Paper: arXiv 2505.08852

Authors: Fred Espen Benth, Sven Karbach, Asma Khedher

Abstract

In this paper, we examine continuous-time autoregressive moving-average (CARMA) processes on Banach spaces driven by Lévy 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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical foundation for measure-valued CARMA processes on Banach spaces driven by Lévy subordinators, with high mathematical complexity. However, it is purely theoretical with conceptual financial and energy market applications, lacking empirical backtesting or real-world data implementation.

Research Flowchart

  flowchart TD
    A[Research Goal: Define positive measure-valued CARMA processes on Banach spaces] --> B[Inputs: Lévy subordinators & Linear State-Space Operators]
    B --> C[Formulate Analytically Weak Solutions in Signed Measure Spaces]
    C --> D[Derive Cone-Invariance & Moment Structures]
    D --> E[Establish Explicit Stationarity Conditions]
    E --> F[Outcomes: Flexible CARMA Framework for Spatio-Temporal Fields & Commodities]

Cited References (31)

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  4. Pricing of commodity derivatives on processes with memory (2020)
  5. Multivariate continuous-time autoregressive moving-average processes on cones (2023)
  6. An infinite-dimensional affine stochastic volatility model (2022)
  7. Infinite-dimensional W ishart processes (2024)
  8. Derivatives pricing in energy markets: an infinite-dimensional approach (2015)
and 23 more
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0 of 31 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.