Technology Category
Reasoning under Uncertainty: Stochastic OptimizationPlanning, Routing, and Scheduling: Optimization of Spatio-temporal SystemsConstraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
Measure-Valued CARMA Processes
🤖 AI Summary
This paper addresses the modeling and analysis of measure-valued CARMA processes in Banach spaces driven by Lévy subordinators. To overcome the limitation of classical CARMA frameworks—confined to Euclidean spaces—it extends CARMA to the Banach space of finite signed measures, constructs an analytic weak solution, and rigorously establishes its existence and invariance of the positive measure cone. Methodologically, it integrates Banach-space operator theory, linear state-space modeling, and Lévy-driven stochastic analysis. The contributions include: (i) necessary and sufficient verifiable conditions for stationarity; (ii) explicit closed-form expressions for first- and second-order moment structures; and (iii) a systematic compatibility framework linking the new model to classical real-valued and Hilbert-space CARMA processes. This work provides a novel paradigm for conically constrained dynamic modeling of spatiotemporal random fields.
Technology Category
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
Existence and cone-invariance of Banach-space CARMA processes
Moment structure and stationarity conditions for CARMA processes
Modeling spatio-temporal fields via measure-valued CARMA dynamics
Innovation
Methods, ideas, or system contributions that make the work stand out.
Measure-valued CARMA processes in Banach spaces
Analytically weak solution in signed measures
Cone-invariance for positive measure dynamics
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