Standard and fractional reflected Ornstein–Uhlenbeck processes as the limits of square roots of Cox–Ingersoll–Ross processes

References

• O.O. Aalen and H.K. Gjessing, Survival models based on the Ornstein-Uhlenbeck process, Lifetime Data Anal. 10 (2004), pp. 407–423.
   PubMed Web of Science ®Google Scholar
• C.A. Ball and A. Roma, Stochastic volatility option pricing, J. Financ. Quantitative Anal. 29 (1994), pp. 589–607.
   Web of Science ®Google Scholar
• L. Bo, D. Tang, Y. Wang and X. Yang, On the conditional default probability in a regulated market: a structural approach, Quantitative Financ. 11 (2011), pp. 1695–1702.
   Web of Science ®Google Scholar
• L. Bo, Y. Wang and X. Yang, Some integral functionals of reflected SDEs and their applications in finance, Quantitative Financ. 11 (2011), pp. 343–348.
   Web of Science ®Google Scholar
• J.C. Cox, J.E. Ingersoll and S.A. Ross, A re-examination of traditional hypotheses about the term structure of interest rates, The J. Financ. 36 (1981), pp. 769–799.
   Web of Science ®Google Scholar
• J.C. Cox, J.E. Ingersoll and S.A. Ross, An intertemporal general equilibrium model of asset prices, Econometrica: J. Econ. Soc. 53 (1985), pp. 363.
   Web of Science ®Google Scholar
• J.C. Cox, J.E. Ingersoll and S.A. Ross, A theory of the term structure of interest rates, Econometrica: J. Econom. Soc. 53 (1985), pp. 385.
   Web of Science ®Google Scholar
• G. Di Nunno, Y. Mishura and A. Yurchenko-Tytarenko, Sandwiched SDEs with unbounded drift driven by Holder noises, preprint (2020), Available at ArXiv 2012.11465.
   Google Scholar
• V. Giorno, A.G. Nobile and R. di Cesare, On the reflected Ornstein-Uhlenbeck process with catastrophes, Appl. Math. Comput. 218 (2012), pp. 11570–11582.
   Web of Science ®Google Scholar
• V. Giorno, A.G. Nobile and L.M. Ricciardi, On some diffusion approximations to queueing systems, Adv. Appl. Probab. 18 (1986), pp. 991–1014.
   Web of Science ®Google Scholar
• R.S. Goldstein and W.P. Keirstead, On the term structure of interest rates in the presence of reflecting and absorbing boundaries, preprint (1997). doi:10.2139/ssrn.19840.
   Google Scholar
• S.L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud. 6 (1993), pp. 327–343.
   Web of Science ®Google Scholar
• J. Hong, C. Huang, M. Kamrani and X. Wang, Optimal strong convergence rate of a backward Euler type scheme for the Cox-Ingersoll-Ross model driven by fractional Brownian motion, Stoch. Process. Appl. 130 (2020), pp. 2675–2692.
   Web of Science ®Google Scholar
• N. Ikeda and S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math. 14 (1977), pp. 619–633.
   Google Scholar
• N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Elsevier, 1981.
   Google Scholar
• I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, Springer, New York, 1998.
   Google Scholar
• P.R. Krugman, Target zones and exchange rate dynamics, The Quarterly J. Econ. 106 (1991), pp. 669–682.
   Web of Science ®Google Scholar
• K. Kubilius, Estimation of the Hurst index of the solutions of fractional SDE with locally Lipschitz drift, Lithuanian Association of Nonlinear Analysts (LANA). Nonlinear Analysis. Modelling and Control 25 (2020), pp. 1059–1078.
   Google Scholar
• C. Lee and J. Song, On drift parameter estimation for reflected fractional Ornstein-Uhlenbeck processes, Stoch. An Int. J. Probab. Stoch. Process. 88 (2016), pp. 751–778.
   Web of Science ®Google Scholar
• V. Linetsky, On the transition densities for reflected diffusions, Adv. Appl. Probab. 37 (2005), pp. 435–460.
   Web of Science ®Google Scholar
• Y. Maghsoodi, Solution of the extended CIR term structure and bond option valuation, Math. Financ. An Int. J. Math., Stat. Financial Econom. 6 (1996), pp. 89–109.
   Google Scholar
• A. Melnikov, Y. Mishura and G. Shevchenko, Stochastic viability and comparison theorems for mixed stochastic differential equations, Methodol. Comput. Appl. Probab. 17 (2015), pp. 169–188.
   Web of Science ®Google Scholar
• Y. Mishura and A. Yurchenko-Tytarenko, Fractional Cox-Ingersoll-Ross process with non-zero “mean”, Modern Stoch.: Theor. Appl. 5 (2018), pp. 99–111.
   Google Scholar
• V.I. Piterbarg, Twenty Lectures About Gaussian Processes, Atlantic Financial Press, 2015.
   Google Scholar
• L.M. Ricciardi, Stochastic population theory: Diffusion processes, Mathematical Ecology, Springer, Berlin Heidelberg, (1986), pp. 191–238.
   Google Scholar
• L.C.G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales, Cambridge University Press, 2000.
   Google Scholar
• R. Schöbel and J. Zhu, Stochastic volatility with an Ornstein-Uhlenbeck process: an extension, Rev. Financ. 3 (1999), pp. 23–46.
   Google Scholar
• T. Shiga and S. Watanabe, Bessel diffusions as a one-parameter family of diffusion processes, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 27 (1973), pp. 37–46.
   Web of Science ®Google Scholar
• A.V. Skorokhod, Stochastic equations for diffusion processes in a bounded region, Theor. Probab. Appl. 6 (1961), pp. 264–274.
   Google Scholar
• A.R. Ward and P.W. Glynn, A diffusion approximation for a Markovian queue with reneging, Queueing Syst. 43 (2003), pp. 103–128.
   Web of Science ®Google Scholar
• A.R. Ward and P.W. Glynn, Properties of the reflected Ornstein-Uhlenbeck process, Queueing Syst.44 (2003), pp. 109–123.
   Web of Science ®Google Scholar
• A.R. Ward and P.W. Glynn, A diffusion approximation for a GI/GI/1 queue with balking or reneging, Queueing Syst. 50 (2005), pp. 371–400.
   Web of Science ®Google Scholar
• X. Yang, X. Li, G. Ren, Y. Wang and L. Bo, Modeling the exchange rates in a target zone by reflected Ornstein-Uhlenbeck process, SSRN Electron. J. (2012), 10.2139/ssrn.2107686.
   Google Scholar
• D. Ylvisaker, A note on the absence of tangencies in Gaussian sample paths, Ann. Math. Stat. 39 (1968), pp. 261–262.
   Google Scholar
• M. Zähle, Integration with respect to fractal functions and stochastic calculus. I, Probability Theory and Related Fields 111 (1998), pp. 333–374.
   Web of Science ®Google Scholar
• S.Q. Zhang and C. YuanStochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, (2020), pp. 1–27.
   Google Scholar
• J. Zhu, Applications of Fourier Transform to Smile Modeling, Springer, Berlin Heidelberg, 2010.
   Google Scholar