Measure-Valued CARMA Processes

Asma Khedher; Fred Espen Benth; Sven Karbach

REVIEW 5 major objections 5 minor 45 references

Measure-Valued CARMA Processes

T0 review · 5 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Measure-valued CARMA processes can be defined as positive, jump-driven, stationary measure flows with explicit Laplace transforms.

desk verdict A genuinely new measure-valued CARMA framework with a coherent L1 core, but the paper's own parameter examples fail the cone-invariance and stationarity assumptions for p≥2 and the current draft needs major revision. read the letter →

arxiv 2505.08852 v1 pith:WBAGV2EB submitted 2025-05-13 math.PR q-fin.MF

Fred Espen Benth , Sven Karbach , Asma Khedher This is my paper · ORCID

classification math.PRq-fin.MF MSC 60G5160G5760G1060H1547D06

keywords measure-valuedCARMALévysubordinatorlinearstate-spacemodelconeinvariancestationarityPettisintegraloperator-valuedtransferfunctionspatio-temporalrandomfields

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

CARMA processes are continuous-time analogues of ARMA time series, and this paper moves them from real or vector values to measures: the state is a density or measure on a spatial domain, and the dynamics are a linear state-space equation driven by a Lévy subordinator. The paper shows that, when the companion operator and output operators preserve positivity, the solution exists as an analytically weak process, stays in the cone of positive measures, and has closed-form Laplace transforms, moments, and autocovariance. It gives a stationarity condition in terms of complete monotonicity of an operator-valued transfer function, and exhibits concrete parameter sets including convolution operators and Poisson-type noise. The point is a tractable jump-driven, cone-valued model for spatio-temporal aggregates such as regional renewable production, flow-forward prices, and power purchase agreements.

What carries the argument

The central object is the companion block operator matrix A_p, built from the operator polynomial P(λ)=Iλ^p−$A_1λ^{{p−1}}$−...−A_p, together with the output polynomial Q(λ)=C_0+C_1λ+...+C_qλ^q. The transfer function λ ↦ Q(λ)P(λ)^{-1} carries the argument: its complete monotonicity with respect to the cone π(K) of positive operators is exactly the condition that makes the stationary kernel K(t) positive, so that the convolution against a Lévy subordinator stays inside the cone. Existence is handled through Pettis stochastic integrals, a weak notion of stochastic integration valid in general Banach spaces, and through quasi-positive semigroups that preserve the cone of positive measures.

What would settle it

Take Section 3.3.1's convolution example with p=2, q=0, C0=I, pick two nonnegative convolution kernels, and compute the kernel K(t) by inverting the Fourier integral in (2.29); if K(t) is not a positive operator for some t while all the paper's stated positivity hypotheses hold, Proposition 2.5's sufficiency claim is false.

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Extended reading notes

Core claim

The paper's central claim is that a pure-jump measure-valued CARMA(p,q) process can be defined as the analytically weak solution of a linear state-space model in a Banach space of measures: dX_t = A_p X_t dt + E_p dL_t, Y_t = C_q X_t, with A_p the companion block operator matrix of P(λ)=Iλ^p−$A_1λ^{{p−1}}$−...−A_p, E_p injecting noise into the last coordinate, and C_q the output operator built from C_0,...,C_q. The process (t,A) ↦ ∫_A Y_t(x)λ(dx) is measure-valued, and under quasi-positivity of A_p, positivity of C_j and E, and a Lévy subordinator L, it exists and remains in the cone of nonnegative measures. The paper derives the Laplace transform of the transition semigroup, explicit first and second moments, conditional covariance and autocovariance, and proves that a unique stationary cone-valued version exists exactly when the operator-valued transfer function λ ↦ Q(λ)P(λ)^{-1} is completely monotone with respect to the positive-operator cone, in which case Y_t = ∫_{−∞}^{∞} K(t−s) dL_s with K given by a Fourier integral. It also connects the construction to classical CARMA, Hilbert-space CARMA, CARMA random fields, and ambit fields.

