On the Exact Distribution of the Sum of Two CIR Processes

TL;DR

This paper derives a closed-form analytical distribution for the sum of two independent CIR processes, extending single-factor models to multifactor frameworks with applications in finance and other fields.

Contribution

It provides the first explicit analytical expression for the sum of two CIR processes, enabling exact likelihood computation and improved modeling in multifactor settings.

Findings

Analytical density and CDF match Monte Carlo simulations accurately.

Closed-form expression involves confluent hypergeometric functions.

Numerical evaluation is stable and computationally efficient.

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