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Original Articles

One-Dimensional Reflected Diffusions with Two Boundaries and an Inverse First-Hitting Problem

Pages 975-991 | Received 21 May 2014, Accepted 26 Aug 2014, Published online: 21 Oct 2014

References

  • Abate, J., Whitt, W. 1987. Transient behavior of regulated Brownian motion. I. Starting at the origin.Adv. Appl. Prob. 19:560–598.
  • Abate, J. Whitt, W. 1987. Transient behavior of regulated Brownian motion. II. Nonzero initial conditions.Adv. Appl. Prob. 19:599–631.
  • Abundo, M. 2013. The double-barrier inverse first-passage problem for Wiener process with random starting point. Statist. Probab. Lett. 83:168–176.
  • Abundo, M. 2012. An inverse first-passage problem for one-dimensional diffusions with random starting point. Statist.Probab. Lett. 82(1):7–14.
  • Abundo, M. 2006. Limit at zero of the first-passage time density and the inverse problem for onedimensional diffusions. Stochastic Anal. Appl. 24:1119–1145.
  • Abundo, M. 2000. On first-passage-times for one-dimensional jump-diffusion processes. Prob. Math. Statis. 20(2):399–423.
  • Abundo, M. 1997. On some properties of one-dimensional diffusion processes on an interval. Prob. Math. Statis. 17(2):235–268.
  • Abundo, M. 2010. On the First Hitting Time of a One-dimensional Diffusion and a Compound Poisson Process.Methodol. Comput. Appl. Probab. 12:473–490. DOI:10.1007/s11009-008-9115-1
  • Abundo, M. 2003. On the first-passage time of diffusion processes over a one-sided stochastic boundary. Stochastic Anal. Appl. 21(1):1–23.
  • Abundo, M. 2002. Some conditional crossing results of Brownian motion over a piecewise-linear boundary. Statis. Probab. Lett. 58(2):131–145.
  • Abundo, M. 2000. On first-crossing times of one-dimensional diffusions over two time-dependent boundaries. Stochastic Anal. Appl. 18(2):179–200.
  • Ball, C.A. Roma, A. 1998. Detecting mean reversions within reflecting barriers: applications to the European exchange rate mechanism. Appl. Math. Finance 5:1–15.
  • Bertolla, G. Caballero, R.J. 1992. Target zones and realignments. Amer. Econom. Rev. 82:520–536.
  • Chuancun, Y. Huiqing, W. 2012. The first passage time and the dividend value function for one-dimensional diffusion processes between two reflecting barriers. International Journal of Stochastic Analysis. DOI:10.1155/2012/971212
  • Daniels, H.E. 1969. The minimum of a stationary Markov process superimposed on a U-shaped trend. J. Appl. Probab. 6:369–408.
  • Darling, D.A. Siegert, A.J.F. 1953. The first passage problem for a continuous Markov process. Ann. Math. Statistics 24:624–639.
  • De Jong, F. 1994. A univariate analysis of European monetary system exchange rates using a target zone model. J. Appl. Econometrics 9:31–45.
  • Crescenzo, A. Giorno, V. Nobile, A.G. Ricciardi, L.M. 2003. On the M/M/1 queue with catastrophes and its continuous approximation. Queueing Sys. 43(4):329–347.
  • Durbin, J. 1971. Boundary crossing probability for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test. J. Appl. Probab. 8:431–453.
  • Gihman, I.I. and Skorohod, A.V. 1972. Stochastic Differential Equations. Springer-Verlag, Berlin.
  • Harrison, M. 1985. Brownian Motion and Stochastic Flow Systems. John Wiley, New York.
  • Has'minskii, R.Z. 1980. Stochastic Stability of Differential Equations. Alphen a/d Rijn, Sijthoff Noordhoff.
  • Ikeda, N. and Watanabe, S. 1981. Stochastic Differential Equations and Diffusion Processes. North- Holland, Amsterdam.
