By an additive process we mean a càdlàg process with independent increments, stochastic continuity, and initial value zero. On [ 0 , T ], let X be the R d -valued additive process with deterministic characteristics ( Γ t , C t , ν ), where Γ is continuous of finite variation, C t = ∫ 0 t σ s σ s ⊤ d s, and ν satisfies the Lévy condition of Assumption 2.1. We use the Lévy–Itô decomposition X t = Γ t + W t + J t , W t = ∫ 0 t σ s d B s , where J t = ∫ ( 0 , t ] × { | z | ≤ 1 } z N ˜ ( d s , d z ) + ∫ ( 0 , t ] × { | z | > 1 } z N ( d s , d z ) . The compensator is non-atomic at deterministic times, so replacing ( 0 , t ] by [ 0 , t ] in predictable representations causes no ambiguity.

Remark 2.4

The main results are formulated on the product Brownian–Poisson filtration. Under the additional nondegeneracy condition d = k, with σ t invertible for Lebesgue-a.e. t and measurable t ↦ σ t − 1 , the usual augmentation of the natural filtration of X coincides with it: the jumps of X recover the Poisson coordinate, and the Brownian coordinate is recovered from ∫ 0 t σ s − 1 d W s after subtracting drift and jumps.

Lemma 2.5

For each fixed r ∈ [ 0 , T ], X r ( ε t , z + ω ) = X r ( ω ) + z 1 { t ≤ r } for ν ⊗ P -a.e. ( t , z , ω ) . Consequently, for every bounded Borel function f and each fixed r ∈ [ 0 , T ], f ( X r ( ε t , z + ω ) ) = f ( X r ( ω ) + z 1 { t ≤ r } ) ν ⊗ P -a.e.

Proof.

Fix r ∈ [ 0 , T ] and set E n : = { z ∈ R 0 d : n − 1 < | z | ≤ n }. Let X r ( n ) be the truncated Lévy–Itô representative obtained by retaining only jumps with marks in E n in the jump part. For every n, the set [ 0 , T ] × E n has finite ν-measure. The finite-window add-one identity holds for the chosen representatives: (2.1) X r ( n ) ( ε t , z + ω ) = X r ( n ) ( ω ) + z 1 { t ≤ r } 1 E n ( z ) for ν ⊗ P-a.e. ( t , z , ω ), after intersecting the corresponding countable family of conull sets. Indeed, on a finite window the identity is pathwise for the Poisson configuration, and Lemma 2.2 transfers it to the fixed canonical representatives.

The truncated variables converge to the fixed Lévy–Itô representative X r in probability. The omitted small compensated part has squared L 2 ( P ) norm equal to ∫ [ 0 , r ] × { | z | ≤ n − 1 } | z | 2 ν ( d s , d z ) → 0 , and the omitted large-jump part is absent with probability tending to one because ν ( [ 0 , r ] × { | z | > n } ) ↓ 0. Choose a subsequence n j such that X r ( n j ) → X r P-a.s., and let A be the exceptional set. By Lemma 2.2, ε t , z + ω ∉ A for ν ⊗ P -a.e. ( t , z , ω ) . Thus, on a common ν ⊗ P-conull set, X r ( n j ) ( ω ) → X r ( ω ) , X r ( n j ) ( ε t , z + ω ) → X r ( ε t , z + ω ) . Passing to the limit in (2.1) and using 1 E n j ( z ) → 1 for every z ≠ 0 gives X r ( ε t , z + ω ) = X r ( ω ) + z 1 { t ≤ r } ν ⊗ P -a.e. The deterministic drift and Brownian coordinate are unchanged by ε t , z + , so this is the add-one identity for the full additive path. The assertion for bounded Borel functions follows by composition. □

For 0 ≤ s < t ≤ T, we write P s , t f ( x ) : = E [ f ( x + X t − X s ) ] , x ∈ R d , for bounded Borel f : R d → R, and denote by μ s , t the law of X t − X s . In the stationary Lévy case, Γ t = tγ, C t = tQ and ν ( d t , d z ) = d t ν 0 ( d z ), with constant γ ∈ R d and symmetric positive semidefinite Q, the evolution family reduces to a one-parameter convolution semigroup.

