Paper: arXiv 2609.37881

Authors: Jongjin Park, David Criens, Hyungbin Park

Abstract

This work studies the relationships among sublinear valuation rules, uncertainty structures, and local specifications in a time-homogeneous Markovian framework with killing. These objects are linked, under finiteness and locality of the upper generator and a Lyapunov condition, by three maps: robust valuation, globalization, and localization. First, we show that the corresponding classes of uncertainty structures and sublinear valuation rules are order-isomorphic via the robust valuation map. Second, our analysis clarifies how local specifications constrain sublinear valuation and uncertainty structures, as well as what information localization and globalization preserve. In particular, the compositions of localization and globalization need not recover the original objects but yield canonical extremal elements. Third, the sublinear valuation generated by a local specification is characterized by the greatest viscosity subsolution of the associated Hamilton–Jacobi–Bellman equation. Finally, each uncertainty structure with killing admits a unique representation by a family of pairs consisting of a cumulative discounting process and an underlying state law. These results provide a framework for order relations, probabilistic and PDE-based representations, and model recovery in sublinear valuation without requiring uniqueness of the underlying martingale problems or a viscosity comparison principle.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper is highly theoretical, focusing on advanced mathematical concepts in sublinear valuation. While it presents novel theoretical relationships, it lacks any empirical validation or practical implementation details. The mathematical rigor is very high, but the absence of empirical work places it in the ‘Lab Rats’ quadrant.

Research Flowchart

  flowchart TD
    A[Research Goal: Relate Sublinear Valuation, Uncertainty, and Local Specifications] --> B(Methodology: Order Isomorphisms, Locality & Globalization, PDE Characterization);
    B --> C{Inputs: Sublinear Valuation Rules, Uncertainty Structures, Local Specifications, Markovian Framework with Killing};
    C --> D[Computational Process: Robust Valuation, Globalization, Localization Maps, HJB Equation, Probabilistic Representations];
    D --> E{Key Findings:
        - Order Isomorphism between Valuation Rules & Uncertainty Structures
        - Local Specifications Constrain Valuation & Uncertainty
        - Localization/Globalization Yield Extremal Elements
        - Valuation by Greatest Viscosity Subsolution of HJB
        - Uncertainty Structures Represented by Discounting & State Law
    };