Exact solutions and asymptotic behaviors for the reflected Wiener, Ornstein-Uhlenbeck and Feller diffusion processes

Virginia Giorno; Amelia G. Nobile; Virginia Giorno; Amelia G. Nobile

doi:10.3934/mbe.2023607

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 8: 13602-13637. doi: 10.3934/mbe.2023607

Previous Article Next Article

Research article Special Issues

Exact solutions and asymptotic behaviors for the reflected Wiener, Ornstein-Uhlenbeck and Feller diffusion processes

Virginia Giorno ,
Amelia G. Nobile ^,

Dipartimento di Informatica, Università degli Studi di Salerno, Via Giovanni Paolo Ⅱ n. 132, 84084 Fisciano (SA), Italy

Received: 30 March 2023 Revised: 17 May 2023 Accepted: 28 May 2023 Published: 14 June 2023

We analyze the transition probability density functions in the presence of a zero-flux condition in the zero-state and their asymptotic behaviors for the Wiener, Ornstein Uhlenbeck and Feller diffusion processes. Particular attention is paid to the time-inhomogeneous proportional cases and to the time-homogeneous cases. A detailed study of the moments of first-passage time and of their asymptotic behaviors is carried out for the time-homogeneous cases. Some relationships between the transition probability density functions for the restricted Wiener, Ornstein-Uhlenbeck and Feller processes are proved. Specific applications of the results to queueing systems are provided.
- reflected diffusion processes,
- transition probability density functions,
- steady-state densities,
- first-passage time moments,
- asymptotic behaviors
Citation: Virginia Giorno, Amelia G. Nobile. Exact solutions and asymptotic behaviors for the reflected Wiener, Ornstein-Uhlenbeck and Feller diffusion processes[J]. Mathematical Biosciences and Engineering, 2023, 20(8): 13602-13637. doi: 10.3934/mbe.2023607

Related Papers:

Abstract

We analyze the transition probability density functions in the presence of a zero-flux condition in the zero-state and their asymptotic behaviors for the Wiener, Ornstein Uhlenbeck and Feller diffusion processes. Particular attention is paid to the time-inhomogeneous proportional cases and to the time-homogeneous cases. A detailed study of the moments of first-passage time and of their asymptotic behaviors is carried out for the time-homogeneous cases. Some relationships between the transition probability density functions for the restricted Wiener, Ornstein-Uhlenbeck and Feller processes are proved. Specific applications of the results to queueing systems are provided.