We next introduce the analytic classes used in the entropy identities. The class C is the standard Chafaï class for conditional Jensen arguments, and the admissible core records the positivity and integrability hypotheses entering the Itô entropy computation.

The assumptions are expressed in terms of the Brownian-driver derivative D B and the raw add-one difference defined by the add-one map. In the L 2 theory these derivatives are the graph-closure derivatives of Definition 2.7. The L 1 Clark–Ocone–Haussmann hypotheses below specify derivative integrability and identification of the representation integrands; Part II verifies them for concrete terminal classes. When a reference uses a mark-normalized jump derivative, the conversion to the present notation is obtained by multiplying by the mark; see Part II for the stationary normalization.

For a measurable functional F : Ω → R, we define the raw jump difference operator by Ψ t , z F ( ω ) : = F ( ε t , z + ω ) − F ( ω ) , ( t , z ) ∈ [ 0 , T ] × R 0 d , where ε t , z + ω denotes the path obtained by adding a jump of size z at time t on the jump coordinate. Lemma 2.2 makes Ψ F an intrinsic L 0 ( ν ⊗ P )-class. We write D t B F for the Gaussian Malliavin derivative, whenever it exists.

Definition 2.6

Let C prod be the linear span of bounded random variables of the form F = φ ( ∫ 0 T ⟨ h 1 ( t ) , d B t ⟩ , … , ∫ 0 T ⟨ h m ( t ) , d B t ⟩ , ∫ E T ρ 1 ( t , z ) N ( d t , d z ) , … , ∫ E T ρ ℓ ( t , z ) N ( d t , d z ) ) , where m , ℓ ≥ 0, h i ∈ L 2 ( [ 0 , T ] ; R k ) are deterministic, each ρ j is a bounded Borel function supported on a set of finite ν-measure, and φ ∈ C b 1 ( R m + ℓ ). Throughout, C b 1 means bounded with bounded first derivative. If vector-valued mark integrals are used, each component is listed as a separate coordinate, so the dimension of φ is the total number of displayed scalar coordinates. Here X T ( n ) denotes the Lévy–Itô variable obtained by retaining marks n − 1 < | z | ≤ n in both jump integrals and leaving the drift and Brownian terms unchanged, as in Lemma 2.5. Thus f ( X T ( n ) ), f ∈ C b 1 ( R d ), belongs to the core by expressing the retained jump part through bounded mark integrals.

Definition 2.7

Let ‖ F ‖ 1 , 2 ; B , N 2 : = ‖ F ‖ L 2 ( P ) 2 + E ∫ 0 T | D t B F | 2 d t + E ∫ E T | Ψ t , z F | 2 ν ( d t , d z ) . We denote by D B , N 1 , 2 the completion, modulo equality in L 2 ( P ), of the core C prod from Definition 2.6 under this graph norm. Thus every element is represented by an L 2 limit F, a Brownian derivative D B F ∈ L 2 ( [ 0 , T ] × Ω ; R k , d t ⊗ P ), and a jump difference Ψ F ∈ L 2 ( ν ⊗ P ) obtained as graph-norm limits of core approximants. The closedness statement for the raw Poisson add-one operator, recalled in Lemma 2.8, identifies this limit with F ( ε t , z + ω ) − F ( ω ) for a Borel representative of F. The pair ( D B , Ψ ) is jointly closable on the core: if F n → 0 in L 2 ( P ) and D B F n → U, Ψ F n → V in the two derivative spaces, then the closability of each component gives U = 0 and V = 0.

Lemma 2.8

The operator Ψ : C prod ⊂ L 2 ( P ) ⟶ L 2 ( ν ⊗ P ) , Ψ t , z F = F ( ε t , z + ⋅ ) − F , is closable. Its closed domain is understood with the graph norm induced by this operator. If F m belongs to this closed domain, F m → F in L 2 ( P ), and Ψ F m → U in L 2 ( ν ⊗ P ), then F belongs to the closed domain and admits a Borel representative, still denoted by F, such that U t , z ( ω ) = F ( ε t , z + ω ) − F ( ω ) in L 2 ( ν ⊗ P ) . Consequently the Ψ component of the graph-norm completion in Definition 2.7 is the raw add-one difference of the chosen representative.

