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Measure-Valued CARMA Processes

Fred Espen Benth, Sven Karbach, Asma Khedher

TL;DR

This work extends continuous-time autoregressive moving-average (CARMA) theory to measure-valued processes on Banach spaces by defining measure-valued CARMA processes as cone-invariant, analytically weak solutions of linear state-space models driven by Lévy subordinators. It develops the existence, positivity, and moment structure (first and second moments) of these processes, and establishes explicit stationarity conditions via two-sided Lévy input and completely monotone transfer functions. The paper then specializes to the Banach space of finite measures, highlights absolutely continuous cases, and provides concrete parameter-sets and examples including CARMA fields and ambit-field connections, thereby enabling spatio-temporal modeling of renewable-energy dynamics and related financial applications. Finally, it presents methods for evaluating expectation functionals and performing measure changes (Esscher transforms) to support risk-adjusted pricing of forward and derivative contracts within this flexible framework. Overall, the results offer a rigorous, scalable approach to infinite-dimensional CARMA modeling with direct relevance to climate, energy markets, and stochastic spatial systems.

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.

Measure-Valued CARMA Processes

TL;DR

This work extends continuous-time autoregressive moving-average (CARMA) theory to measure-valued processes on Banach spaces by defining measure-valued CARMA processes as cone-invariant, analytically weak solutions of linear state-space models driven by Lévy subordinators. It develops the existence, positivity, and moment structure (first and second moments) of these processes, and establishes explicit stationarity conditions via two-sided Lévy input and completely monotone transfer functions. The paper then specializes to the Banach space of finite measures, highlights absolutely continuous cases, and provides concrete parameter-sets and examples including CARMA fields and ambit-field connections, thereby enabling spatio-temporal modeling of renewable-energy dynamics and related financial applications. Finally, it presents methods for evaluating expectation functionals and performing measure changes (Esscher transforms) to support risk-adjusted pricing of forward and derivative contracts within this flexible framework. Overall, the results offer a rigorous, scalable approach to infinite-dimensional CARMA modeling with direct relevance to climate, energy markets, and stochastic spatial systems.

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.
Paper Structure (30 sections, 13 theorems, 161 equations)

This paper contains 30 sections, 13 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. CARMA Processes and Linear-State Space Models
  3. Measure-valued Processes
  4. Applications to Modeling Dynamics of Renewable Energy Markets
  5. Climate Data
  6. Flow Forwards
  7. Power Purchase Agreements and Renewable Portfolios
  8. Layout of the article
  9. Linear State-Space Models in Banach Spaces
  10. Lévy Processes in Banach Spaces
  11. Lévy Processes on Cones in Banach Spaces
  12. Linear State-Space Models in Banach Spaces
  13. Lévy-Driven Banach-Valued CARMA Processes
  14. Stationary CARMA Processes on Cones in Banach Spaces
  15. Lévy-Driven Measure-Valued CARMA Processes
  16. ...and 15 more sections

Key Result

Theorem 2.1

Let $K$ be a proper convex cone of a separable Banach space $B$. Let $(L_t)_{t\geq 0}$ be a Lévy process in $B$ with characteristic triplet $(\gamma, Q, \ell)$. Assume the following three conditions: Then the process $(L_t)_{t\geq 0}$ is a subordinator.

Theorems & Definitions (33)

  • Theorem 2.1
  • Definition 2.1: Linear State-Space Model in Banach Spaces
  • Definition 2.2: cf. Lem98
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • ...and 23 more