ) , so that there exists a b ounded contin uous function ¯ f : (0 , T ] × Y → K suc h th at s 1 / 2 ∇ B g ( s, z ) = ¯ f  s, Γ S ( s ) z  , s ∈ (0 , T ] , z ∈ H . Note also that for x ∈ H , S ( t − s ) x liv es in H for s < t . Note that, since the integral runs o ver s ∈ (0 , t ) , w e ha ve s > 0 . Thus, S ( s ) maps the noise y ∈ H in to H , allo wing the op erator Γ to b e applied. Similarly , S ( t ) x is in H for x ∈ H . T o compute the B -directional deriv ative in a direction k ∈ K , we lo ok at the limit of the diﬀerence quotient: lim α → 0 1 α [ ˆ g ( t, x + αB k ) − ˆ g ( t, x )] . 8 Note that we use the action B k whic h lands in H . This is v alid as x ∈ H . The p erturbation on x propagates to the argumen t of ¯ f . Sp eciﬁcally , replacing x with x + αB k , the argument b ecomes Γ S ( s )  S ( t − s )( x + α B k ) + y  = Γ S ( t ) x + α Γ S ( t ) B k + Γ S ( s ) y . Th us, the diﬀerence quotient corresp onds to the integral: lim α → 0 1 α Z t 0 Z H " ψ  s − 1 / 2 ¯ f  s, Γ S ( t ) x + α Γ S ( t ) B k + Γ S ( s ) y   − ψ  s − 1 / 2 ¯ f  s, Γ S ( t ) x + Γ S ( s ) y   # N (0 , Q t − s )(d y ) d s. W e no w pro ceed with a change of measure. Let ν t,s b e the image measure of N (0 , Q t − s ) under the linear map y 7→ Γ S ( s ) y . This is a Gaussian measure on Y with cov ariance op erator Q Γ t − s : = Γ S ( s ) Q t − s S ( s ) ∗ Γ ∗ . W e identify the shift vector h = Γ S ( t ) B k . By Hyp othesis 3.3 , the shift satisﬁes the range inclusion required for the Cameron- Martin theorem (relative to the pro jected cov ariance structure at time t ). Let d ( t, t − s, αB k , ξ ) denote the Radon- Nik o dym deriv ative of the shifted measure with respect to the original measure ν t,s . Explicitly d ( t, t − s, αB k , ξ ) = exp n α  Λ Γ ,B ( t − s ) k , ( Q Γ t − s ) − 1 / 2 ξ  Y − α 2 2   Λ Γ ,B ( t − s ) k   2 Y o . Using this densit y , we can rewrite the ﬁrst term of the diﬀerence quotient as an integral ov er the unp erturb ed measure, but multiplied b y the density . The limit b ecomes: lim α → 0 Z t 0 Z H ψ  s − 1 / 2 ¯ f  s, Γ S ( t ) x + Γ S ( s ) y    d ( t, t − s, αB k , Γ y ) − 1 α  N (0 , Q t − s )(d y ) d s. Diﬀeren tiating the densit y with resp ect to α at α = 0 , i.e. lim α → 0 d ( t,t − s,αB k, Γ y ) − 1 α =  Λ Γ ,B ( t − s ) k , ( Q Γ t − s ) − 1 / 2 Γ y  Y , w e obtain the explicit form ula for the deriv ative: ⟨∇ B ˆ g ( t, x ) , k ⟩ K = Z t 0 Z H ψ  s − 1 / 2 ¯ f  s, Γ S ( t ) x + Γ S ( s ) y   ⟨ Λ Γ ,B ( t − s ) k , ( Q Γ t − s ) − 1 / 2 Γ y ⟩ Y N (0 , Q t − s ) (d y ) d s. (5.4) This formula conﬁrms that ∇ B ˆ g ( t, x ) dep ends on x ∈ H only through Γ S ( t ) x , preserving the structure of Σ 1 T , 1 / 2 . Finally , we pro ve the estimate ( 5.3 ) . Using the representation ( 5.4 ) and the Hölder inequality with resp ect to the measure N (0 , Q t − s ) , we hav e:   ⟨∇ B ˆ g ( t, x ) , k ⟩ K   ≤ Z t 0  Z H    ψ  s − 1 / 2 ¯ f  s, Γ S ( t ) x + Γ S ( s ) y      2 N (0 , Q t − s )(d y )  1 / 2 ×  Z H    ⟨ Λ Γ ,B ( t − s ) k , ( Q Γ t − s ) − 1 / 2 Γ y ⟩ Y    2 N (0 , Q t − s )(d y )  1 / 2 d s. Using the Lipschitz prop erty of ψ (i.e., | ψ ( z ) | ≤ C ψ (1 + | z | ) ) and the norm deﬁnition of g ∈ Σ 1 T , 1 / 2 , the ﬁrst factor is b ounded by: sup y ∈ H   ψ  s − 1 / 2 ¯ f  s, Γ S ( t ) x + Γ S ( s ) y    ≤ C ψ  1 + s − 1 / 2 ∥ g ∥ C 0 , 1 ,B 1 / 2  . The second factor equals ∥ Λ Γ ,B ( t − s ) k ∥ Y . Using Hyp othesis 3.3 (Eq. ( 3.6 )) with γ = 1 / 2 , we get:   ⟨∇ B ˆ g ( t, x ) , k ⟩ K   ≤ C Z t 0  1 + s − 1 / 2 ∥ g ∥ C 0 , 1 ,B 1 / 2    Λ Γ ,B ( t − s )   L ( K ; Y ) | k | K d s ≤ C κ 0  Z t 0 ( t − s ) − 1 / 2 d s + ∥ g ∥ C 0 , 1 ,B 1 / 2 Z t 0 s − 1 / 2 ( t − s ) − 1 / 2 d s  | k | K . Computing the time in tegrals, we hav e R t 0 ( t − s ) − 1 / 2 d s = 2 t 1 / 2 and R t 0 s − 1 / 2 ( t − s ) − 1 / 2 d s = β (1 / 2 , 1 / 2) = π , where β ( · , · ) is the Euler Beta function. Thus: sup x ∈ H ∥∇ B ˆ g ( t, x ) ∥ K ≤ C T  t 1 / 2 + ∥ g ∥ C 0 , 1 ,B 1 / 2  , whic h concludes the proof. 6 The Abstract HJB Equation W e study the inﬁnite-dimensional Hamilton-Jacobi-Bellman equation asso ciated with the optimal control problem ( 2.5 ) . Deﬁne the Hamiltonian H min : K → R b y H min ( p ) := inf u ∈ U {⟨ p, u ⟩ K +  1 ( u ) } . T o correctly handle the singular nature of the control op erator B , we solve the HJB equation for the v alue function v ( t, x ) deﬁned on the extended space H . The equation is formally written as: ( − ∂ v ( t, x ) ∂ t = L [ v ( t, · )]( x ) + H min ( ∇ B v ( t, x )) , t ∈ [0 , T ] , x ∈ H , v ( T , x ) = φ ( x ) , x ∈ H (6.1) where L is the generator of the Ornstein-Uhlenbeck pro cess. Note how ev er that, coheren tly with ( 2.5 ) (and our application to SVIEs later), the ﬁnal datum φ is only deﬁned on H . W e interpret this equation in the mild sense on H . 9 Deﬁnition 6.1 (Mild Solution of HJB on H ) . A fun ction v : [0 , T ] × H → R is a mild solution of ( 6.1 ) if: 1. v ( T − · , · ) ∈ C 0 , 1 ,B 1 / 2  [0 , T ] × H  , i.e., it is contin uous on H and has a B -directional deriv ative satisfying the singularit y b ound ∥∇ B v ( t, x ) ∥ K ≤ C ( T − t ) − 1 / 2 . 2. It satisﬁes the integral equation for all ( t, x ) ∈ [0 , T ] × H : v ( t, x ) = P T − t [ φ ]( x ) + Z T t P s − t  H min ( ∇ B v ( s, · ))  ( x ) d s. (6.2) Note that for x ∈ H , P T − t [ φ ]( x ) is well deﬁned b ecause S maps H to H . Theorem 6.2 (Existence and Uniqueness) . L et Hyp otheses 2.1 and 3.3 (with γ = 1 / 2 ) hold. Assume that the ﬁnal datum φ b elongs to B Γ b ( H ) and is Lipschitz c ontinuous, and that  1 is such that the Hamiltonian H min is Lipschitz c ontinuous. Then the HJB e quation ( 6.2 ) admits a mild solution v on H ac c or ding to Deﬁnition 6.1 . Mor e over, v is unique among the functions w such that w ( T − · , · ) ∈ Σ 1 T , 1 / 2 and it satisﬁes, for a suitable C T > 0 , the estimate: ∥ v ( T − · , · ) ∥ C 0 , 1 ,B 1 / 2 ≤ C T ∥ ¯ φ ∥ ∞ . (6.3) Final ly, if the initial datum φ is also c ontinuously B -F r é chet diﬀer entiable, then v ∈ C 0 , 1 ,B b ([0 , T ] × H ) and, for a suitable C T > 0 , ∥ v ∥ C 0 , 1 ,B b ≤ C T  ∥ φ ∥ ∞ + ∥∇ B φ ∥ ∞  . (6.4) Pr o of. W e use a ﬁxed p oin t argument in Σ 1 T , 1 / 2 (whic h consists of functions on H ). T o this aim, ﬁrst w e rewrite ( 6.2 ) in a forward w ay . Namely , if v satisﬁes ( 6.2 ) then, setting w ( t, x ) := v ( T − t, x ) for any ( t, x ) ∈ [0 , T ] × H , we get that w satisﬁes the mild form of the forward HJB equation: w ( t, x ) = P t [ φ ]( x ) + Z t 0 P t − s  H min ( ∇ B w ( s, · ))  ( x ) d s, t ∈ [0 , T ] , x ∈ H . (6.5) Deﬁne the map K on Σ 1 T , 1 / 2 b y setting, for g ∈ Σ 1 T , 1 / 2 , K ( g )( t, x ) := P t [ φ ]( x ) + Z t 0 P t − s  H min ( ∇ B g ( s, · ))  ( x ) d s, t ∈ [0 , T ] . (6.6) By Theorem 3.4 and Lemma 5.3 - (i) we deduce that K is w ell deﬁned in Σ 1 T , 1 / 2 and tak es its v alues in Σ 1 T , 1 / 2 . Since Σ 1 T , 1 / 2 is a closed subspace of the Banach space C 0 , 1 ,B 1 / 2 ([0 , T ] × H ) , once we hav e pro ved that K is a contraction, by the Con traction Mapping Principle there exists a unique (in Σ 1 T , 1 / 2 ) ﬁxed p oin t of the map K , which gives a mild solution of ( 6.1 ). Let g 1 , g 2 ∈ Σ 1 T , 1 / 2 . W e ev aluate ∥K ( g 1 ) − K ( g 2 ) ∥ Σ T , 1 / 2 . First, we estimate the uniform norm diﬀerence. Arguing as in the proof of Lemma 5.3 , and using the Lipschitz con tinuit y of H min with constan t L , w e ha ve for ev ery ( t, x ) ∈ [0 , T ] × H : |K ( g 1 )( t, x ) − K ( g 2 )( t, x ) | =     Z t 0 P t − s  H min  ∇ B g 1 ( s, · )  − H min  ∇ B g 2 ( s, · )  ( x ) d s     ≤ Z t 0 s − 1 / 2 L sup y ∈ H | s 1 / 2 ∇ B ( g 1 − g 2 )( s, y ) | K d s ≤ 2 Lt 1 / 2 ∥ g 1 − g 2 ∥ C 0 , 1 ,B 1 / 2 . Similarly , regarding the weigh ted gradient norm, arguing exactly as in the pro of of estimate ( 5.3 ) (Lemma 5.3 ), we get: t 1 / 2 |∇ B K ( g 1 )( t, x ) − ∇ B K ( g 2 )( t, x ) | K = t 1 / 2     ∇ B Z t 0 P t − s  H min  ∇ B g 1 ( s, · )  − H min  ∇ B g 2 ( s, · )  ( x ) d s     K ≤ t 1 / 2 C T L ∥ g 1 − g 2 ∥ C 0 , 1 ,B 1 / 2 Z t 0 ( t − s ) − 1 / 2 s − 1 / 2 d s ≤ t 1 / 2 Lβ (1 / 2 , 1 / 2) C T ∥ g 1 − g 2 ∥ C 0 , 1 ,B 1 / 2 . Hence, if T is suﬃciently small, summing the t wo estimates w e get ∥K ( g 1 ) − K ( g 2 ) ∥ C 0 , 1 ,B 1 / 2 ≤ C ∥ g 1 − g 2 ∥ C 0 , 1 ,B 1 / 2 (6.7) with C < 1 . So the map K is a contraction in Σ 1 T , 1 / 2 and, denoting by w its unique ﬁxed p oin t, v := w ( T − · , · ) is the unique mild solution of the HJB equation on H . 10 Since the Lipschitz constant L is indep enden t of t , the case of generic T > 0 follows by dividing the in terv al [0 , T ] in to a ﬁnite num b er of subin terv als of length δ suﬃcien tly small, or equiv alen tly , by taking an equiv alen t norm with an adequate exp onen tial weigh t. The estimate ( 6.3 ) follo ws from Theorem 3.4 and Lemma 5.3 . Sp eciﬁcally , since w is the ﬁxed p oin t of K , we hav e w = K ( w ) . Thus, taking the norm in Σ 1 T , 1 / 2 : ∥ w ∥ C 0 , 1 ,B 1 / 2 ≤ ∥ P · [ φ ] ∥ C 0 , 1 ,B 1 / 2 +     Z · 0 P ·− s [ H min ( ∇ B w ( s, · ))] d s     C 0 , 1 ,B 1 / 2 . By Theorem 3.4 , the ﬁrst term satisﬁes ∥ P · [ φ ] ∥ C 0 , 1 ,B 1 / 2 ≤ C T ∥ ¯ φ ∥ ∞ . F or the integral term, we use the linear growth assumption on the Hamiltonian (Hyp othesis 2.1 ), i.e., | H min ( p ) | ≤ L (1 + | p | ) . Applying the con volution estimate ( 5.3 ) from Lemma 5.3 to the function ψ ( p ) = H min ( p ) , we obtain:   R · 0 P ·− s [ H min ( ∇ B w )] d s   C 0 , 1 ,B 1 / 2 ≤ C T  T 1 / 2 + ∥ w ∥ C 0 , 1 ,B 1 / 2  , where the constant C T comes from the Beta function integral (as seen in the pro of of Lemma 5.3 ). Combining these estimates, we get: ∥ w ∥ C 0 , 1 ,B 1 / 2 ≤ C T ∥ ¯ φ ∥ ∞ + C T T 1 / 2 + ˜ C T T 1 / 2 ∥ w ∥ C 0 , 1 ,B 1 / 2 . F or T suﬃcien tly small, the term ˜ C T T 1 / 2 ∥ w ∥ can be absorb ed in to the left-hand side, yielding ( 6.3 ) . T o extend the result to arbitrary T > 0 , we equip Σ 1 T , 1 / 2 with the equiv alent norm ∥ w ∥ λ, Σ w eighted by e − λt . Exploiting the prop erties of the Laplace transform on the singular conv olution kernel, the integral term scales with λ − 1 / 2 . Consequently , choosing λ suﬃciently large ensures the contraction property globally , yielding: ∥ w ∥ λ, Σ ≤  1 − C λ − 1 / 2  − 1 C T ∥ ¯ φ ∥ ∞ . The ﬁnal estimate ( 6.3 ) follo ws immediately from the equiv alence of the weigh ted and standard norms on the compact in terv al [0 , T ] . Finally , the pro of of the last statemen t follows observing that, if φ is con tinuously B -F réc het diﬀerentiable, then P t [ φ ] is con tinuously B -F réc het diﬀerentiable with ∇ B P t [ φ ] bounded in [0 , T ] × H , see Theorem 3.4 . This allows to p erform the ﬁxed p oint argument exactly as done in the ﬁrst part of the pro of, but in the space C 0 , 1 ,B b ([0 , T ] × H ) and to pro ve estimate ( 6.4 ). Theorem 6.3 (Regularity) . L et v b e the unique mild solution to the HJB e quation ( 6.2 ) on H . Assume Hyp otheses 2.1 and 3.3 hold with singularity exp onent γ = 1 / 2 . (i) If the ﬁnal c ost φ is c ontinuously F r é chet diﬀer entiable with b ounde d derivative ( φ ∈ C 1 b ( H ) ∩ B Γ b ( H ) ) and ∇ φ is Lipschitz c ontinuous, then the value function v b elongs to the sp ac e Σ 2 T , 1 / 2 (gener alize d to H ). Conse quently, the se c ond-or der derivatives ∇ B ∇ v and ∇∇ B v exist and c oincide. Mor e over, ther e exists a c onstant C > 0 such that, for al l ( t, x ) ∈ [0 , T ) × H : |∇ v ( t, x ) | ≤ C ∥∇ φ ∥ ∞ , (6.8) |∇ B ∇ v ( t, x ) | L ( H ,K ) ≤ C ( T − t ) − 1 / 2 ∥∇ φ ∥ ∞ . (6.9) (ii) If φ is mer ely b ounde d and c ontinuous ( φ ∈ B Γ b ( H ) ), then the function ( t, x ) 7→ ( T − t ) 1 / 2 v ( t, x ) b elongs to Σ 2 T , 1 / 2 . Mor e over, ther e exists a c onstant C > 0 such that: |∇ v ( t, x ) | ≤ C ( T − t ) − 1 / 2 ∥ φ ∥ ∞ , (6.10) |∇ B ∇ v ( t, x ) | L ( H ,K ) ≤ C ( T − t ) − 1 ∥ φ ∥ ∞ . (6.11) Pr o of. W e start b y proving (i) by applying the Con traction Mapping Theorem in the space Σ 2 T , 1 / 2 . Recall that Σ 2 T , 1 / 2 is the space of functions g ∈ C 0 , 1 b ([0 , T ] × H ) suc h that ∇ B ∇ g exists, is con tinuous, and satisﬁes a weigh ted gro wth condition. W e consider the map K deﬁned ab o ve. By Lemma 5.3 (extended to second deriv atives), K maps Σ 2 T , 1 / 2 in to itself. W e verify the contraction prop ert y by estimating the deriv atives. Let g ∈ Σ 2 T , 1 / 2 . First, for the gradien t ∇K ( g ) , using the diﬀerentiabilit y of the semigroup (Theorem 3.4 ) and diﬀerentiating under the in tegral sign: |∇K ( g )( t, x ) | ≤ |∇ P t [ φ ]( x ) | +     ∇ Z t 0 P t − s  H min  ∇ B g ( s, · )  ( x ) d s     ≤ M ∥∇ φ ∥ ∞ + M Z t 0   ∇  H min  ∇ B g ( s, · )    ∞ d s = M ∥∇ φ ∥ ∞ + M Z t 0   ∇ H min ( ∇ B g ) ∇∇ B g ( s, · )   ∞ d s ≤ M ∥∇ φ ∥ ∞ + M L H Z t 0 ∥∇ B ∇ g ( s, · ) ∥ L ( H ,K ) d s, (6.12) 11 where M = sup t ∈ [0 ,T ] ∥ S ( t ) ∥ and we used the iden tity ∇∇ B g = ∇ B ∇ g v alid in Σ 2 . Second, for the mixed deriv ative ∇ B ∇K ( g ) , we use the smoothing prop ert y (Theorem 3.4 ) on the initial term and the conv olution smo othing (analogous to Lemma 5.3 ) on the in tegral term: t 1 / 2 |∇ B ∇K ( g )( t, x ) | L ( H ,K ) ≤ t 1 / 2   ∇ B P t [ ∇ φ ]( x )   + t 1 / 2     ∇ B Z t 0 P t − s  ∇  H min  ∇ B g ( s, · )  ( x ) d s     ≤ C ∥∇ φ ∥ ∞ + C t 1 / 2 Z t 0 ( t − s ) − 1 / 2   ∇ H min ( ∇ B g ) ∇ B ∇ g ( s, · )   ∞ d s ≤ C ∥∇ φ ∥ ∞ + C L H t 1 / 2 Z t 0 ( t − s ) − 1 / 2 ∥∇ B ∇ g ( s, · ) ∥ L ( H ,K ) d s. (6.13) Let g 1 , g 2 ∈ Σ 2 T , 1 / 2 . By taking the diﬀerence K ( g 1 ) − K ( g 2 ) and using the estimates ab o ve (replacing linear terms with diﬀerences), we observe that the term ∥∇ B ∇ ( g 1 − g 2 ) ∥ app ears inside the in tegral with a k ernel ( t − s ) − 1 / 2 . Sp eciﬁcally , deﬁning the norm ∥ g ∥ 2 ,λ = sup t e − λt ( ∥∇ g ∥ + t 1 / 2 ∥∇ B ∇ g ∥ ) , and choosing λ suﬃcien tly large (or T small), the map K b ecomes a contraction. Let w b e the unique ﬁxed p oin t. The a priori estimates ( 6.8 ) and ( 6.9 ) follo w directly from ( 6.12 ) and ( 6.13 ) by applying the generalized Gronw all lemma (since the forcing terms dep end only on ∥∇ φ ∥ ∞ ). W e now pro ve (ii). Let v b e the mild solution of ( 6.2 ) (whic h exists by Theorem 6.2 ). F or any  ∈ (0 , T ) , deﬁne φ ϵ ( x ) := v ( T − , x ) . By the semigroup