Substochastic Path Operators for Refinement-Consistent Network Risks

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.