Load-bearing premise

The load-bearing premise is that the companion operator A_p generates a positivity-preserving semigroup and that the operator-valued transfer function λ↦Q(λ)P(λ)^{-1} is completely monotone in the positive-operator order, a condition the paper assumes abstractly rather than verifying for its concrete convolution examples.

Editorial extensions

If this is right

Under the positivity assumptions, every measure-valued CARMA(p,q) process remains in the cone of positive densities for all times, so it can model non-negative quantities like capacity factors and forward prices.
The Laplace transform of any spatial average ∫_A Y_t(x)λ(dx) has a closed form, so aggregate quantities inherit explicit exponential-affine dynamics.
First and second moments, conditional covariance, and autocovariance are given by explicit operator integrals, enabling moment-based statistical inference.
When the transfer function is completely monotone, the process has a unique stationary version Y_t=∫_{−∞}^{∞}K(t−s)dL_s, so all temporal dependence is captured by the kernel K.
Evaluating the process on indicator functions yields multi-parameter CARMA random fields, and evaluating it on spatial sets recovers classical real-valued CARMA processes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension the paper leaves implicit is that the Laplace-transform formalism would support characteristic-function estimation of measure-valued CARMA parameters from aggregate spatial observations, without observing the full measure field.
The complete-monotonicity condition is probably restrictive; even natural positive convolution kernels can be checked case by case, and failures would rule out stationarity (not positivity) for those parameter sets.
Because the existence proof needs a separable space, arbitrary finite signed measures are not covered; an integration theory for non-separable measure-valued Lévy processes would be needed to reach the full space of finite signed measures advertised in the title.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The core of this paper is a real extension: measure-valued CARMA processes built from analytically weak solutions of linear state-space equations in L1 or in finite measures, with explicit Laplace transforms, moment formulas, a stationarity criterion in terms of complete monotonicity of an operator-valued transfer function, and an Esscher transform. That is worth taking seriously. The L1-based theory is mostly coherent, and the authors are honest about the non-separability of M(E) and about using finite-dimensional noise there.

The soft spots are concentrated in the examples and in a few definitions. Section 3.1 states that C0(E) is the dual of M(E); the duality runs the other way, C0(E)* = M(E), and the dual of M(E) under total variation is strictly larger. Since the paper then moves to L1, this is not fatal, but it should be corrected. Definition 2.2 is vacuous as written: requiring equality under all f in K* forces x=y, so the quasi-monotonicity condition is never tested. The intended condition is presumably the standard one with a single f and f(x-y)=0; as it stands, Proposition 2.3(i) does not actually establish cone invariance.

The bigger problem is the parameter sets. In §3.3.1 all Ai are positive convolution operators; then Ap is a nonnegative companion matrix with no negative feedback, its spectral bound is not negative, and the stationary representation (2.28) cannot converge. In §3.4.2 the authors take Ai = -ai I with ai > 0. For p=2 the (2,1) block is -a2 I, so applying e^{tAp} to (α,0) produces a negative second component for small t; the semigroup is not positive on the product cone. Demanding a2 ≤ 0 for positivity makes the spectral bound nonnegative, so stationarity is lost. Thus the paper contains no non-degenerate p≥2 tuple satisfying both cone-invariance and stationarity, and the advertised CARMA(2,q) examples do not fall under the theory as stated. This is a genuine gap, not a missing computation. There are also typos in (2.12), Proposition 3.2(iii), and Proposition 4.2, and the covariance operator Q in Proposition 2.6 is ambiguous for a pure-jump subordinator.

Who should read it: people building infinite-dimensional CARMA or measure-valued affine models, and energy/weather modelers. The ideas are useful, but the current examples overstate what is established. It deserves a serious referee, with the expectation of substantial revision: fix the definitions, correct the dual-space claim, and either exhibit feasible p≥2 parameters or state clearly that the method currently yields only p=1-type examples. I would not desk-reject it.