Proof.

First consider a core sequence. Let G m ∈ C prod satisfy G m → G in L 2 ( P ) and Ψ G m → V in L 2 ( ν ⊗ P ). Choose a subsequence m j such that G m j → G P-a.s. and, after passing to a further subsequence if necessary, Ψ G m j → V ν ⊗ P-a.e. Let A be the exceptional P-null set for the first convergence. By the Mecke null-set transfer in Lemma 2.2, ε t , z + ω ∉ A for ν ⊗ P-a.e. ( t , z , ω ). Hence, on a common ν ⊗ P-conull set, G m j ( ε t , z + ω ) → G ( ε t , z + ω ) , G m j ( ω ) → G ( ω ) . For core variables, Ψ t , z G m j = G m j ( ε t , z + ω ) − G m j ( ω ) pointwise outside the fixed representative null set. Taking limits along the subsequence gives V t , z = G ( ε t , z + ω ) − G ( ω ) ν ⊗ P -a.e. If G = 0 and G m → 0, this identity gives V = 0, so Ψ is closable. It also identifies the graph limit of every core graph-convergent sequence with the raw add-one representative.

Now let F m be a sequence in the closed domain such that F m → F in L 2 ( P ) and Ψ F m → U in L 2 ( ν ⊗ P ). For each m, choose G m , j ∈ C prod with G m , j → F m and Ψ G m , j → Ψ F m in graph norm. Choose j ( m ) so large that ‖ G m , j ( m ) − F m ‖ L 2 ( P ) + ‖ Ψ G m , j ( m ) − Ψ F m ‖ L 2 ( ν ⊗ P ) < m − 1 . Then the core diagonal sequence G m : = G m , j ( m ) satisfies G m → F in L 2 ( P ) and Ψ G m → U in L 2 ( ν ⊗ P ). Applying the core case just proved identifies U with the raw add-one difference of a Borel representative of F, and shows that F belongs to the closed domain. □

Three domains are used. The graph domain D B , N 1 , 2 is the L 2 domain for the Brownian–Poisson projection formula and for terminal observables. The analytic entropy core A T , Φ ε combines bounded positivity with the L 1 jump-difference and entropy-remainder assumptions needed in derivative inequalities. The COH domain D P COH is the representation class on which the L 1 Clark–Ocone–Haussmann identification of predictable integrands is imposed.

For a convex function Φ : R + → R of class C 1 on ( 0 , ∞ ), define A Φ ( u , v ) : = Φ ( u + v ) − Φ ( u ) − Φ ′ ( u ) v , u > 0 , u + v > 0 . For nonnegative F with F , Φ ( F ) ∈ L 1 ( P ), define the Φ-entropy by Ent Φ ( F ) : = E [ Φ ( F ) ] − Φ ( E [ F ] ) . For Φ ( u ) = u log u, with 0 log 0 : = 0, we write Ent ( F ) : = E [ F log F ] − E [ F ] log E [ F ] . The exact identities below use the integrability assumptions in their stated domains.

Definition 2.9

Let C denote the class of functions Φ : R + → R such that Φ is continuous on R + , convex, of class C 2 on ( 0 , ∞ ), and either affine or else Φ ′ ′ ( u ) > 0 for all u > 0 and 1 / Φ ′ ′ is concave on ( 0 , ∞ ).

Remark 2.10

For convex Φ, Jensen’s inequality gives Ent Φ ( F ) ≥ 0 whenever the terms are integrable; if Φ is strictly convex on the convex hull of the essential range of F, equality holds only for a.s. constant F.