References

[1]	V. Giorno, A. G. Nobile, L. M. Ricciardi, On some time non homogeneous diffusion approximations to queueing systems, Adv. Appl. Prob., 19 (1987), 974–994. https://doi.org/10.2307/1427111 doi: 10.2307/1427111
[2]	A. R. Ward, P. W. Glynn, A diffusion approximation for a Markovian queue with reneging, Queueing Syst., 43 (2003), 103–128. https://doi.org/10.1023/A:1021804515162 doi: 10.1023/A:1021804515162
[3]	A. Di Crescenzo, B. Martinucci, A. Rhandi, A multispecies birth-death-immigration process and its diffusion approximation, J. Math. Anal. Appl., 442 (2016), 291–316. https://doi.org/10.1016/j.jmaa.2016.04.059 doi: 10.1016/j.jmaa.2016.04.059
[4]	A. Di Crescenzo, V. Giorno, B. K. Kumar, A. G. Nobile, $M/M/1$ queue in two alternating environments and its heavy traffic approximation, J. Math. Anal. Appl., 465 (2018), 973–1001. https://doi.org/10.1016/j.jmaa.2018.05.043 doi: 10.1016/j.jmaa.2018.05.043
[5]	V. Linetsky, Computing hitting time densities for CIR and OU diffusions, Applications to mean-reverting models, J. Comput. Finance, 7 (2004), 1–22. https://doi.org/10.21314/JCF.2004.120 doi: 10.21314/JCF.2004.120
[6]	D. Veestraeten, Valuing stock options when prices are subject to a lower boundary, Journal of Futures Markets, 28 (2008), 231–247. https://doi.org/10.1002/fut.20299 doi: 10.1002/fut.20299
[7]	P. Lánský, S. Ditlevsen, A review of the methods for signal estimation in stochastic diffusion leaky integrate-and-fire neuronal models, Biol. Cybern., 99 (2008), 253–262. https://doi.org/10.1007/s00422-008-0237-x doi: 10.1007/s00422-008-0237-x
[8]	A. Buonocore, L. Caputo, A. G. Nobile, E. Pirozzi, Gauss-Markov processes in the presence of a reflecting boundary and applications in neuronal models, Appl. Math. Comp., 232 (2014), 799–809. https://doi.org/10.1016/j.amc.2014.01.143 doi: 10.1016/j.amc.2014.01.143
[9]	A. Buonocore, L. Caputo, A. G. Nobile, E. Pirozzi, Restricted Ornstein-Uhlenbeck process and applications in neuronal models with periodic input signals, J. Comp. Appl. Math., 285 (2015), 59–71. https://doi.org/10.1016/j.cam.2015.01.042 doi: 10.1016/j.cam.2015.01.042
[10]	G. D'Onofrio, P. Lánský, E. Pirozzi, On two diffusion neuronal models with multiplicative noise: the mean first-passage time properties, Chaos, 28 (2018), Article number 043103. https://doi.org/10.1063/1.5009574 doi: 10.1063/1.5009574
[11]	A. Di Crescenzo, V. Giorno, A. G. Nobile, Analysis of reflected diffusions via an exponential time-based transformation, J. Stat. Phys., 163 (2016), 1425–1453. https://doi.org/10.1007/s10955-016-1525-9 doi: 10.1007/s10955-016-1525-9
[12]	V. Giorno, A. G. Nobile, Restricted Gompertz-type diffusion processes with periodic regulation functions, Mathematics, 7 (2019), Article number 555. https://doi.org/10.3390/math7060555 doi: 10.3390/math7060555
[13]	V. Giorno, A. G. Nobile, On a time-inhomogeneous diffusion process with discontinuous drift. Appl. Math. Comp., 451 (2023), Article number 128012. https://doi.org/10.1016/j.amc.2023.128012 doi: 10.1016/j.amc.2023.128012
[14]	Y. Mishura, A. Yurchenko-Tytarenko, Standard and fractional reflected Ornstein-Uhlenbeck processes as the limits of square roots of Cox-Ingersoll-Ross processes, Stochastics, 95 (2023), 99–117. https://doi.org/10.1080/17442508.2022.2047188 doi: 10.1080/17442508.2022.2047188
[15]	L. M. Ricciardi, A. Di Crescenzo, V. Giorno, A. G. Nobile, An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling, Math. Japonica, 50 (1999), 247–322.
[16]	J. Masoliver, J. Perelló, First-passage and escape problems in the Feller process, Phys. Rev. E, 86 (2012), Article number 041116. https://doi.org/10.1103/PhysRevE.86.041116 doi: 10.1103/PhysRevE.86.041116
[17]	L. Bo, G. Ren, Y. Wang, X. Yang, First passage times of reflected generalized Ornstein-Uhlenbeck processes, Stochastics Dyn., 13 (2013), 1–16. https://doi.org/10.1142/S0219493712500141 doi: 10.1142/S0219493712500141
[18]	M. Abundo, E. Pirozzi, Integrated stationary Ornstein-Uhlenbeck process, and double integral processes, Phys. A, 494 (2018), 265–275. https://doi.org/10.1016/j.physa.2017.12.043 doi: 10.1016/j.physa.2017.12.043
[19]	V. Giorno, A. G. Nobile, On the absorbing problems for Wiener, Ornstein-Uhlenbeck and Feller diffusion processes: Similarities and differences, Fractal Fract., 7 (2023), Article number 11. https://doi.org/10.3390/fractalfract7010011 doi: 10.3390/fractalfract7010011
[20]	E. B. Dynkin, Kolmogorov and the theory of Markov processes, Ann. Probab., 17 (1989), 822–832. https://www.jstor.org/stable/2244385
[21]	S. Karlin, H. W. Taylor, A Second Course in Stochastic Processes, Academic Press, New York, 1981.
