Abstract
In this work, the first hitting times for skew CIR processes are investigated. We compute the Laplace transforms and the means of the first hitting times of some given levels. The solutions for Laplace transforms are in terms of Tricomi and Kummer confluent hypergeometric functions. We also exhibit the hitting time densities numerically at the end of this paper.
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References
Abundo M (2009) First-passage problems for asymmetric diffusions and skew-diffusion processes. Open Syst Inf Dyn 16:325–350
Appuhamillage T A, Bokil V A, Thomann E, Waymire E, Wood B D (2010) Solute transport across an interface: a fickian theory for skewness in breakthrough curves. Water Resour Res 46:13pp
Appuhamillage T, Sheldon D (2012) First passage time of skew Brownian motion. J Appl Probab 49:685–696
Barlow M T (1988) Skew Brownian motion and a one dimensional stochastic differential equation. Stochastics 25:1–2
Borodin A N, Salminen P (2002) Handbook of Brownian motion: facts and formulae. Birkhäuser, Basel
Buchholz H, Lichtblau H, Wetzel K (1969) The confluent hypergeometric function: with special emphasis on its applications. Springer, Berlin
Chou C S, Lin H J (2006) Some properties of CIR processes. Stochast Anal Appl 24:901–912
Chuancun Y, Huiqing W (2012) The first passage time and the dividend value function for one-dimensional diffusion processes between two reflecting barriers. Int J Stochast Anal 2012:1–15
Engelbert HJ, Schmidt W (1991) Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations (Part III). Mathematische Nachrichten 151:149–197
Freidlin M I, Wentzell A D (1993) Diffusion processes on graphs and the averaging principle. Ann Probab 21:2215–2245
Harrison J M, Shepp L A (1981) On skew Brownian motion. Ann Probab 9:309–313
Karlin S, Taylor H M (1981) A second course in stochastic processes. Academic Press, New York
Lang R (1995) Effective conductivity and skew Brownian motion. J Stat Phys 80:125–146
Le Gall JF (1983) Temps locaux et equations différentielles stochastiques, vol 986. Springer, Berlin
Portenko N I (2000) A probabilistic representation for the solution of one problem of mathematical physics. Ukr Math J 52:1457–1469
Protter P (2004) Stochastic integration and differential equations. Springer, New York
Ramirez J M, Thomann E A, Waymire E C, Haggerty R, Wood B (2006) A generalized Taylor-Aris formula and skew diffusion. Multiscale Model Simul 5:786–801
Trutnau G (2010) Weak existence of the squared Bessel and CIR processes with skew reflection on a deterministic time-dependent curve. Stochast Process Appl 120:381–402
Trutnau G (2011) Pathwise uniqueness of the squared Bessel and CIR processes with skew reflection on a deterministic time dependent curve. Stochast Process Appl 121:1845–1863
Valko P P, Abate J (2004) Comparison of sequence accelerators forthe Gaver method of numerical Laplace transform inversion. Comput Math Appl 48:629–636
Walsh J B (1978) A diffusion with a discontinuous local time. Astérisque 52:37–45
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This work was partially supported by the National Natural Science Foundation of China (No. 11101223 and 11271203).
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Song, S., Xu, G. & Wang, Y. On First Hitting Times for Skew CIR Processes. Methodol Comput Appl Probab 18, 169–180 (2016). https://doi.org/10.1007/s11009-014-9406-7
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DOI: https://doi.org/10.1007/s11009-014-9406-7