  • Itô, K. and McKean, H.P. 1974. Diffusion Processes and their Sample Paths. Springer-Verlag, New York.
  • Jackson, K. Kreinin, A. Zhang, W. 2009. Randomization in the first hitting problem. Statis. Probab. Lett. 79:2422–2428.
  • Karlin, S. and Taylor, H.M. 1975. A Second Course in Stochastic Processes. Academic Press, New York.
  • Krugman, P.R. 1991. Target zones and exchange rate dynamics. Quart. J. Econom. 106:669–682.
  • Lanska, V. Lansky, P. Smiths, C.E. 1994. Synaptic transmission in a diffusion model for neural activity. J. Theor. Biol. 166:393–406.
  • Lansky, P. Smith, C.E. 1989. The effect of a random initial value in neural first- passage-time models. Math. Biosci. 93(2):191–215.
  • Lijun, B. Lidong, Z. and Yongjin, W. 2006. On the first passage times of reflected O-U processes with two-sided barriers. Queueing Sys. 54:313–316.
  • Linetsky, V. 2005. On the transition densities for reflected diffusions. Adv. Appl. Prob. 37:435–460.
  • Linetsky, V. 2004. Computing hitting time densities for CIR and OU diffusions: applications to mean-reverting models. J. Comput. Finance 7:1–22.
  • Linetsky, V. 2004. Lookback options and diffusion hitting times: a spectral expansion approach. Finance and Stochastics 8(3):373–398.
  • Lions, P.L. Sznitman, A.S. 1984. Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37(4):511–537.
  • Martin-Lof, A. 1998. The final size of a nearly critical epidemic, and the first-passage time of a Wiener process to a parabolic barrier. J. Appl. Probab. 85:671–682.
  • McKean, H.P. 1963. A Skorohod's integral equation for a reflecting barrier diffusion. J. Math. Kyoto Univ. 3:86–88.
  • McKean, H.P.. 1969. Stochastic Integrals. Academic Press, New York.
  • Hu, Q. Wang, Y.J. Yang, X. 2012. The hitting time density for a reflected Brownian motion. Comput. Econ. 40(1):1–18.
  • Ricciardi, L.M. Sacerdote, L. 1987. On the probability densities of a Ornstein-Uhlenbeck process with a reflecting boundary. J. Appl. Prob. 24:355–369.
  • Ricciardi, L.M. and Sato, S. 1990. Diffusion processes and first-passage-time problems. In Ricciardi, L. M. (Ed.) Lectures in Applied Mathematics and Informatics. Manchester University Press, Manchester, UK.
  • Salminen, P. 1998. On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary. Adv. Appl. Prob. 20:411–426.
  • Skorohod, A.V. 1962. Stochastic equations for diffusion processes in a bounded region 2. Theor. Veroyatnost. i Primenen. 7:3–23. [Stochastic equations for diffusion processes in a bounded region 1; Theor. Veroyatnost. i Primenen. 69(1961):264–274]
  • Srikant, R. Whitt, W. 1996. Simulation run lengths to estimate blocking probabilities. ACM Trans. Model. Comput. Simul. 6:7–52.
  • Svensson, L.E.O. 1991. The term structure of interest rate differentials in a target zone. Theory and Swedish data. J. Monetary Econom. 28:87–116.
  • Veestraeten, D. 2004. The conditional probability density function for a reflected Brownian motion. Comput. Econom. 23:185–207.
  • Zucca, C. Sacerdote, L. 2009. On the inverse first-passage-time problem for a Wiener process. Ann. Appl. Prob. 19(4):1319–1346.
  • Ward, A.R. Glynn, P.W. 2003. A diffusion approximation for a Markovian queue with reneging. Queueing Sys. 43:103–128.
  • Ward, A.R. Glynn, P.W. 2003. Properties of the reflected Ornstein-Uhlenbeck process. Queueing Sys. 44:109–123.

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