[22]	A. J. F. Siegert, On the first passage time probability problem, Phys. Rev., 81 (1951), 617—623. https://doi.org/10.1103/PhysRev.81.617 doi: 10.1103/PhysRev.81.617
[23]	V. Giorno, A. G. Nobile, L. M. Ricciardi, On neuronal firing modeling via specially confined diffusion processes, Scientiae Mathematicae Japonicae, 58 (2003), 265–294.
[24]	J. F. C. Kingman, On queue in heavy traffic, J. R. Stat. Soc. B, 24 (1962), 383–392. http://www.jstor.org/stable/2984229
[25]	J. M. Harrison, The diffusion approximation for tandem queues in heavy traffic, Adv. Appl. Prob., 10(4) (1978), 886–905. https://doi.org/10.2307/1426665 doi: 10.2307/1426665
[26]	V. Linetsky, On the transition densities for reflected diffusions, Adv. Appl. Prob., 37 (2005), 435–460. http://www.jstor.org/stable/30037335
[27]	A. Molini, P. Talkner, G. G. Katul, A. Porporato, First passage time statistics of Brownian motion with purely time dependent drift and diffusion, Phys. A, 390 (2011), 1841–1852. https://doi.org/10.1016/j.physa.2011.01.024 doi: 10.1016/j.physa.2011.01.024
[28]	D. R. Cox, H. D. Miller, The Theory of Stochastic Processes, Chapman & Hall/CRC, Boca Raton, Florida, 1996.
[29]	A. Buonocore, A. G. Nobile, E. Pirozzi, Simulation of sample paths for Gauss-Markov processes in the presence of a reflecting boundary, Cogent Math., 4 (2017), Article number 1354469. https://doi.org/10.1080/23311835.2017.1354469 doi: 10.1080/23311835.2017.1354469
[30]	V. Giorno, A. G. Nobile, R. di Cesare, On the reflected Ornstein-Uhlenbeck process with catastrophes, Appl. Math. Comp., 218 (2012), 11570–11582. https://doi.org/10.1016/j.amc.2012.04.086 doi: 10.1016/j.amc.2012.04.086
[31]	A. R. Ward, P. W. Glynn, Properties of the reflected Ornstein-Uhlenbeck process, Queueing Syst., 44 (2003), 109–123. https://doi.org/10.1023/A:1024403704190 doi: 10.1023/A:1024403704190
[32]	Y. Nie, V. Linetsky, Sticky reflecting Ornstein-Uhlenbeck diffusions and the Vasicek interest rate model with the sticky zero lower bound, Stochastic Models, 36 (2020), 1–19. https://doi.org/10.1080/15326349.2019.1630287 doi: 10.1080/15326349.2019.1630287
[33]	I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press Inc., 2014. https://doi.org/10.1016/C2010-0-64839-5
[34]	V. Giorno, A. G. Nobile, Time-inhomogeneous Feller-type diffusion process in population dynamics, Mathematics, 9 (2021), Article number 1879. https://doi.org/10.3390/math9161879 doi: 10.3390/math9161879
[35]	A. Di Crescenzo, A. G. Nobile, Diffusion approximation to a queueing system with time-dependent arrival and service rates, Queueing Syst., 19 (1995), 41–62. https://doi.org/10.1007/BF01148939 doi: 10.1007/BF01148939
[36]	V. Giorno, P. Lánský, A. G. Nobile, L. M. Ricciardi, Diffusion approximation and first-passage-time problem for a model neuron. Ⅲ. A birth-and-death process approach, Biol. Cyber., 58(6) (1988), 387–404. https://doi.org/10.1007/BF00361346 doi: 10.1007/BF00361346
[37]	S. Ditlevsen, P. Lánský, Estimation of the input parameters in the Feller neuronal model, Phys. Rev. E, 73 (2006), Article number 061910. https://doi.org/10.1103/PhysRevE.73.061910 doi: 10.1103/PhysRevE.73.061910
[38]	Y. Tian, H. Zhang, Skew CIR process, conditional characteristic function, moments and bond pricing, Appl. Math. Comput., 329 (2018), 230–238. https://doi.org/10.1016/j.amc.2018.02.013 doi: 10.1016/j.amc.2018.02.013
[39]	J. C. Cox, J. E. Ingersoll Jr., S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385–407. https://doi.org/10.2307/1911242 doi: 10.2307/1911242
[40]	E. Di Nardo, G. D'Onofrio, A cumulant approach for the first-passage-time problem of the Feller square-root process, Appl. Math. Comput., 391 (2021), Article Number 125707. https://doi.org/10.1016/j.amc.2020.125707 doi: 10.1016/j.amc.2020.125707
[41]	J. Masoliver, Nonstationary Feller process with time-varying coefficients, Phys. Rev. E, 93 (2016), Article number 012122. https://doi.org/10.1103/PhysRevE.93.012122 doi: 10.1103/PhysRevE.93.012122
[42]	W. Feller, Two singular diffusion problems, Ann. Math., 5 (1951), 173–182. https://doi.org/10.2307/1969318 doi: 10.2307/1969318
[43]	V. Giorno, A. G. Nobile, On the first-passage time problem for a Feller-type diffusion process, Mathematics, 9 (2021), Article number 2470. https://doi.org/10.3390/math9192470 doi: 10.3390/math9192470
[44]	V. Giorno, A. G. Nobile, L. M. Ricciardi, On the densities of certain bounded diffusion processes, Ricerche Mat., 60 (2011), 89–124. https://doi.org/10.1007/s11587-010-0097-2 doi: 10.1007/s11587-010-0097-2

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)