Cox–Ingersoll–Ross Process and Skorokhod Problems

Giulia Di Nunno (), Yuliya Mishura () and Anton Yurchenko-Tytarenko ()
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Giulia Di Nunno: University of Oslo, Department of Mathematics
Yuliya Mishura: Taras Shevchenko National University of Kyiv, Department of Probability Theory, Statistics and Actuarial Mathematics
Anton Yurchenko-Tytarenko: Statkraft Energi AS

Chapter 2 in Volterra Volatility Models, 2026, pp 51-78 from Springer

Abstract: Abstract In this chapter, we begin our exploration of volatility models by revisiting the classical Cox–Ingersoll–Ross (CIR) process, a key component of the classical Heston model. Although the CIR process is Markovian and thus lacks memory effects, it exhibits very interesting structural features. In particular, under the Feller condition, the square root of the CIR process satisfies a stochastic differential equation with a drift that diverges near zero—this creates a “repelling” effect that ensures a.s. positivity of the solution. We argue that this mechanism can be extended to construct broader classes of positive volatility models with memory. To complete the picture, we also examine the behavior of CIR processes that do not satisfy the Feller condition and establish connections between this regime and the reflected Ornstein–Uhlenbeck (ROU) model.

Date: 2026
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DOI: 10.1007/978-3-032-26576-0_2

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