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On the Exact Distribution of the Sum of Two CIR Processes

Bilgi Yilmaz, Alper Hekimoglu

TL;DR

This work solves the long-standing problem of the exact distribution for the sum of two independent CIR processes by deriving a closed-form density expressed as a double infinite Poisson–Gamma mixture of a Kummer-type Gamma convolution, with the kernel $f_Z(s;\nu_1,\nu_2,\beta_1,\beta_2)$ involving the confluent hypergeometric function ${}_1F_1$. The key construction uses the Poisson–Gamma representation of each noncentral $\chi^2$ marginal and averages over the Poisson counts to obtain $f_S(s)=\sum_{n_1,n_2} w_{n_1} w_{n_2} f_Z(s;...)$, providing a computationally stable avenue for exact likelihoods in multifactor affine models. The paper also introduces practical truncation schemes (Poisson-tail and weight-window) for efficient numerical evaluation and proves limiting Gaussian behavior as $\Delta t\to 0$, with the equal-scale case recovering the classical single-factor result. Practically, the results enable exact calibration and pricing in multi-factor term-structure, stochastic volatility, and credit-risk models, while offering a framework potentially applicable to other interacting mean-reverting systems beyond finance.

Abstract

This paper derives the exact transition density and cumulative distribution function of a linear combination of two independent Cox-Ingersoll-Ross (CIR) processes. By combining the Poisson Gamma mixture representation of the noncentral chi-square law with the Kummer type convolution of Gamma densities, we obtain a closed-form analytical expression involving confluent hypergeometric functions. This result extends the classical single-factor CIR transition law to a multifactor framework, providing the first explicit analytical characterization of the sum of two independent CIR diffusions. The proposed density admits stable numerical evaluation and facilitates exact likelihood computation, enabling rigorous parameter estimation in multifactor affine term-structure, stochastic volatility, and credit risk models. Numerical experiments confirm that the analytical density and CDF closely match Monte Carlo simulations across various parameter regimes, demonstrating high accuracy and computational efficiency. Beyond financial mathematics, the derived distribution has potential applications in fields involving interacting mean-reverting processes, such as insurance mathematics, reliability theory, and biophysical modeling

On the Exact Distribution of the Sum of Two CIR Processes

TL;DR

This work solves the long-standing problem of the exact distribution for the sum of two independent CIR processes by deriving a closed-form density expressed as a double infinite Poisson–Gamma mixture of a Kummer-type Gamma convolution, with the kernel involving the confluent hypergeometric function . The key construction uses the Poisson–Gamma representation of each noncentral marginal and averages over the Poisson counts to obtain , providing a computationally stable avenue for exact likelihoods in multifactor affine models. The paper also introduces practical truncation schemes (Poisson-tail and weight-window) for efficient numerical evaluation and proves limiting Gaussian behavior as , with the equal-scale case recovering the classical single-factor result. Practically, the results enable exact calibration and pricing in multi-factor term-structure, stochastic volatility, and credit-risk models, while offering a framework potentially applicable to other interacting mean-reverting systems beyond finance.

Abstract

This paper derives the exact transition density and cumulative distribution function of a linear combination of two independent Cox-Ingersoll-Ross (CIR) processes. By combining the Poisson Gamma mixture representation of the noncentral chi-square law with the Kummer type convolution of Gamma densities, we obtain a closed-form analytical expression involving confluent hypergeometric functions. This result extends the classical single-factor CIR transition law to a multifactor framework, providing the first explicit analytical characterization of the sum of two independent CIR diffusions. The proposed density admits stable numerical evaluation and facilitates exact likelihood computation, enabling rigorous parameter estimation in multifactor affine term-structure, stochastic volatility, and credit risk models. Numerical experiments confirm that the analytical density and CDF closely match Monte Carlo simulations across various parameter regimes, demonstrating high accuracy and computational efficiency. Beyond financial mathematics, the derived distribution has potential applications in fields involving interacting mean-reverting processes, such as insurance mathematics, reliability theory, and biophysical modeling

Paper Structure

This paper contains 8 sections, 4 theorems, 52 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Transition Density of the Sum of Two Independent CIR Processes
  3. 1. Poisson-tail truncation.
  4. 2. Weight window truncation
  5. Numerical Illustration
  6. Conclusion
  7. Section title of first appendix
  8. Proof via the product semigroup

Key Result

Theorem 1

Let us consider two independent CIR diffusions $X^{1}$ and $X^{2}$ evolving on a common probability space $(\Omega,\mathcal{F},\mathbb{P})$ equipped with the filtration $(\mathcal{F}_t)_{t\in[0,T]}$. Suppose both processes satisfy the stochastic differential equation where $\kappa_i>0$, $\theta_i>0$, and $\sigma_i>0$ denote the mean-reversion rate, the long-term mean level, and the instantaneous

Figures (2)

  • Figure 1: Comparison of two CIR-sum densities for different discretization/time steps.
  • Figure 2: Comparison of two CIR-sum cumulative densities (CDF) for different discretization/time steps.

Theorems & Definitions (10)

  • Theorem 1: Transition density of a linear combination of two independent CIR transitions
  • proof
  • Corollary 2
  • Remark 3: Gaussian limit of the two-factor CIR sum
  • proof
  • Theorem 4: CDF of a two–factor CIR process sum with packed Poisson weights
  • proof
  • Corollary 5
  • proof
  • proof