3e0476bd-5ca6-4a15-bc87-350276be3d8d20250523032751079wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=4051120202512020252410.37394/23206.2025.24https://wseas.com/journals/mathematics/2025.phpMarioLefebvreDepartment of Mathematics and Industrial Engineering, Polytechnique Montréal, 2500, chemin de Polytechnique, Montréal (Québec) H3T 1J4, CANADARomainMradDepartment of Mathematics and Industrial Engineering, Polytechnique Montréal, 2500, chemin de Polytechnique, Montréal (Québec) H3T 1J4, CANADALet {X(t), t ≥ 0} be a CIR diffusion process, and τ (x) be the first time that X(t) = 0 or c, given that X(0) = x ∈ (0, c). First, we compute the moment-generating function and the expected value of τ (x). Then, an optimal control problem is considered for {X(t), t ≥ 0}. Finally, we add jumps to the diffusion process and we calculate in a particular case the probability that X(τ (x)) = 0, as well as the expected time needed to leave the interval (0, c). Explicit and exact results are obtained.5232025523202538238836https://wseas.com/journals/mathematics/2025/a725106-018(2025).pdf10.37394/23206.2025.24.36https://wseas.com/journals/mathematics/2025/a725106-018(2025).pdf10.1007/s11009-011-9223-1M. Abundo, On the first-passage area of a one-dimensional jump-diffusion process, Methodology and Computing in Applied Probability, Vol. 15, No. 1, 2013, pp. 85–103. https://doi.org/10.1007/s11009-011-9223-1 10.1103/physreve.83.051115E. Martin, U. Behn and G. Germano, First-passage and first-exit times of a Bessel-like stochastic process, Physical Review E, Vol. 83, No. 5, 2011, 051115. https://doi.org/10.1103/PhysRevE.83.051115 10.1103/physreve.86.041116J. Masoliver and J. Perelló, First-passage and escape problems in the Feller process, Physical Review E, Vol. 86, No. 4, 2012, 049906. https://doi.org/10.1103/PhysRevE.86.049906 10.1016/j.amc.2020.125707E. Di Nardo and G. D’Onofrio, A cumulant approach for the first-passage-time problem of the Feller square-root process, Applied Mathematics and Computation, Vol. 391, 2021, 125707. https://doi.org/10.1016/j.amc.2020.125707 10.3390/math9192470V. Giorno and A. G. Nobile, On the first-passage-time problem for a Feller-type diffusion process, Mathematics, Vol. 9, No. 19, 2021, 2470. https://doi.org/10.3390/math9192470 10.3390/fractalfract7010011V. Giorno and A. G. Nobile, On the absorbing problems for Wiener, Ornstein–Uhlenbeck, and Feller diffusion processes: similarities and differences, Fractal and Fractional, Vol. 7, No. 1, 2023, 11. https://doi.org/10.3390/fractalfract7010011 10.1016/j.amc.2024.128911E. Di Nardo, G. D’Onofrio and T. Martini, Orthogonal gamma-based expansion for the CIR’s first passage time distribution, Applied Mathematics and Computation, Vol. 480, No. C, 2024, 20 pages. https://doi.org/10.1016/j.amc.2024.128911 D. R. Cox and H. D. Miller, The Theory of Stochastic Processes, Methuen, London, 1965. M. Lefebvre, Applied Stochastic Processes, Springer, New York, 2007. 10.1007/978-3-662-11761-3W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd Ed., Springer-Verlag, New York, 1966. 10.2307/1266136M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1965. P. Whittle, Optimization over Time, Vol. I, Wiley, Chichester, 1982. 10.1239/aap/1051201658S. G. Kou and H. Wang, First passage times of a jump diffusion process, Advances in Applied Probability, Vol. 35, No. 2, 2003, pp. 504–531. https://doi.org/ 10.1239/aap/1051201658 10.4064/ba190812-11-6M. Lefebvre, Exact solutions to first-passage problems for jump-diffusion processes, Bulletin of the Polish Academy of Sciences − Mathematics, Vol. 72, 2024, pp. 81–95. https://doi.org/10.4064/ba190812-11-6