Substochastic Path Operators for Refinement-Consistent Network Risks
- 1Department of Economics, University of Foggia, 71121 Foggia, Italy
- 2Department of Economics, Statistics and Business, Faculty of Technological and Innovation Sciences, Universitas Mercatorum, 00186 Rome, Italy
Abstract
A network risk associated with persistent exposure depends on both the history of the exposure and the points at which decisions can be made. These two sources of dependence must remain distinguishable when an arc is divided into shorter segments. Substochastic operators provide a finite-state description in which homogeneous subdivision preserves propagation, while changes in exposure and in the available continuations remain visible. Building on continuous semigroups and linear realization theory, this paper studies the interaction between refinement and the information required for network decisions. At each physical vertex, the pairing of reachable states with admissible continuation observations determines the dimension of an exact linear state. Its transitions respect arc subdivision when the physical decision and stopping locations are preserved. The same observations determine which partial paths can be discarded without worsening any permitted continuation. Worked finite-state networks demonstrate why a smaller prefix risk may lead to a larger final risk and how recovery, exposure ordering, and generator uncertainty affect that comparison. They also show that allowing uncertainty to change at an artificial breakpoint alters the decision problem even when local propagation remains consistent. These relationships connect physical exposure, state reduction, and path optimization within a formulation whose invariance and computational requirements can be stated explicitly.
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BibTeX
@article{grilli2026substochastic,
title = {{Substochastic Path Operators for Refinement-Consistent Network Risks}},
author = {Luca Grilli and Domenico Santoro and Giacinto Angelo Sgarro},
journal = {Axioms},
year = {2026},
month = oct,
doi = {10.3390/axioms15100775},
url = {https://www.mdpi.com/2075-1680/15/10/775},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Substochastic Path Operators for Refinement-Consistent Network Risks
AU - Luca Grilli
AU - Domenico Santoro
AU - Giacinto Angelo Sgarro
PY - 2026
DA - 2026/10/16
JO - Axioms
DO - 10.3390/axioms15100775
UR - https://www.mdpi.com/2075-1680/15/10/775
AB - A network risk associated with persistent exposure depends on both the history of the exposure and the points at which decisions can be made. These two sources of dependence must remain distinguishable when an arc is divided into shorter segments. Substochastic operators provide a finite-state description in which homogeneous subdivision preserves propagation, while changes in exposure and in the available continuations remain visible. Building on continuous semigroups and linear realization theory, this paper studies the interaction between refinement and the information required for network decisions. At each physical vertex, the pairing of reachable states with admissible continuation observations determines the dimension of an exact linear state. Its transitions respect arc subdivision when the physical decision and stopping locations are preserved. The same observations determine which partial paths can be discarded without worsening any permitted continuation. Worked finite-state networks demonstrate why a smaller prefix risk may lead to a larger final risk and how recovery, exposure ordering, and generator uncertainty affect that comparison. They also show that allowing uncertainty to change at an artificial breakpoint alters the decision problem even when local propagation remains consistent. These relationships connect physical exposure, state reduction, and path optimization within a formulation whose invariance and computational requirements can be stated explicitly.
SN - 2075-1680
ER -