prop erty , v restricted to [0 , T −  ] is the unique mild solution of the HJB equation with terminal condition φ ϵ at time T −  . F rom Theorem 6.2 (sp eciﬁcally estimate ( 6.3 ) ), we know that φ ϵ is B -diﬀerentiable and: ∥∇ B φ ϵ ∥ ∞ = ∥∇ B v ( T − , · ) ∥ ∞ ≤ C T  − 1 / 2 ∥ φ ∥ ∞ . (6.14) Ho wev er, to apply Part (i), we need φ ϵ to b e fully F réchet diﬀerentiable ( ∇ φ ϵ ). W e observe that: v ( T − , x ) = P ϵ [ φ ]( x ) + Z T T − ϵ P s − ( T − ϵ )  H min ( ∇ B v ( s, · ))  ( x ) d s. (6.15) The term P ϵ [ φ ] is F réc het diﬀerentiable for any  > 0 thanks to the global smoothing of P t on H (Theorem 3.4 ), with gradien t b ounded by: ∥∇ P ϵ [ φ ] ∥ ∞ ≤ C  − 1 / 2 ∥ φ ∥ ∞ . (6.16) The integral term inherits this regularity . Th us, φ ϵ satisﬁes the assumptions of P art (i) with ∥∇ φ ϵ ∥ ∞ ≤ C  − 1 / 2 ∥ φ ∥ ∞ . Applying the estimates from Part (i) to the problem on [0 , T −  ] : |∇ v ( t, x ) | ≤ C ∥∇ φ ϵ ∥ ∞ ≤ C  − 1 / 2 ∥ φ ∥ ∞ , (6.17) |∇ B ∇ v ( t, x ) | ≤ C ( T −  − t ) − 1 / 2 ∥∇ φ ϵ ∥ ∞ ≤ C ( T −  − t ) − 1 / 2  − 1 / 2 ∥ φ ∥ ∞ . (6.18) Since  is arbitrary , w e can c ho ose  = ( T − t ) / 2 (so that t is far from the terminal time T ). Substituting this in to the estimates: |∇ v ( t, x ) | ≤ C  T − t 2  − 1 / 2 ∥ φ ∥ ∞ = C ′ ( T − t ) − 1 / 2 ∥ φ ∥ ∞ , (6.19) |∇ B ∇ v ( t, x ) | ≤ C  T − t 2  − 1 / 2  T − t 2  − 1 / 2 ∥ φ ∥ ∞ = C ′ ( T − t ) − 1 ∥ φ ∥ ∞ . (6.20) This prov es estimates ( 6.10 ) and ( 6.11 ) . Finally , the fact that ( T − t ) 1 / 2 v ∈ Σ 2 T , 1 / 2 follo ws from the derived b ounds on the second deriv atives. 7 V eriﬁcation theorem In this section, w e introduce the notion of K -strong solutions. This concept serves as a rigorous bridge b et ween the mild solutions derived in Section 6 and the analytical requirements of V eriﬁcation Theorems, bypassing the lack of full C 1 , 2 -regularit y inherent to inﬁnite-dimensional problems with singular controls. This concept of solution follo ws the standard metho dology for such problems. 7.1 F unctional Analytic Setting In this subsection, w e establish the functional analytic framework required to in terpret the HJB equation as a parabolic partial diﬀeren tial equation on the Hilb ert space H . W e recall that while the state dynamics are driven by singular inputs, the asso ciated transition semigroup P t acts on the space of b ounded, uniformly contin uous functions U C b ( H ) . W e ﬁrst introduce the inﬁnitesimal generator of the uncon trolled Ornstein-Uhlen b ec k pro cess. Due to the un b ounded nature of the drift op erator A , the domain of the generator requires careful deﬁnition inv olving the adjoint op erator A ∗ . 12 Deﬁnition 7.1 (The Ornstein-Uhlenbeck Generator L ) . Let U C 2 b ( H ) denote the space of functions in U C b ( H ) with uniformly contin uous and b ounded F réc het deriv atives up to order tw o. W e deﬁne the domain D ( L ) ⊂ U C 2 b ( H ) as the set of functions ϕ satisfying the following regularity conditions: (i) F or all x ∈ H , the gradient ∇ ϕ ( x ) b elongs to the domain of the adjoint op erator D ( A ∗ ) ⊂ H , and the map x 7→ A ∗ ∇ ϕ ( x ) is b ounded and uniformly con tinuous on H . (ii) The map x 7→ T r( Q ∇ 2 ϕ ( x )) is well-deﬁned, b ounded, and uniformly contin uous. F or any ϕ ∈ D ( L ) , the inﬁnitesimal generator L is deﬁned by L [ ϕ ]( x ) : = 1 2 T r  Q ∇ 2 ϕ ( x )  + ⟨ x, A ∗ ∇ ϕ ( x ) ⟩ H . With this op erator, the optimal con trol problem is formally asso ciated with the follo wing inﬁnite-dimensional non-linear parab olic equation (HJB equation): ( − ∂ t v ( t, x ) − L [ v ( t, · )]( x ) − H min ( ∇ B v ( t, x )) = 0 , ( t, x ) ∈ [0 , T ) × H , v ( T , x ) = φ ( x ) . (7.1) A key technical c hallenge is that mild solutions to ( 7.1 ) are not necessarily smo oth enough to b elong to D ( L ) , particularly near the terminal time T where the gradient ∇ B v ma y b ecome singular. T o o vercome this, w e rely on approximations that con verge uniformly on compact sets. Deﬁnition 7.2 ( K -Con vergence) . Let ( f n ) n ∈ N b e a sequence of functions in C b ([0 , T ] × H ) . W e say that ( f n ) is K -c onver gent to a function f , denoted by f n K − → f , if the sequence is uniformly b ounded and con verges uniformly on ev ery compact subset K ⊂⊂ H , uniformly with resp ect to time: sup n ∈ N ∥ f n ∥ ∞ < ∞ and lim n →∞ sup t ∈ [0 ,T ] sup x ∈ K | f n ( t, x ) − f ( t, x ) | = 0 , ∀ K ⊂⊂ H . This top ology is suﬃciently strong to pass to the limit in the in tegral terms of the veriﬁcation theorem, while b eing w eak enough to allo w for the approximation of functions on inﬁnite-dimensional spaces. 7.2 Appro ximation via Lo cal Strong Solutions The singularity of the v alue function’s gradient at the terminal time T (where ∥∇ B v ( t, · ) ∥ ∼ ( T − t ) − 1 / 2 ) preven ts the mild solution from b eing a classical solution on the closed interv al [0 , T ] . How ever, the smo othing prop erties of the transition semigroup ensure regularit y on any interv al b ounded aw ay from T . W e thus introduce the notion of lo cal strong solutions. Deﬁnition 7.3 (Lo cal K -Strong Solution) . A function v ∈ C b ([0 , T ] × H ) is a lo c al K -str ong solution to the HJB equation ( 7.1 ) if, for any  ∈ (0 , T ) , there exist sequences ( v ϵ n ) ⊂ C 1 , 2 ([0 , T −  ] × H ) and ( g ϵ n ) ⊂ C b ([0 , T −  ] × H ) suc h that: (i) F or each t ∈ [0 , T −  ] , v ϵ n ( t, · ) ∈ D ( L ) ; (ii) v ϵ n satisﬁes p oin twise the approximated equation on the restricted in terv al [0 , T −  ] : − ∂ t v ϵ n ( t, x ) − L [ v ϵ n ( t, · )]( x ) = H min ( ∇ B v ϵ n ( t, x )) + g ϵ n ( t, x ); (7.2) (iii) As n → ∞ , the follo wing conv ergences hold in the K -sense on the domain [0 , T −  ] × H : v ϵ n K − → v , ∇ B v ϵ n K − → ∇ B v , g ϵ n K − → 0 . F or the follo wing results we explicitly assume the following analytic smo othing prop ert y . Hyp othesis 7.4. for any t > 0 , the extended semigroup S ( t ) maps H in to the domain of the generator in the smaller space, i.e., S ( t )( H ) ⊆ D ( A ) ⊂ H . R emark 7.5 . 1. Comparing with the analytic semigroup prop ert y we know that a necessary condition is that the semigroup maps istantaneously on D ( A ) . How ev er, here w e require more regularity , b eing the image of the semigroup con tained in the smallest susbspace D ( A ) . Nevertheless, this do es not require that the semigroup is analytic. 2. W e introduced this analytic smo othing prop ert y as it is satisﬁed in our application to SVIEs (see Prop osition 9.3 ). This condition can be probably relaxed, generalizing the approach of [ 33 ]. How ever, this would require more eﬀort in the pro of of the veriﬁcation theorem. 