Referee Report

5 major / 5 minor

Summary. The paper proposes a Banach-space framework for Lévy-driven CARMA processes with values in convex cones, with primary focus on measure-valued processes. It defines the state process as an analytically weak solution of a linear state-space model driven by a Lévy subordinator, proves existence and cone-invariance under quasi-positivity conditions, derives Laplace transforms and moment formulas, and gives stationarity conditions via complete monotonicity of the operator-valued transfer function Q(λ)P(λ)^{-1}. The authors then specialize to the space L1(E) of absolutely continuous finite signed measures and to M+(E) with finite-dimensional noise, calling the output a pure-jump measure-valued CARMA(p,q) process, and discuss applications to renewable energy and flow forwards.

Significance. If the technical gaps were repaired, the framework would be a useful contribution: it provides explicit variation-of-constants representations, Laplace transforms, moment formulas, and a clean link between measure-valued CARMA processes and ambit fields. The analytical-weak-solution approach for cone-valued state processes driven by subordinators is a natural and potentially generalizable idea, and the explicit formulas in Propositions 2.4, 3.1, and 4.1 would be valuable for applications. However, the impact is currently limited by several load-bearing issues: the quasi-monotonicity definition as stated is vacuous, the duality statement for M(E) is reversed, and the paper does not exhibit a feasible stationary cone-valued parameter set for p≥2. These issues must be resolved before the advertised class of stationary positive measure-valued CARMA processes can be considered established.

major comments (5)

[Section 2.2, Definition 2.2] As written, the quasi-monotonicity condition is vacuous for the cones considered in the paper. The definition requires that for all x,y in the domain, if x≤K y and ⟨f,x⟩=⟨f,y⟩ for all f∈K*, then ⟨f,Ax⟩≤⟨f,Ay⟩ for all f∈K*. For K=L1_+(E), the premises force x=y, so the implication carries no information. Consequently, the citation to [32, Theorem 1] in support of St(K^p)⊆K^p is not justified, and the cone-invariance asserted in Proposition 2.1, Proposition 2.3(i), and Definition 3.2 is not established. The standard formulation, in which one requires f(Ax)≤f(Ay) for individual f∈K* satisfying f(x)=f(y), should replace the current statement.
[Section 3.1] The statement “C0(E) is the dual of M(E)” reverses the Riesz–Markov–Kakutani theorem: for a locally compact Polish space E, C0(E)* is isometrically isomorphic to M(E), while the dual of M(E) under the total variation norm is not C0(E). This error matters because Proposition 3.2 computes Laplace transforms and moments only for g∈C0(E), whereas the appropriate dual cone in the absolutely continuous case is L∞_+(E). The weak-solution and Laplace-transform claims are therefore established only for a proper subspace of admissible test functions, not for the full dual cone needed in Definition 3.1 and Proposition 3.1.
[Sections 3.3.1 and 3.4.2; Proposition 2.5] The manuscript never exhibits a non-degenerate parameter tuple satisfying both cone-invariance (Definition 3.2) and stationarity (Proposition 2.5) for p≥2, and the two concrete families are mutually incompatible. In Section 3.3.1 the convolution blocks Ai and Cj are positive, so Ap has a nonnegative spectral bound; the stationary representation (2.28) cannot converge and the transfer Q(λ)P(λ)^{-1} cannot be completely monotone in the required sense. In Section 3.4.2 the choice Ai=−aiI with ai>0 gives, for p=2, the block (Ap)_{2,1}=−a2I; applying the semigroup to (α,0) with α∈L1_+\{0} yields second component −t a2 α+O(t^2) for small t, which is outside the positive cone. Setting a2≤0 to restore quasi-positivity destroys Hurwitz stability. The paper must either provide feasible parameters for stationary cone-valued CARMA(p,q) with p≥2 or explicitly separate the stationarity result for output positivity from the state-cone-invariance requirement in Definition 3.2.
[Proposition 2.5] The proof of the stationary representation cites exponential bounds ‖K(±u)‖≤η e^{−w(S)|u|} from [23], but the standing assumption (2.27) only excludes singularities on the imaginary axis; it does not imply that the spectral bound of Ap is negative, nor that the kernel is exponentially integrable. Thus the claimed “if and only if” between complete monotonicity of Q(λ)P(λ)^{-1} and the existence of a stationary K-valued solution is not fully demonstrated. In particular, the proof should show that complete monotonicity, together with the rational decay of the transfer function, forces the poles into the open left half-plane and yields the required integrability of the kernel.
[Proposition 2.6] The covariance formulas in Proposition 2.6 are inconsistent with the driving noise used in the paper. Theorem 2.1 and Proposition 2.1 concern pure-jump subordinators with Q=0, whereas Proposition 2.6 assumes a square-integrable Lévy process with covariance operator Q and refers to Proposition 2.1 for its existence. For a jump-driven process the conditional covariance of Yt given Fs is not simply CqΣ_{t,s}Cq* with Σ built from a Gaussian covariance operator; the jump part contributes an additional integral of squares with respect to the Lévy measure, as Proposition 2.4 itself shows. Moreover, in a general Banach space such as L1(E), the existence of a covariance operator in this form is not automatic. The proposition should be restated under assumptions that are compatible with the pure-jump setting, or its formulas should be corrected.