Definition 2.11

Fix Φ ∈ C and ε > 0. We denote by A T , Φ ε the class of all F T -measurable functionals F : Ω → ( 0 , ∞ ) such that: ε ≤ F ≤ M ⋆ a.s. for some M ⋆ < ∞ , F ∈ L 1 ( Ω , F T , P ) and admits a Brownian derivative representative D B F , Ψ F is the raw add-one difference, with Ψ F ∈ L 1 ( E T × Ω , ν ⊗ P ) , E [ ∫ 0 T | D t B F | 2 d t ] < ∞ , E [ ∫ 0 T Φ ′ ′ ( F ) | D t B F | 2 d t ] < ∞ , A Φ ( F , Ψ F ) ∈ L 1 ( E T × Ω , ν ⊗ P ) .

If Φ is affine, both the Φ ′ ′ -term and the A Φ -term vanish. Otherwise, the bound ε ≤ F ≤ M ⋆ and the unweighted Gaussian energy condition imply the weighted Gaussian condition, since Φ ′ ′ ( F ) | D t B F | 2 d t ≤ sup u ∈ [ ε , M ⋆ ] Φ ′ ′ ( u ) | D t B F | 2 d t as measures. We keep the weighted term in the definition because it is the Gaussian contribution appearing in the derivative inequality.

The class A T , Φ ε supports the derivative inequalities. Its assumptions encode the Gaussian energy, the jump remainder, and the positivity of the arguments of Φ and A Φ . The main entropy identity is formulated on a representation domain stated directly in terms of the predictable integrands of the conditional-expectation martingale.

Definition 2.12

Fix Φ ∈ C and ε > 0.

(i)

We denote by R T , Φ ε ( P ) the class of all F T -measurable functionals F : Ω → ( 0 , ∞ ) such that ε ≤ F ≤ M ⋆ a.s. for some M ⋆ < ∞, and the martingale M t : = E [ F | F t ] , 0 ≤ t ≤ T , admits a predictable representation M t = E [ F ] + ∫ 0 t ⟨ θ s , d B s ⟩ + ∫ [ 0 , t ] × R 0 d ξ s , z N ˜ ( d s , d z ) for a predictable R k -valued process θ, identified up to d t ⊗ P-null sets, and a predictable real-valued function ξ on [ 0 , T ] × R 0 d × Ω, identified up to ν ⊗ P-null sets, where the stochastic integrals are integrable martingales on [ 0 , T ]. We require these integrands to satisfy M t − + ξ t , z > 0 for ν ⊗ P -a.e. ( t , z , ω ) , and E [ ∫ 0 T Φ ′ ′ ( M t − ) | θ t | 2 d t ] < ∞ , E [ ∫ [ 0 , T ] × R 0 d A Φ ( M t − , ξ t , z ) ν ( d t , d z ) ] < ∞ .

Remark 2.13

Definition 2.12(i) records the positivity and integrability hypotheses used in the martingale entropy balance. For bounded functionals the product L 2 predictable representation property (PRP) gives a square-integrable representation, and Lemma 2.15 identifies its integrands with those of any finite-horizon integrable representation. Thus all representation energies depend only on the corresponding d t ⊗ P- and ν ⊗ P-classes. The condition M t − + ξ t , z > 0 is included because A Φ ( u , v ) requires u > 0 and u + v > 0. When ξ = p ( Ψ F ), Lemma 2.24 gives M t − + ξ t , z = p ( F ( ε t , z + ⋅ ) ), so positivity of the canonical representative of F transfers to the compensator class.

The following results give the representation inputs for the entropy balance. The Brownian projection part is the classical Clark–Ocone mechanism [5, 12, 14], while the Poisson projection and add-one duality are based on the martingale representation and Mecke calculus for Poisson random measures [9, 10]. The finite-window localization, density, and closedness arguments are given in the proofs below.

Proposition 2.14

Let M = ( M t ) 0 ≤ t ≤ T be a square-integrable martingale on the usual augmented Brownian–Poisson product filtration. Then there exist predictable integrands θ and ξ such that M t = M 0 + ∫ 0 t ⟨ θ s , d B s ⟩ + ∫ [ 0 , t ] × R 0 d ξ s , z N ˜ ( d s , d z ) , with θ ∈ L 2 ( [ 0 , T ] × Ω ; R k , d t ⊗ P ) , ξ ∈ L 2 ( ν ⊗ P ) . The two stochastic-integral martingales are orthogonal, and the Itô isometry gives E [ ( M T − M 0 ) 2 ] = E ∫ 0 T | θ t | 2 d t + E ∫ [ 0 , T ] × R 0 d | ξ t , z | 2 ν ( d t , d z ) . The Brownian integrand θ and the jump integrand ξ are unique in these L 2 equivalence classes.