13 Prop osition 7.6 (Lo cal Strong Regularity) . L et Hyp otheses 2.1 , 3.3 and 7.4 hold. Assume that the terminal datum φ satisﬁes the assumptions of The or em 6.2 and that the Hamiltonian H min is Lipschitz c ontinuous. Then, the unique mild solution v of e quation ( 6.2 ) is a lo c al K -str ong solution. Pr o of. Fix  ∈ (0 , T ) and consider the restricted time interv al [0 , T −  ] . W e aim to construct a sequence of regularized functions approx imating the mild solution v that satisﬁes the requirements of Deﬁnition 7.3 . Construction of ( φ ϵ n , F n ) . Let us deﬁne the auxiliary terminal datum φ ϵ ∈ C b ( H ) and the forcing term F ∈ C b ([0 , T −  ] × H ) b y φ ϵ ( x ) : = v ( T − , x ) , F ( s, x ) : = H min ( ∇ B v ( s, x )) . By virtue of Theorem 6.3 , b oth v and its B -directional gradient are b ounded and contin uous on [0 , T −  ] × H , ensuring that φ ϵ and F are well-deﬁned. W e generate the appro ximating sequence ( φ ϵ n , F n ) via a simultaneous regularization pro cedure in volving ﬁnite- dimensional pro jection, molliﬁcation, and a smo othing shift by the semigroup. Let { e k } k ∈ N b e an orthonormal basis for H , and denote by P n the orthogonal pro jection onto the subspace H n : = span { e 1 , . . . , e n } . Let ρ n ∈ C ∞ c ( R n ) b e a standard molliﬁer supp orted in the ball of radius 1 /n . W e asso ciate to φ ϵ a smo oth cylindrical approximation ψ n ∈ U C ∞ b ( H ) deﬁned b y ψ n ( x ) : =  ( φ ϵ ◦ I − 1 n ) ∗ ρ n  ( ⟨ x, e 1 ⟩ , . . . , ⟨ x, e n ⟩ ) , where I n : R n → H n is the canonical isomorphism. Similarly , let Ψ n ∈ C 1 ([0 , T −  ]; U C ∞ b ( H )) b e a smooth cylindrical appro ximation of F , obtained by applying spatial molliﬁcation as ab o ve and a standard temporal regularization. T o guaran tee compatibilit y with the unbounded op erator A ∗ , we in tro duce the semigroup shift δ n : = 1 /n . W e deﬁne the candidate approximations as φ ϵ n ( x ) : = ψ n ( S ( δ n ) x ) , F n ( s, x ) : = Ψ n ( s, S ( δ n ) x ) . W e no w rigorously v erify that φ ϵ n ∈ D ( L ) (the argumen t for F n is identical). Let z : = S ( δ n ) x . By the Chain Rule, the F réchet deriv ative of φ ϵ n is ∇ φ ϵ n ( x ) = S ( δ n ) ∗ ∇ ψ n ( z ) . Note that ∇ ψ n ( z ) lies in H n and is uniformly b ounded. T o chec k the regularity condition for the drift term, w e recall that the analytic smo othing prop ert y (Hyp othesis 2.1 ) implies that the adjoint semigroup maps H in to D ( A ∗ ) for any t > 0 . Consequen tly , the gradien t ∇ φ ϵ n ( x ) = S ( δ n ) ∗ ∇ ψ n ( z ) b elongs to D ( A ∗ ) . Moreo ver, the map x 7→ A ∗ ∇ φ ϵ n ( x ) is b ounded and uniformly contin uous on H . Regarding the diﬀusion term, the Hessian is given by ∇ 2 φ ϵ n ( x ) = S ( δ n ) ∗ ∇ 2 ψ n ( z ) S ( δ n ) . Since ψ n is cylindrical, ∇ 2 ψ n ( z ) has ﬁnite rank. Therefore, the trace T r ( Q ∇ 2 φ ϵ n ( x )) is well-deﬁned and contin uous. Thus, φ ϵ n ∈ D ( L ) and F n ( s, · ) ∈ D ( L ) . Classic al solutions of p erturb e d HJB e quations. Let v ϵ n b e the unique mild solution to the linear parab olic equation asso ciated with the data ( φ ϵ n , F n ) , deﬁned by the integral representation: v ϵ n ( t, x ) = P T − ϵ − t [ φ ϵ n ]( x ) + Z T − ϵ t P s − t [ F n ( s, · )]( x ) d s, t ∈ [0 , T −  ] . (7.3) W e observe that, b y construction, the terminal datum φ ϵ n b elongs to D ( L ) . F urthermore, the source term s 7→ F n ( s, · ) is con tinuously diﬀeren tiable and tak es v alues in D ( L ) . Under these regularity assumptions, standard results on abstract ev olution equations (see, e.g., [ 26 , Chapter 4, pro of of Theorem 4.135, step 2]) ensure that the mild solution upgrades to a classical solution. Th us, v ϵ n ∈ C 1 , 2 ([0 , T −  ] × H ) and satisﬁes the point wise equation: − ∂ t v ϵ n ( t, x ) − L [ v ϵ n ( t, · )]( x ) = F n ( t, x ) , v ϵ n ( T − , x ) = φ ϵ n ( x ) . (7.4) Rewriting the source term as F n = H min ( ∇ B v ϵ n ) + g ϵ n , with err or term g ϵ n : = F n − H min ( ∇ B v ϵ n ) , w e recov er the structure required by Deﬁnition 7.3 . Conver genc e . The prop erties of pro jections, molliﬁers, and the strong contin uity of the semigroup ensure that φ ϵ n K − → φ ϵ and F n K − → F (uniform conv ergence on compact sets). Standard stability estimates for mild solutions of linear ev olution equations imply v ϵ n K − → v . F or the gradients, diﬀerentiating the mild representation yields ∥∇ B v ϵ n ( t, · ) − ∇ B v ( t, · ) ∥ ∞ ,K ≤ ∥∇ B P T − ϵ − t [ φ ϵ n − φ ϵ ] ∥ ∞ ,K + Z T − ϵ t ∥∇ B P s − t [ F n ( s, · ) − F ( s, · )] ∥ ∞ ,K d s. (7.5) Exploiting the smo othing estimate (Theorem 3.4 ) and the lo cal in tegrability of the kernel, the K -con vergence of the data implies ∇ B v ϵ n K − → ∇ B v . Since H min is Lipschitz contin uous, it follows that g ϵ n K − → 0 , concluding the pro of. Prop osition 7.7 (F undamental Iden tity) . L et the assumptions of Pr op osition 7.6 hold. L et v b e the unique mild solution to the HJB e quation ( 6.2 ) extende d to H . Then, for any initial datum ( t, x ) ∈ [0 , T ] × H and for any admissible c ontr ol u ∈ U such that the inte gr al term b elow is wel l-deﬁne d, the fol lowing identity holds: v ( t, x ) = J ( t, x, u ) + E " Z T t  H min ( ∇ B v ( s, X ( s ))) − H C V ( ∇ B v ( s, X ( s )); u ( s ))  d s # , (7.6) wher e X ( · ) denotes the mild solution of the state e quation ( 2.2 ) driven by u , and we deﬁne the c ontr ol Hamiltonian as H C V ( p ; u ) : = ⟨ p, u ⟩ K +  1 ( u ) . 14 Pr o of. Fix ( t, x ) ∈ [0 , T ] × H and let u ∈ U b e an arbitrary admissible con trol. Let X ( · ) b e the asso ciated state pro cess. The pro of pro ceeds by regularization via lo cal strong solutions, follo wed by limiting arguments. Fix  ∈ (0 , T − t ) . By Prop osition 7.6 , the mild solution v is a local K -strong solution. Th us, there exists an approximating sequence ( v ϵ n ) n ∈ N ⊂ C 1 , 2 ([0 , T −  ] × H ) ∩ D ( L ) satisfying the p erturbed PDE ( 7.2 ) p oin t wise, where the error term g ϵ n con verges to zero uniformly on compact sets as n → ∞ . Applying Itô’s formula to the pro cess s 7→ v ϵ n ( s, X ( s )) on the interv al [ t, T −  ] yields v ϵ n ( T − ϵ, X ( T − ϵ )) − v ϵ n ( t, x ) = Z T − ϵ t ( ∂ s + L ) v ϵ n ( s, X ( s )) d s + Z T − ϵ t ⟨∇ B v ϵ n ( s, X ( s )) , u ( s ) ⟩ K d s + Z T − ϵ t ⟨∇ v ϵ n ( s, X ( s )) , G d W ( s ) ⟩ H . Substituting the PDE relation ( 7.2 ) in to the ﬁrst integral and adding/subtracting the running cost  1 ( u ( s )) , w e rearrange terms to obtain: v ϵ n ( t, x ) = v ϵ n ( T − , X ( T −  )) + Z T − ϵ t  1 ( u ( s )) d s + Z T − ϵ t  H min ( ∇ B v ϵ n ) − H C V ( ∇ B v ϵ n ; u ( s )) + g ϵ n  ( s, X ( s )) d s − Z T − ϵ t ⟨∇ v ϵ n ( s, X ( s )) , G d W ( s ) ⟩ H . (7.7) W e take the exp ectation in ( 7.2 ) . The sto c hastic integral v anishes since v ϵ n has b ounded deriv atives on the compact in terv al. Regarding the deterministic terms, recall that X ( · ) is contin uous , so the image X ([ t, T −  ]) is compact almost surely . By Prop osition 7.6 , v ϵ n → v and ∇ B v ϵ n → ∇ B v uniformly on compacts, while g ϵ n → 0 . In v oking the Dominated Con vergence Theorem (justiﬁed b y the uniform bounds on the appro ximations), we pass to the limit n → ∞ : v ( t, x ) = E  v ( T − ϵ, X ( T − ϵ )) + Z T − ϵ t ℓ 1 ( u ( s )) d s  + E  Z T − ϵ t  H min ( ∇ B v ( s, X ( s ))) − H C V ( ∇ B v ( s, X ( s )); u ( s ))  d s  . W e extend the identit y to [ t, T ] . First, consider the terminal term. Due to the analytic smo othing prop ert y we ha ve that X ( T ) ∈ H almost surely . Since v is con tinuous on [0 , T ] × H and coincides with φ on H , we hav e the almost sure conv ergence lim ϵ → 0 v ( T − , X ( T −  )) = φ ( X ( T )) . Since v is b ounded, conv ergence in L 1 (Ω) follows. Second, consider the integral term inv olving the Hamiltonians. By Theorem 6.3 , the gradient satisﬁes the singular estimate ∥∇ B v ( s, · ) ∥ K ≤ C ( T − s ) − 1 / 2 . Using the linear growth of H min and H C V in the momentum v ariable p , the in tegrand is b ounded b y   H min ( ∇ B v ) − H C V ( ∇ B v ; u )   ≤ C ( T − s ) − 1 / 2 (1 + ∥ u ( s ) ∥ K ) . Since the con trol set U is b ounded (Hyp othesis 2.1 ), the term ∥ u ( s ) ∥ K is uniformly b ounded. This guaran tees the integrabilit y of the pro duct with the singular kernel ( T − s ) − 1 / 2 , and the Dominated Conv ergence Theorem allows us to extend the in tegral to T . Letting  → 0 and identifying the cost functional J ( t, x, u ) , we reco ver the identit y ( 7.6 ). Theorem 7.8 (V eriﬁcation Theorem) . L et the assumptions of Pr op osition 7.7 hold. L et v b e the unique mild solution of the HJB e quation ( 6.2 ) extende d to H . Then: (i) F or any ( t, x ) ∈ [0 , T ] × H , u ∈ U , we have v ( t, x ) ≤ J ( t, x, u ) . Conse quently, v ( t, x ) ≤ V ( t, x ) , wher e V ( t, x ) : = inf u ∈U J ( t, x, u ) is the value function of the optimal c ontr ol pr oblem deﬁne d on H . (ii) L et ( t, x ) ∈ [0 , T ] × H b e ﬁxe d. Supp ose ther e exists an admissible c ontr ol u ∗ ∈ U such that, denoting by X ∗ ( · ) the c orr esp onding state tr aje ctory, the fol lowing fe e db ack c ondition holds for ds ⊗ d P -almost al l ( s, ω ) ∈ [ t, T ] × Ω : u ∗ ( s ) ∈ arg min u ∈ U H C V ( ∇ B v ( s, X ∗ ( s )); u ) . (7.8) Then the p air ( u ∗ , X ∗ ) is optimal, and the value function c oincides with the mild solution of the HJB e quation: v ( t, x ) = V ( t, x ) = J ( t, x, u ∗ ) . (7.9) Pr o of. The pro of follows directly from the F undamen tal Identit y ( 7.6 ) . F or (i) , observe that H min ( p ) ≤ H C V ( p ; u ) by deﬁnition, making the in tegrand in ( 7.6 ) non-p ositiv e. F or (ii) , the feedback condition implies H min = H C V almost ev erywhere, making the in tegral term zero. R emark 7.9 (Extension to Lipschitz Pa y oﬀs) . The v alidity of Theorem 7.8 relies on the integrable singularity of the gradien t ∥∇ B v ( s, · ) ∥ K ≤ C ( T − s ) − 1 / 2 . This condition holds even if the terminal pay oﬀ φ is merely Lipsc hitz con tinuous (rather than C 1 ), as shown in Theorem 6.3 - (ii) . Th us, the V eriﬁcation Theorem extends to standard option pay oﬀs via the density argument detailed in Prop osition 7.7 . 15 8 Optimal F eedbac k Con trols This section addresses the construction of optimal controls in feedbac k form. Building up on the V eriﬁcation Theorem 7.8 , w e introduce the multivalue d fe e db ack map Ψ : [0 , T ) × H ⇒ U deﬁned b y the minimization of the Hamiltonian: Ψ( s, y ) : = arg min u ∈ U H C V ( ∇ B v ( s, y ); u ) = arg min u ∈ U  ⟨∇ B v ( s, y ) , u ⟩ K +  1 ( u )  , (8.1) where v denotes the unique mild solution of the HJB equation ( 6.2 ) . F or a given initial pair ( t, x ) ∈ [0 , T ) × H , the Close d L o op Equation is formally described b y the sto c hastic diﬀerential inclusion: d Y ( s ) ∈ ( AY ( s ) + B Ψ( s, Y ( s ))) d s + G d W ( s ) , s ∈ [ t, T ) , Y ( t ) = x. (8.2) The connection b et ween the solv ability of the closed lo op system and optimalit y is established by the following corollary . Corollary 8.1 (Optimality of F eedbac k Controls) . L et the assumptions of The or em 7.8 hold. L et v b e the unique mild solution of ( 6.2 ) . Fix ( t, x ) ∈ [0 , T ) × H . Assume that the map Ψ admits a me asur able sele ction ψ : [0 , T ) × H → U such that the c orr esp onding Close d L o op Equation: d Y ( s ) = ( AY ( s ) + B ψ ( s, Y ( s ))) d s + G d W ( s ) , s ∈ [ t, T ) , Y ( t ) = x. (8.3) admits a mild solution Y ψ ( · ) in H (in the sense of Deﬁnition 2.2 ). Then, the fe e db ack c ontr ol u ψ ( s ) : = ψ ( s, Y ψ ( s )) is optimal for the pr oblem starting at ( t, x ) , and V ( t, x ) = v ( t, x ) . F urthermor e, if Ψ is single-value d and the mild solution to ( 8.3 ) is unique, the optimal c ontr ol is unique. T o ensure the well-posedness of ( 8.3 ) , w e require regularity conditions on the minimizer of the Hamiltonian. Let Γ( p ) denote the set of minimizers for a momentum p ∈ K : Γ( p ) : = { u ∈ U : ⟨ p, u ⟩ K +  1 ( u ) = H min ( p ) } . (8.4) Consequen tly , Ψ( t, x ) = Γ( ∇ B v ( t, x )) . While Γ( p ) is non-empty under standard compactness or co ercivit y assumptions, the existence of a strong solution requires Lipschitz regularit y of the selection. Hyp othesis 8.2 (Regular F eedback) . The m ultiv alued map Γ : K ⇒ U admits a Lipschitz contin uous selection γ : K → U . Speciﬁcally , there exists L γ > 0 such that ∥ γ ( p 1 ) − γ ( p 2 ) ∥ K ≤ L γ ∥ p 1 − p 2 ∥ K , p 1 , p 2 ∈ K. R emark 8.3 . Hyp othesis 8.2 is satisﬁed, for example, when  1 is strictly conv ex and smo oth, where γ is the unique minimizer. W e consider the Closed Lo op Equation driven b y this regular selection: d Y ( s ) = ( AY ( s ) + B γ ( ∇ B v ( s, Y ( s )))) d s + G d W ( s ) , s ∈ [ t, T ] , Y ( t ) = x. (8.5) Theorem 8.4 (Existence of Optimal F eedbac k Control) . L et Hyp otheses 2.1 , 3.3 (with γ = 1 / 2 ), and 8.2 hold. Assume that the ﬁnal datum φ satisﬁes the r e gularity c onditions of Pr op osition 7.6 : (i) φ ∈ B Γ b ( H ) ; (ii) φ ∈ C 1 b ( H ) and ∇ φ is Lipschitz c ontinuous. Then, for any initial data ( t, x ) ∈ [0 , T ) × H , the close d lo op e quation ( 8.5 ) admits a unique mild solution Y γ ( · ) in the sp ac e of c ontinuous p aths C ([ t, T ]; H ) . Conse quently, the fe e db ack c ontr ol pr o c ess deﬁne d by u ∗ ( s ) : = γ ( ∇ B v ( s, Y γ ( s ))) , s ∈ [ t, T ] , is an optimal c ontr ol for the pr oblem starting at ( t, x ) , and the optimal c ost is given by J ( t, x, u ∗ ) = v ( t, x ) . R emark 8.5 (Integrabilit y and Pa yoﬀ Regularity) . The assumption that φ ∈ C 1 b ( H ) is critical for the well-posedness of the closed lo op equation up to the te rminal time T . Sp eciﬁcally , under this assumption, Theorem 6.3 - (i) ensures that the second-order deriv ative ∇ B ∇ v ( s, · ) satisﬁes the estimate ∥∇ B ∇ v ( s, · ) ∥ ≤ C ( T − s ) − 1 / 2 . This singularity is in tegrable in time, allo wing for the application of the singular Gronw all lemma. Con versely , if φ w ere merely contin uous (as in typical option pay oﬀs), the Hessian singularit y would scale as ( T − s ) − 1 (Theorem 6.3 - (ii) ), rendering the feedback term non-in tegrable and the con trol p oten tially unbounded as s → T . Pr o of. W e prov e the existence and uniqueness of the solution to the closed lo op equation via a ﬁxed-p oin t argument in the Banac h space Z : = C ([ t, T ]; H ) equipp ed with the uniform norm. The mild formulation of ( 8.5 ) is giv en by the op erator T : Z → Z : T ( Y )( s ) : = S ( s − t ) x + Z s t S ( s − r ) B γ ( ∇ B v ( r , Y ( r ))) d r + Z s t S ( s − r ) G d W ( r ) . 