Paper: arXiv 2509.15752

Authors: Claudio Fontana, Simone Pavarana, Thorsten Schmidt

Abstract

We study an extension of the Cox-Ingersoll-Ross (CIR) process that incorporates jumps at deterministic dates, referred to as stochastic discontinuities. Our main motivation stems from short-rate modelling in the context of overnight rates, which often exhibit jumps at predetermined dates corresponding to central bank meetings. We provide a formal definition of a CIR process with stochastic discontinuities, where the jump sizes depend on the pre-jump state, thereby allowing for both upwarrd and downward movements as well as potential autocorrelation among jumps. Under mild assumptions, we establish existence of such a process and identify sufficient and necessary conditions under which the process inherits the affine property of its continuous counterpart. We illustrate our results with practical examples that generate both upward and downward jumps while preserving the affine property and non-negativity. In particular, we show that a stochastically discontinuous CIR process can be constructed by applying a determinisitic cadlag time-change of a classical CIR process. Finally, we further enrich the affine framework by characterizing conditions that ensure infinite divisibility of the extended CIR process.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical extension of the Cox-Ingersoll-Ross (CIR) process incorporating stochastic discontinuities at deterministic jump times. While mathematically sound with formal existence proofs and measure-theoretic characterizations, it is a preliminary theoretical work with limited empirical backtesting and no detailed calibration to market data.

Research Flowchart

  flowchart TD
    A["Research Goal: Extend CIR process with stochastic discontinuities (jumps) for short-rate modeling"] --> B["Data/Inputs: Deterministic dates (Central Bank meetings) & pre-jump state values"]
    B --> C["Define CIR Process with Jumps"]
    C --> D["Apply Cadlag Time-Change & Mathematical Proofs"]
    D --> E["Analyze Properties: Non-negativity, Affine Structure, & Infinite Divisibility"]
    E --> F["Outcomes: Proven existence of affine extended CIR process with bi-directional jumps"]

Cited References (18)

  1. SOFR Term Structure Dynamics - Discontinuous Short Rates and Stochastic Volatility Forward Rates (2024) — link
  2. Expected and Unexpected Jumps in the Overnight Rate: Consistent Management of the Libor Transition (2022) — link
  3. Caplet pricing in affine models for alternative risk-free rates (2022) — link
  4. Term Structure Modelling With Overnight Rates Beyond Stochastic Continuity (2022) — link
  5. The martingale problem method revisited (2021) — link
  6. Affine processes beyond stochastic continuity (2018) — link
  7. General dynamic term structures under default risk (2016) — link
  8. DYNAMIC DEFAULTABLE TERM STRUCTURE MODELING BEYOND THE INTENSITY PARADIGM (2014) — link
and 10 more
  1. Term-Structure Models: A Graduate Course (2009) — link
  2. Risk and Valuation of Collateralized Debt Obligations (2001) — link
  3. Affine Diffusions and Related Processes: Simulation, Theory and Applications (2015) — link
  4. Quantitative Risk Management: Concepts, Techniques and Tools Revised edition (2015)
  5. Brownian Motion and Stochastic Calculus (2008) — link
  6. Characteristic Functions (2005) — link
  7. Limit Theorems for Stochastic Processes , 2nd edn (2003)
  8. AFFINE PROCESSES AND APPLICATIONS IN FINANCE (2002)
  9. An econometric model of the yield curve with macroeconomic jump effects (2001)
  10. Calcul stochastique et problèmes de martingales (1979) — link

0 of 18 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.