Proof.

The Brownian and compensated-Poisson stochastic integrals are continuous and purely discontinuous martingales, respectively, and hence are orthogonal. Their isometries show that the map sending a pair of square-integrable predictable integrands to the sum of its terminal integrals is an isometry. Its range H is therefore a closed subspace of centered L 2 ( F T ).

Fix a finite-ν window A. Let U be a bounded Brownian-coordinate variable and V a bounded variable measurable with respect to N | A . In their coordinate filtrations, the classical Brownian and Poisson representation theorems give U t = E [ U ] + ∫ 0 t ⟨ h s , d B s ⟩ , V t = E [ V ] + ∫ A ∩ ( [ 0 , t ] × R 0 d ) k s , z N ˜ ( d s , d z ) , where U t and V t are the coordinate conditional-expectation martingales; see [12] and [10, Theorem 3.1]. Independence of the coordinates and of the Poisson restrictions to A and A c gives E [ UV | F t ] = U t V t . The coordinate martingales retain their martingale property in the product filtration. Since [ U , V ] = 0, integration by parts yields U t V t − E [ U ] E [ V ] = ∫ 0 t V s − ⟨ h s , d B s ⟩ + ∫ A ∩ ( [ 0 , t ] × R 0 d ) U s − k s , z N ˜ ( d s , d z ) . The integrands are square-integrable because U t and V t are bounded. Thus UV − E [ UV ] ∈ H. Finite linear combinations of these products are dense in the finite-window product L 2 -space: bounded coordinate indicators generate its sigma-field, and the monotone-class theorem followed by truncation gives density. Closedness of H therefore gives the representation on each finite window.

Choose increasing finite-ν windows A n ↑ E T , and let G n be the sigma-field generated by the full Brownian coordinate and N | A n . For Y ∈ L 2 ( F T ), set Y n = E [ Y | G n ]. Then Y n → Y in L 2 , since ⋁ n G n generates F T up to completion. Let ( θ n , ξ n ) represent Y n − E [ Y ], extending ξ n by zero outside A n . Orthogonality and the two isometries give ‖ θ n − θ m ‖ L 2 ( d t ⊗ P ) 2 + ‖ ξ n − ξ m ‖ L 2 ( ν ⊗ P ) 2 = ‖ Y n − Y m ‖ L 2 ( P ) 2 . The integrands therefore converge in their completed predictable L 2 -spaces, and their integrals represent Y − E [ Y ]. Taking Y = M T and conditioning at time t proves the assertion; F 0 is trivial up to null sets. The same isometry gives the stated energy identity and uniqueness. □

Lemma 2.15

Let ( θ , ξ ) and ( θ ′ , ξ ′ ) be predictable integrands such that the stochastic integrals appearing below are integrable martingales on [ 0 , T ]. If ∫ 0 t ⟨ θ s , d B s ⟩ + ∫ [ 0 , t ] × R 0 d ξ s , z N ˜ ( d s , d z ) = ∫ 0 t ⟨ θ s ′ , d B s ⟩ + ∫ [ 0 , t ] × R 0 d ξ s , z ′ N ˜ ( d s , d z ) for every 0 ≤ t ≤ T, up to indistinguishability, then θ = θ ′ d t ⊗ P -a.e. , ξ = ξ ′ ν ⊗ P -a.e. Consequently, if a bounded martingale has a finite-horizon representation in the sense of Definition 2.12(i), its integrands agree with the integrands of the square-integrable representation of Proposition 2.14 on the corresponding null-set classes.

Proof.