16 W e verify the con traction prop erty . Let Y 1 , Y 2 ∈ Z . W e estimate the diﬀerence in the H -norm: ∥T ( Y 1 )( s ) − T ( Y 2 )( s ) ∥ H ≤ Z s t ∥ S ( s − r ) ∥ L ( H ) ∥ B ∥ L ( K,H ) ∥ γ ( ∇ B v ( r , Y 1 ( r ))) − γ ( ∇ B v ( r , Y 2 ( r ))) ∥ K d r . Using the Lipschitz contin uit y of the selection γ (Hyp othesis 8.2 ) with constant L γ , and the uniform b ound on the semigroup M T = sup r ≤ T ∥ S ( r ) ∥ , w e hav e: ∥T ( Y 1 )( s ) − T ( Y 2 )( s ) ∥ H ≤ M T ∥ B ∥ L γ Z s t ∥∇ B v ( r , Y 1 ( r )) − ∇ B v ( r , Y 2 ( r )) ∥ K d r . Since φ ∈ C 1 b ( H ) , Theorem 6.3 - (i) implies that v ( r , · ) is twice diﬀerentiable with ∇ B ∇ v ( r , · ) contin uous. By the Mean V alue Theorem, the Lipschitz constant of the map y 7→ ∇ B v ( r , y ) is b ounded by the norm of the second deriv ative. Using the estimate ( 6.9 ) from Theorem 6.3 : ∥∇ B v ( r , Y 1 ( r )) − ∇ B v ( r , Y 2 ( r )) ∥ K ≤ ∥∇ B ∇ v ( r , · ) ∥ L ( H ,K ) ∥ Y 1 ( r ) − Y 2 ( r ) ∥ H ≤ C ( T − r ) − 1 / 2 ∥ Y 1 ( r ) − Y 2 ( r ) ∥ H . Substituting this back into the integral inequalit y: ∥T ( Y 1 )( s ) − T ( Y 2 )( s ) ∥ H ≤ ˜ C Z s t ( T − r ) − 1 / 2 ∥ Y 1 ( r ) − Y 2 ( r ) ∥ H d r , (8.6) where ˜ C dep ends on the system data and ∥∇ φ ∥ ∞ . Since the kernel k ( s, r ) = ( T − r ) − 1 / 2 is in tegrable on [ t, T ] , the generalized singular Gronw all lemma guarantees that the map T (or a suﬃciently high p ow er of it) is a strict con traction on Z . Th us, there exists a unique ﬁxed p oint Y γ ∈ C ([ t, T ]; H ) . The optimality of the feedbac k control u ∗ ( s ) = γ ( ∇ B v ( s, Y γ ( s ))) then follo ws directly from the V eriﬁcation Theorem (Theorem 7.8 - (ii) ), as the pair ( u ∗ , Y γ ) satisﬁes the feedback condition by construction. 9 Optimal Con trol of Sto c hastic V olterra In tegral Equations Here, we apply the abstract inﬁnite-dimens ional control framework developed in previous sections to the optimal con trol of Sto c hastic V olterra Integral Equations (SVIEs) with completely monotone kernels. W e recov er Marko vianity b y lifting the state equation on to suitable Hilb ert spaces [ 27 ] (see also [ 7 , 52 ]). Consider the one-dimensional controlled SVIE for the state proc ess y ( s ) ∈ R : y ( s ) = z ( s ) + Z s 0 K ( s − r )[ cy ( r ) + bu ( r )] d r + Z s 0 K ( s − r ) g d W ( r ) , (9.1) where g : [0 , T ] → R denotes an admissible initial forw ard curve (deﬁned b elow), W is a one-dimensional standard Wiener process, and the co eﬃcien ts c, b, g ∈ R are real c onstan ts. W e assume the ge ometric c ontr ol lability c ondition holds, i.e., b  = 0 and g  = 0 . The set of control v alues U is a compact subset of R . The space of admissible controls U consists of all ( F s ) -progressiv ely measurable pro cesses u : [ t, ∞ ) × Ω → U . The goal is to minimize the cost functional: ˜ J ( t, z ; u ) = E " Z T t  1 ( u ( s )) d s + ˜ φ ( y ( T )) # , (9.2) where  1 : U → R and ˜ φ : R → R are the running cost and the terminal cost, resp ectiv ely . 9.1 Case c = 0 In this subsection, w e assume that c = 0 . How ever, in Subsection 9.2 , we will drop this assumption by working with the p erturbation theory of analytic semigroups. T o resolve the non-Mark ovian nature of ( 9.1 ) , we rely on a Marko vian lifting determined by the analytic prop erties of the kernel K . Deﬁnition 9.1 (Admissible Kernel) . A measurable function K : (0 , ∞ ) → [0 , ∞ ) is termed an admissible kernel if it admits the Laplace represen tation K ( t ) = K ( ∞ ) + R (0 , ∞ ) e − xt µ (d x ) , t > 0 , where µ is a Borel measure on (0 , ∞ ) . W e deﬁne the critical in tegrability exponent η ∗ as: η ∗ : = inf n η ∈ R : Z (0 , ∞ ) (1 + x ) − η µ (d x ) < ∞ o . (9.3) W e require η ∗ ∈ [ −∞ , 1 2 ) to ensure square-integrabilit y of the controlled tra jectories. 17 Lemma 9.2 (Asymptotic Gro wth Condition) . L et K b e an admissible kernel in the sense of Deﬁnition 9.1 and let I K ( t ) : = R t 0 K ( s ) d s denote its primitive. Then, 0 ≤ tK ( t ) I K ( t ) ≤ 1 , ∀ t > 0 . Henc e, the singularity of the kernel at the origin is strictly c ontr ol le d by its aver age, i.e., ρ 0 : = lim sup t → 0 +  tK ( t ) I K ( t )  ≤ 1 . Pr o of. Since K is an admissible kernel, it admits a represen tation via a Borel measure µ on [0 , ∞ ) . Consequen tly , K is completely monotone, which implies K is non-increasing on (0 , ∞ ) and non-negative. Assume K is not identically zero (otherwise the statement is trivial). Then I K ( t ) > 0 for all t > 0 . By the monotonicit y of K , for any s ∈ (0 , t ] , w e ha ve K ( s ) ≥ K ( t ) . Integrating this inequality o ver the interv al (0 , t ] yields I K ( t ) = R t 0 K ( s ) d s ≥ R t 0 K ( t ) d s = tK ( t ) . Rearranging the terms (since I K ( t ) > 0 ), we obtain 0 ≤ tK ( t ) I K ( t ) ≤ 1 for all t > 0 . T aking the limit sup erior as t → 0 + immediately yields ρ 0 ≤ 1 . W e now rigorously deﬁne the Hilb ert spaces and op erators that transform the SVIE ( 9.1 ) in to a Sto c hastic Ev olution Equation (SEE). Spaces H η . Let ¯ µ : = δ 0 + µ b e the extended measure on [0 , ∞ ) , where δ 0 accoun ts for the constan t part of the k ernel (if any) or acts as an auxiliary co ordinate. W e introduce a family of w eighted Hilb ert spaces H η parameterized by η ∈ R : H η : = L 2 ( R + , (1 + x ) η ¯ µ (d x ); R ) , ⟨ f , g ⟩ H η : = Z ∞ 0 f ( x ) g ( x ) (1 + x ) η ¯ µ (d x ) . (9.4) T o handle singular k ernels where the "input" direction ma y not b e in the state space, we distinguish b et w een: 1. H : = H η , where we choose η ∈ ( η ∗ , 1) . This choice ensures the reconstruction op erator is b ounded. 2. H : = H η ′ , where we choose η ′ < η ∗ . This larger space con tains the singular inputs. Note that η ′ < η implies H  → H with contin uous and dense embedding. Semigroup. The operator A describ es the relaxation of the forward curv e mo des. It is deﬁned as the multiplication op erator D ( A ) =  ϕ ∈ H : Z ∞ 0 x 2 ∥ ϕ ( x ) ∥ 2 (1 + x ) η ¯ µ (d x ) < ∞  , ( Aϕ )( x ) = − xϕ ( x ) , ¯ µ -a.e. x ∈ R + . Since x ≥ 0 , A is dissipative and generates an analytic and self-adjoint semigroup of contractions S ( t ) on H (and clearly on H as w ell). The semigroup is explicitly given by ( S ( t ) ϕ )( x ) = e − tx ϕ ( x ) . The follo wing result establishes the analytic smo othing property of the extended semigroup. Sp eciﬁcally , it demonstrates that the action of S ( t ) is suﬃcient to instantaneously recov er regularit y , mapping elements from the extended space H directly in to the domain of the op erator A for all t > 0 . Prop osition 9.3 (Analytic Smo othing of the Lifted Semigroup) . L et η , η ′ b e chosen such that η ′ < η ∗ < η < 1 . F or any t > 0 , the extende d semigr oup S ( t ) (the extension of S ( t ) to H ) maps H c ontinuously into D ( A ) ⊂ H . Sp e ciﬁc al ly, ther e exists a c onstant C > 0 dep ending on η , η ′ such that: ∥ AS ( t ) z ∥ H ≤ C t − (1+ η − η ′ 2 ) ∥ z ∥ H , ∀ z ∈ H . (9.5) Pr o of. Let z ∈ H . W e compute the norm of AS ( t ) z in H . ∥ AS ( t ) z ∥ 2 H = Z ∞ 0   − xe − tx z ( x )   2 R n (1 + x ) η ¯ µ (d x ) = Z ∞ 0 x 2 e − 2 tx (1 + x ) η − η ′ ∥ z ( x ) ∥ 2 R n (1 + x ) η ′ ¯ µ (d x ) . Deﬁne the weigh ting function ψ t ( x ) : = x 2 e − 2 tx (1 + x ) η − η ′ . W e estimate its supremum ov er x ≥ 0 . Note that for x ≥ 1 , (1 + x ) η − η ′ ≤ (2 x ) η − η ′ . Thus, the dominant term b eha ves as x 2+ η − η ′ e − 2 tx . Standard calculus shows that sup y ≥ 0 y k e − y = ( k /e ) k . Letting y = 2 tx , w e ha ve x = y / (2 t ) , and the function scales as (2 t ) − (2+ η − η ′ ) . Therefore, sup x ≥ 0 ψ t ( x ) ≤ ˜ C t − (2+ η − η ′ ) . Substituting this back into the in tegral: ∥ AS ( t ) z ∥ 2 H ≤ ˜ C t − (2+ η − η ′ ) Z ∞ 0 ∥ z ( x ) ∥ 2 (1 + x ) η ′ ¯ µ (d x ) = ˜ C t − (2+ η − η ′ ) ∥ z ∥ 2 H . T aking the square ro ot yields the result. This conﬁrms that the singularity of the kernel is smo othed b y the semigroup ev olution, allowing the state to enter the domain of the generator instan tly . 