minor comments (5)

[Definition 3.2] The sentence defining the measure-valued CARMA process contains a duplicated article: “a a pure-jump measure-valued CARMA(p,q) process.”
[Equation (2.12)] The displayed Laplace transform in Proposition 2.2(i) has unbalanced parentheses and a misplaced “du” after the second exponential; the plus sign before “exp” also appears inconsistent with the product formula derived in the preceding lines.
[Proposition 3.2(iii)] The variance formula is missing the time integral: as printed, Var[⟨g,Y_t⟩] is time-independent and does not match Proposition 2.4, which contains ∫0^t∫K ... ds.
[Example 3.3.2] In the Laplace transform for the L1_+(Rd)-valued subordinator, the displayed expression contains e^{⟨g,uφ(·−y)⟩} where the exponent should be −⟨g,uφ(·−y)⟩ for a Laplace transform with g∈L∞_+; as printed the expression is not a Laplace transform.
[Definition 3.1] The notation A:D(A)⊂L1_+(E)^p→L1_+(E)^p is problematic for a linear operator: a linear generator has a domain that is a subspace of L1(E)^p, not a subset of the positive cone, unless the operator is trivial.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's construction and theorems follow from explicit operator-semigroup, Lévy, and complete-monotonicity assumptions, with no fitted inputs or self-citation chain carrying the central claim.

full rationale

The central objects are defined rather than empirically predicted: Definition 3.2 calls (t,A) ↦ ∫_A Y_t(x)λ(dx) a measure-valued CARMA(p,q) process when Y_t = C_q X_t and X_t solves dX_t = A_p X_t dt + E_p dL_t, which is a definition, not a derived conclusion. Existence and cone-invariance are established in Proposition 2.1 through explicit Pettis-integrability arguments from the Lévy subordinator's characteristics and the quasi-monotonicity/positivity of the semigroup; no parameter is fitted to data and later renamed as a prediction. The Laplace-transform formulas in Proposition 2.2 and Proposition 3.2 are direct computations from the Lévy-Khintchine representation and the variation-of-constants formula. Stationarity in Proposition 2.5 is stated as an iff condition involving complete monotonicity of Q(λ)P(λ)^{-1}, with the kernel representation derived from the spectral/semigroup apparatus; this is an explicitly assumed hypothesis, not a hidden input recycled as an output. The self-citations to [5] and [10] are used as ordinary mathematical references; Proposition 2.3 does cite the same authors' Lemma 3.13 for an auxiliary converse statement, but that statement is not the load-bearing route to existence or stationarity, and the cited work is an external published theorem rather than an unverified premise. Potential technical concerns, such as the wording of Definition 2.2 making the quasi-monotonicity condition formally vacuous or the lack of a concrete feasible parameter tuple satisfying both cone-invariance and stationarity in Section 3.3, are correctness or feasibility issues, not circularity: they do not amount to a claim reducing by construction to its own assumptions. The derivation chain is therefore self-contained apart from standard mathematical citations, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim rests on standard stochastic analysis results (Lévy-Khintchine, stochastic integration theory, semigroup generation, Bernstein's theorem) and on modeling assumptions about the driving subordinator and the operators. No constants are fitted to data; the parameters A_p, E_p, C_q and the Lévy characteristics are inputs. The restriction to L1-absolutely continuous measures or finite-dimensional noise is an explicit workaround for the non-separability of M(E) under total variation norm, acknowledged in Section 3.1.