Subtract the two representations and write the difference as Z = Z c + Z d , where Z t c = ∫ 0 t ⟨ θ s − θ s ′ , d B s ⟩ , Z t d = ∫ [ 0 , t ] × R 0 d ( ξ s , z − ξ s , z ′ ) N ˜ ( d s , d z ) . The local martingale Z c is continuous and Z d is purely discontinuous. Since Z ≡ 0, uniqueness of the decomposition of a local martingale into its continuous and purely discontinuous parts gives Z c ≡ 0 and Z d ≡ 0; see, for instance, Jacod–Shiryaev [8, Theorem I.4.18]. Localizing the continuous stochastic integral and applying the Brownian isometry yield E ∫ 0 T ∧ τ n | θ s − θ s ′ | 2 d s = 0 for a reducing sequence ( τ n ), and monotone convergence gives θ = θ ′ d t ⊗ P-a.e.

We first verify that no two Poisson atoms share a time coordinate. On a finite-ν window A with ν ( A ) > 0, conditional on N ( A ) = m, the points have the law of m independent samples from ν | A / ν ( A ). Its time marginal is atomless by Assumption 2.1, so any two sampled times agree with probability zero. The assertion is immediate if ν ( A ) = 0. Applying this argument to increasing finite windows covering E T gives a single event of probability one on which every pair of distinct atoms has distinct times.

For the jump part, Z d ≡ 0 implies that every jump is zero. The jump formula defining compensated random-measure integrals [8, Definition II.1.27], together with time-diffuseness and the preceding time-simplicity, gives Δ Z t d = ξ t , z − ξ t , z ′ at each atom ( t , z ) of N. With H t , z : = 1 ∧ | ξ t , z − ξ t , z ′ | , we therefore have ∫ E T H t , z N ( d t , d z ) = 0 a.s. The compensation formula gives E ∫ E T H t , z ν ( d t , d z ) = E ∫ E T H t , z N ( d t , d z ) = 0 , and hence ξ = ξ ′ ν ⊗ P-a.e. The final assertion follows because a bounded terminal martingale is square-integrable, so Proposition 2.14 supplies an L 2 representation and the preceding uniqueness argument compares it with the given integrable representation. □

Lemma 2.16

Let K : [ 0 , T ] × R 0 d × Ω → R be bounded and P ⊗ B ( R 0 d )-measurable, where P is the predictable σ-field of the usual augmented product filtration. Then K t , z ( ε t , z + ω ) = K t , z ( ω ) for ν ⊗ P -a.e. ( t , z , ω ) .

Proof.

First work with the unaugmented product filtration, denoted by ( F t 0 ). Its predictable sigma-field is generated by sets of the form ( s , u ] × A, with A ∈ F s 0 , together with the time-zero sets. For an elementary field K t , z ( ω ) = 1 ( s , u ] ( t ) 1 S ( z ) 1 A ( ω ) , A ∈ F s 0 , S ∈ B ( R 0 d ) , the equality is pointwise apart from the harmless endpoint t = 0: if t ∈ ( s , u ], then the map ε t , z + adds a Poisson atom strictly after time s and leaves the Brownian coordinate unchanged, so it does not change the F s 0 -measurable event A. The time-zero exceptional set is ν-null because the deterministic compensator has no fixed time atoms under the standing product realization. A monotone-class argument extends the identity to all bounded unaugmented predictable fields.

For the usual augmentation, the raw-version argument in the proof of Lemma 2.19 gives a bounded unaugmented predictable field K 0 and a P-null set D such that K t , z = K t , z 0 for all t > 0 , z whenever ω ∉ D. Enlarge D to a raw measurable null set. Lemma 2.2, applied to 1 D , gives ε t , z + ω ∉ D for ν ⊗ P-a.e. ( t , z , ω ). Thus K and K 0 agree both before and after insertion, and the raw identity proves the claim. □

Lemma 2.17

Let F be bounded and F T -measurable, and let K be bounded predictable with support contained in a deterministic set A ⊂ E T satisfying ν ( A ) < ∞. Then E [ F ∫ E T K t , z N ˜ ( d t , d z ) ] = E ∫ E T Ψ t , z F K t , z ν ( d t , d z ) .