18 Reconstruction op erator Γ . The state y ( t ) of the original SVIE will b e recov ered from the inﬁnite-dimensional state X ( t ) ∈ H via the op erator Γ ∈ L ( H ; R ) , Γ ϕ : = R ∞ 0 ϕ ( x ) ¯ µ (d x ) . The b oundedness of Γ on H is guaran teed by the Cauch y-Sch warz inequality and the condition η > η ∗ , i.e., | Γ ϕ | ≤  R ∞ 0 (1 + x ) − η ¯ µ (d x )  1 / 2 ∥ ϕ ∥ H . Represen tation of K . T aking into account Deﬁnition 9.1 and the notations ab o ve, we hav e K ( t ) = K ( ∞ ) + R (0 , ∞ ) e − xt µ (d x ) = K ( ∞ )Γ S ( t ) I { x =0 } ( · ) + Γ S ( t ) I x> 0 ( · ) , i.e., the kernel K has the representation K ( t ) = Γ S ( t ) ξ K , wher e ξ K ( x ) := K ( ∞ ) I { x =0 } ( x ) + I x ∈ (0 , ∞ ) ( x ) . (9.6) The function ξ K can b e understo o d as the lift of K to H (similarly to the lift ξ g of initial curves later). Crucially , for singular k ernels, ξ K / ∈ H b ecause R (1 + x ) η µ (d x ) div erges for η > η ∗ . How ever, by construction, ξ K ∈ H . Un b ounded con trol and diﬀusion op erators. W e deﬁne the control op erator B : R → H and diﬀusion op erator G : R → H as ( B u )( x ) : = ξ K ( x ) bu, ( Gw )( x ) : = ξ K ( x ) g w, ∀ x ∈ R + . These act as rank- 1 op erators mapping in to the extrap olation space. Lifted SDE. The lifted pro cess X ( t ) is the mild solution to the sto chastic evolution equation in H : d X ( t ) = AX ( t )d t + B u ( t )d t + G d W ( t ) , X (0) = ξ g . (9.7) The lifting of the Sto c hastic V olterra Integral Equation ( 9.1 ) in to a Marko vian framework requires the initial signal g ( · ) to b e consistent with the dissipative structure of the inﬁnitesimal generator A . In this setting, the forward curve is not merely a b oundary condition but the pro jection of an initial functional state. Deﬁnition 9.4 (Admissible Initial Curve) . A measurable function g : [0 , T ] → R n is said to be an admissible initial curv e if there exists a unique element ξ g ∈ H (the lifting of g ) such that, for almost every t ∈ [0 , T ] , the curve admits the representation z ( t ) = Γ S ( t ) ξ z = R ∞ 0 e − xt ξ z ( x ) ¯ µ (d x ) . W e denote b y Z the set of admissible curves. W e refer to [ 2 , 7 ] for Theorems relating solutions to SVIEs and Mild solutions of the lifted SDE ( 9.7 ) . Here, we only giv e a quic k intuition of the lifting pro cedure. In tuition for the lift. Consider the SVIE ( 9.1 ) with c = 0 and z ( t ) = Γ S ( t ) ξ z . W e seek an H η -v alued pro cess X ( t ) suc h that Γ X ( t ) = y ( t ) ; b y the representation ( 9.6 ), w e hav e Γ X ( t ) = y ( t ) = z ( t ) + Z t 0 K ( t − s ) bu ( s ) d s + Z t 0 K ( t − s ) g d W s = Γ S ( t ) ξ z + Z t 0 Γ S ( t − s ) ξ K bu ( s ) d s + Z t 0 Γ S ( t − s ) ξ K g d W s = Γ  S ( t ) ξ z + Z t 0 S ( t − s ) B u ( s ) d s + Z t 0 S ( t − s ) G d W s  . Hence we are led to X ( t ) = S ( t ) ξ g + R t 0 S ( t − s ) B u ( s ) d s + R t 0 S ( t − s ) G d W s , which is the mild solution of ( 9.7 ). Lifted cost functional. F or t ∈ [0 , T ] and z ∈ G , the cost functional ˜ J ( t, g ; u ) is rewritten as: J ( t, ξ z , u ) := E " Z T t  1 ( u ( s )) d s + φ ( X ( T )) # = ˜ J ( t, z ; u ) , (9.8) where φ = ˜ φ ◦ Γ . Since the state equation ( 9.7 ) and functional ( 9.8 ) are of the same form as those considered in Section 2 , they are well-deﬁned for all x ∈ H = H η ′ . Therefore, with the ab o ve notations, the abstract control problem w e are led to solve is the one in Section 2 . Smo othing. Finally , we chec k that Hyp othesis 3.3 is satisﬁed. W e do this by means of Theorem 4.1 . Prop osition 9.5 (Universal Energy Bound) . L et K b e an admissible kernel in the sense of Deﬁnition 9.1 . Assume that the ge ometric c ontr ol lability c ondition holds (i.e., b  = 0 and g  = 0 ). Then, the p artial r e gularization op er ator Λ Γ ,B ( t ) , deﬁne d in ( 3.5 ) , satisﬁes the fol lowing norm estimate: ∥ Λ Γ ,B ( t ) ∥ L ( R , R ) ≤ C t − 1 / 2 , ∀ t ∈ (0 , T ] , (9.9) wher e C is a p ositive c onstant dep ending on the system c o eﬃcients and the asymptotic b ehaviour of the kernel at the origin. 19 Pr o of. The pro of relies on the isometric isomorphism established in Section 4 , which relates the op erator norm of the singular kernel to a minimum energy control problem. W e ﬁrst derive the fundamen tal link b et ween the abstract displacemen t and the scalar k ernel K ( t ) . F or an y direction k ∈ R : Γ S ( t ) B k = Z ∞ 0 e − tx ( B k ) ¯ µ (d x ) =  Z ∞ 0 e − tx ¯ µ (d x )  bk = K ( t ) bk , (9.10) where the last equality follows from the representation of K . W e now inv oke Theorem 4.1 (part (ii) ), which identiﬁes the op erator norm with the v alue function of a deterministic optimal control problem. Speciﬁcally: | Λ Γ ,B ( t ) k | = inf  ∥ v ∥ L 2 (0 ,t ; R ) : L Γ t v = − Γ S ( t ) B k  . (9.11) Here, L Γ t represen ts the input-to-pro jected-state map for the virtual system driven b y the diﬀusion op erator G . Analogously to ( 9.10 ), the action of the conv olution is given by L Γ t v = Z t 0 Γ S ( t − s ) Gv ( s ) d s = Z t 0 K ( t − s ) g v ( s ) d s. Th us, the abstract constrain t L Γ t v = − Γ S ( t ) B k is equiv alen t to the V olterra in tegral equation of the ﬁrst kind: Z t 0 K ( t − s ) g v ( s ) d s = − K ( t ) bk . (9.12) T o establish the upp er b ound ( 9.9 ) , it suﬃces to ev aluate th e energy of a sub optimal admissible control. W e prop ose a constan t strategy ansatz ¯ v ( s ) ≡ ¯ v ∈ R . Substituting this in to ( 9.12 ) yields, with usual notation I K ( t ) := R t 0 K ( τ ) d τ , I K ( t ) g ¯ v =  Z t 0 K ( s ) d s  g ¯ v = − K ( t ) bk . i.e., we are led to the linear algebraic equation I K ( t ) g ¯ v = − K ( t ) bk . Under the geometric con trollability assumption (if b  = 0 then g  = 0 ), this equation admits solutions. The minimal norm solution is simply: ¯ v = − K ( t ) I K ( t ) b g k . The L 2 (0 , t ; R ) -norm of the constant strategy is: ∥ ¯ v ∥ L 2 = √ t | ¯ v | = √ t K ( t ) I K ( t )     b g k     ≤ √ t · κt − 1     b g     | k | = C t − 1 / 2 | k | , where we hav e applied Lemma 9.2 and set C = κ | b/g | . Finally , since the optimal control ˆ v has minimal energy , w e ha ve | Λ Γ ,B ( t ) k | = ∥ ˆ v ∥ L 2 ≤ ∥ ¯ v ∥ L 2 . T aking the supremum ov er unit scalars k (i.e., | k | = 1 ), we conclude: ∥ Λ Γ ,B ( t ) ∥ L ( R , R ) ≤ C t − 1 / 2 . R emark 9.6 (Comparison with Standard and P artial Smo othing) . The Γ -smo othing prop erty established here departs signiﬁcan tly from existing literature. Unlik e the classical framew ork of Da Prato and Zab czyk (see [ 13 , Section 9.4.1]), which requires inﬁnite-dimensional noise to provide regularization o ver the entire space H , our setting inv olves ﬁnite-rank noise, making global smoothing unattainable. Moreo ver, the partial smo othing approac h of [ 31 , Section 4] is not applicable for structural reasons. W e are therefore in the setup of the previous sections. W e can then apply our results to show existence and uniqueness of mild solutions of the HJB equation ( 6.1 ), state v eriﬁcation theorems, and construct feedbac k controls. 