assumptions (8)

standard math Lévy-Khintchine representation for Banach-valued Lévy processes with characteristic triplet (γ,Q,ℓ)
Invoked in Section 2.1, equation (2.1), and used throughout for subordinator Laplace transforms.
standard math Theorem 2.1 from Rocha-Arteaga (2006) characterizing subordinators in cones of Banach spaces
Used in Theorem 2.1 to define the subordinator class that drives the state equation.
standard math Stochastic integration theory for Banach-valued Lévy processes (Riedle and van Gaans 2009, Theorems 5.2 and 7.2)
Used in the proof of Proposition 2.1 to justify the Pettis integrals in the variation-of-constants formula.
standard math Quasi-monotone generators produce positive semigroups (Lemmert and Volkmann 1998, Theorem 1)
Used in Section 2.3 to convert quasi-monotonicity of A_p into cone invariance of S_t.
standard math Operator Bernstein theorem for completely monotone functions (Arendt 1984, Theorem 5.5)
Used in the proof of Proposition 2.5 to pass from complete monotonicity of Q(λ)P(λ)^{-1} to positivity of the convolution kernel.
domain assumption The driving Lévy process is a subordinator concentrated on the cone K, with characteristic triplet satisfying Theorem 2.1 (iii)
This is the model class chosen for cone-valued CARMA; it rules out Gaussian noise and negative jumps.
domain assumption The companion operator A_p generates a strongly continuous quasi-positive semigroup and the transfer function Q(λ)P(λ)^{-1} is completely monotone
This is the main parametric condition for existence of positive solutions and stationarity; it is assumed in Propositions 2.3 and 2.5 and Definition 3.2.
domain assumption In the non-separable space of finite signed measures, either the measures are absolutely continuous with respect to a σ-finite λ, or the driving noise is finite-dimensional
Introduced in Section 3.1 to ensure the separability required by the Banach-valued Lévy theory.

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Pith. "Pith review of Measure-Valued CARMA Processes." pith.science (2026). https://pith.science/paper/WBAGV2EB

@misc{pith2026250508852,
  author       = {Pith},
  title        = {Pith review of: Measure-Valued CARMA Processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WBAGV2EB}},
  note         = {Machine review of arXiv:2505.08852}
}

read the original abstract

In this paper, we examine continuous-time autoregressive moving-average (CARMA) processes on Banach spaces driven by L\'evy subordinators. We show their existence and cone-invariance, investigate their first and second order moment structure, and derive explicit conditions for their stationarity. Specifically, we define a measure-valued CARMA process as the analytically weak solution of a linear state-space model in the Banach space of finite signed measures. By selecting suitable input, transition, and output operators in the linear state-space model, we show that the resulting solution possesses CARMA dynamics and remains in the cone of positive measures defined on some spatial domain. We also illustrate how positive measure-valued CARMA processes can be used to model the dynamics of functionals of spatio-temporal random fields and connect our framework to existing CARMA-type models from the literature, highlighting its flexibility and broader applicability.

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