A Superprocess-based Approach to Rough CIR Processes and Feller Random Measures

Authors: Gordan Žitković

Abstract

Feller random measures generalize the Feller diffusion (the CIR process) by giving it memory. They arise as the scaling limits of nearly unstable Hawkes processes, and include the rough CIR process and its hyper-rough and discontinuous relatives. We show that every Feller random measure is the occupation measure of a Dawson–Watanabe superprocess whose spatial motion is a killed Lévy subordinator, integrated over the branching time. This is a continuum analogue of the Hawkes–Oakes cluster representation. It yields existence, stability in the parameters, and stochastic equations driven by an explicit noise. The structure of this noise depends on whether the kernel has an atom at the origin. Without an atom, the noise is a Brownian motion time-changed by the distribution function of the measure. This leads to a martingale characterization and to sharp results on densities and their Hölder regularity. With an atom, the noise is a compensated inverse-Gaussian process time-changed by the compensator of that distribution function, and the measure is purely atomic.

Complexity vs Empirical Score

Math Complexity: 9.0/10
Empirical Rigor: 2.0/10
Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly theoretical and mathematically dense approach to Feller random measures, offering significant novelty in its superprocess construction. While it provides strong theoretical results and characterizations, it lacks empirical validation or practical implementation details, focusing purely on mathematical foundations. The clarity is good for its target audience, but the reproducibility is low due to the absence of code or data.

Research Flowchart

  flowchart TD
    A[Research Goal: Generalize Feller Diffusion & Characterize Feller Random Measures] --> B{Key Methodology: Superprocess-based Approach};
    B --> C[Inputs: Rough CIR Processes, Unstable Hawkes Processes, Lévy Subordinators];
    C --> D[Computational Process: Integrate Superprocess over Branching Time, Analyze Noise Structure];
    D --> E{Key Finding 1: Feller Random Measure = Dawson-Watanabe Superprocess (killed Lévy subordinator)};
    E --> F{Key Finding 2: Existence, Stability, Stochastic Equations (Noise depends on kernel atom)};
    F --> G{Key Finding 3: Noise = Brownian (no atom) / Inverse-Gaussian (with atom) -> Martingale, Regularity / Purely Atomic};

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