9.2 The case c  = 0 via p erturbation of analytic semigroups In the general case where the mean-reversion parameter c  = 0 , the standard lift describ ed ab o ve must be adapted to incorp orate the state-dep enden t drift directly into the inﬁnite-dimensional dynamics. F ollowing the approach in [ 52 ], w e redeﬁne the system dynamics through a rank-one p erturbation of the underlying semigroup. 20 The P erturb ed Semigroup S ( t ) . W e introduce the p erturbed evolution on the extended space H . The drift term in the scalar SVIE ( 9.1 ) is giv en by R s 0 K ( s − r ) cy ( r ) d r . Recalling that the lift of the constant direction is ξ K ∈ H and the observ ation is y ( r ) = Γ X ( r ) , the generator A is formally deﬁned as A ϕ = Aϕ + c ξ K Γ( ϕ ) , ϕ ∈ D ( A ) . Although Γ acts on H , the v ector ξ K b elongs to the extended space H . Consequently , this constitutes a rank-one p erturbation. Rigorously , the associated strongly con tinuous semigroup S ( t ) on H is deﬁned via the v ariation of constants formula [ 25 , Corollary 1.7]: S ( t ) x = S ( t ) x + c Z t 0 S ( t − s ) ξ K Γ( S ( s ) x ) d s, x ∈ H . (9.13) Thanks to the analytic smo othing prop ert y of S ( t ) , the integrand is w ell-deﬁned, and S ( t ) inherits the smo othing prop erties of the original semigroup. Ornstein-Uhlen b ec k Dynamics. W e deﬁne the con trol and diﬀusion op erators as B : = ξ K b and G : = ξ K g , resp ectiv ely . The lifted dynamics ( 9.7 ) can then b e rewritten in terms of the p erturbed generator as an Ornstein- Uhlen b ec k pro cess: d X ( t ) = A X ( t ) d t + B u ( t ) d t + G d W ( t ) , X (0) = ξ g . (9.14) This formulation eﬀectively absorbs the path-dep enden t drift of the SVIE in to the Mark ovian dynamics, restoring the structure required by the abstract control framew ork derived in Section 2 . Relation b et w een K and the Resolv ent Kernel K . The eﬀectiv e kernel go verning the input-output resp onse of the system ( 9.14 ) is no longer K , but the so-called r esolvent kernel K , deﬁned via the p erturbed semigroup: K ( t ) : = Γ S ( t ) ξ K . (9.15) By applying Γ to the v ariation of constan ts formula ( 9.13 ) on the left and ξ K on the right, and recalling that K ( t ) = Γ S ( t ) ξ K , we derive the scalar resolven t equation of the second kind linking the original kernel K and the new k ernel K : K ( t ) = Γ S ( t ) ξ K + c Z t 0 Γ S ( t − s ) ξ K Γ( S ( s ) ξ K = K ( t ) + c Z t 0 K ( t − s ) K ( s ) d s. (9.16) Since the Laplace transform of the conv olution is the pro duct of the Laplace transforms, we get ˆ K ( λ ) = ˆ K ( t ) + c ˆ K ( λ ) ˆ K ( λ ) i.e. ˆ K ( λ ) = ˆ K ( λ ) 1 − c ˆ K ( λ ) (9.17) Monotonicit y Assumption. The theory developed in subsection 9.1 (minimum energy analysis and the Γ -smo othing) can b e applied once the eﬀectiv e kernel is an admissible kernel. Hyp othesis 9.7 (Admissibilit y of the Resolven t Kernel) . W e assume that the parameter c and the original kernel K are such that the resolven t kernel K , deﬁned b y ( 9.16 ) , is completely monotone and satisﬁes the conditions of Deﬁnition 9.1 . In particular, K is non-negative and non-increasing. The following lemma ensures this hypothesis holds in the standard mean-reverting case. Lemma 9.8 (Complete Monotonicit y of the Resolven t Kernel) . L et K b e a c ompletely monotone kernel satisfying the L aplac e r epr esentation in Deﬁnition 9.1 . If c < 0 , then the r esolvent kernel K is also a c ompletely monotone kernel. Pr o of. The pro of relies on the characterization of completely monotone functions via their Laplace transforms. Recall that a function f : (0 , ∞ ) → R is completely monotone if and only if its Laplace transform ˆ f ( λ ) is a Stieltjes function . Since K is an admissible kernel, ˆ K ( λ ) is a Stieltjes function. F rom the resolven t equation ( 9.17 ) , we ha ve ˆ K ( λ ) = ˆ K ( λ ) 1+ α ˆ K ( λ ) = 1 1 ˆ K ( λ ) + α , with α : = − c > 0 , whic h is a Stieltjes function b y Remark A.4 . Hence its in verse Laplace transform K ( t ) is an admissible k ernel. 9.3 P opular k ernels in v olatility mo deling W e examine sp eciﬁc kernels widely used in volatilit y mo deling, as w ell as other applications, which satisfy Deﬁnition 9.1 [ 27 , 52 ]. Example 9.9 (Rough Riemann-Liouville and Gamma k ernels) . Let K ( t ) = t α − 1 e − β t Γ( α ) with α ∈ (1 / 2 , 1) and β ≥ 0 , whic h cov ers b oth Riemann-Liouville and gamma k ernels with α = H + 1 / 2 . Then it follows that K ( ∞ ) = 0 and η ∗ = 1 − α ∈ (0 , 1 / 2) with µ (d x ) = ( x − β ) − α Γ(1 − α )Γ( α ) I ( β , ∞ ) ( x )d x. Example 9.10 (Logarithmic kernel) . Let K ( t ) = log (1 + 1 /t ) , then K ( ∞ ) = 0 and η ∗ = 0 with µ (d x ) = 1 − e − x x d x 21 Example 9.11 (Finite-Spectrum Kernels) . Let K ( t ) = c 0 + P N i =1 c i e − λ i t with c 0 , c 1 , . . . , c N ≥ 0 , N ≥ 1 , and λ 1 , . . . , λ N > 0 . Then η ∗ = −∞ , K ( ∞ ) = c 0 and µ is giv en by µ (d x ) = P N i =1 c i δ λ i ( d x ) , where δ w denotes the Dirac measure concentrated in { w } . Example 9.12 (Shifted k ernel) . F or ev ery K satisfying Deﬁnition 9.1 and ε > 0 , the shifted kernel deﬁned via K ε := K ( · + ε ) fulﬁlls again De ﬁnition 9.1 with η ∗ = −∞ , K ε ( ∞ ) = K ( ∞ ) , and µ ε ≪ µ is given by µ ε (d x ) = e − εx µ ( d x ) . A Completely monotone kernels Deﬁnition A.1 (Completely Monotone F unctions) . A function f ∈ C ∞ ((0 , ∞ ); R ) is called c ompletely monotone if ( − 1) n f ( n ) ( t ) ≥ 0 , ∀ t > 0 , ∀ n ∈ N 0 . W e denote this class by C M . Theorem A.2 (Bernstein’s Theorem [ 46 , Theorem 1.4]) . A function f b elongs to C M if and only if ther e exists a non-ne gative R adon me asur e µ on [0 , ∞ ) such that f ( t ) = R [0 , ∞ ) e − xt µ (d x ) , ∀ t > 0 , with the inte gr al c onver ging for al l t > 0 . Deﬁnition A.3 (Stieltjes Class) . A function f : (0 , ∞ ) → R b elongs to the Stieltjes class S if it admits the represen tation f ( t ) = a t + b + R (0 , ∞ ) 1 x + t ν (d x ) , where a, b ≥ 0 and ν is a non-negative Borel measure on (0 , ∞ ) satisfying the integra bility condition R (0 , ∞ ) (1 + x ) − 1 ν (d x ) < ∞ . R emark A.4 . Let f ∈ S . Then the function 1 1 f + α b elongs to S for any α ≥ 0 by [ 46 , Theorem 7.3]. 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Optimal control of stochastic Volterra integral equations with completely monotone kernels and stochastic differential equations on Hilbert spaces with unbounded control and diffusion operators

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