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On first passage times of sticky reflecting diffusion processes with double exponential jumps

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  04 May 2020

Shiyu Song Open the ORCID record for Shiyu Song [Opens in a new window]  and
Yongjin Wang
Shiyu Song*
Affiliation:
Tianjin University
Yongjin Wang*
Affiliation:
Nankai University
*
*Postal address: School of Mathematics, Tianjin University, Tianjin, 300354, China. Email address: [email protected]
**Postal address: School of Business, Nankai University, Tianjin 300071, China.
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Abstract

We explore the first passage problem for sticky reflecting diffusion processes with double exponential jumps. The joint Laplace transform of the first passage time to an upper level and the corresponding overshoot is studied. In particular, explicit solutions are presented when the diffusion part is driven by a drifted Brownian motion and by an Ornstein–Uhlenbeck process.

Keywords

First passage timesticky reflecting processesdouble exponential jumps

MSC classification

Primary: 60G55: Point processes
Secondary: 60J60: Diffusion processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 221 - 236
Copyright
© Applied Probability Trust 2020

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References

Asmussen, S. and Pihlsgård, M. (2007). Loss rates for Lévy processes with two reflecting barriers. Math. Operat. Res. 32, 308–321.10.1287/moor.1060.0226CrossRefGoogle Scholar
Atar, R. and Budhiraja, A. (2002). Stability properties of constrained jump-diffusion processes. Electron. J. Prob. 7, 1–31.10.1214/EJP.v7-121CrossRefGoogle Scholar
Avram, F., Kyprianou, A. E. and Pistorius, M. R. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Prob. 14, 215–238.Google Scholar
Bo, L. (2013). First passage times of reflected Ornstein–Uhlenbeck processes with two-sided jumps. Queueing Systems 73, 105–118.10.1007/s11134-012-9308-8CrossRefGoogle Scholar
Bo, L., Ren, G., Wang, Y. and Yang, X. (2013). First passage times of reflected generalized Ornstein–Uhlenbeck processes. Stoch. Dynam. 13, 1250014.10.1142/S0219493712500141CrossRefGoogle Scholar
Cai, N. (2009). On first passage times of a hyper-exponential jump diffusion process. Operat. Res. Lett. 37, 127–134.10.1016/j.orl.2009.01.002CrossRefGoogle Scholar
Doney, R. A. (1991). Hitting probabilities for spectrally positive Lévy processes. J. London Math. Soc. 2, 566–576.10.1112/jlms/s2-44.3.566CrossRefGoogle Scholar
Gapeev, P. V. and Stoev, Y. I. (2019). On some functionals of the first passage times in jump models of stochastic volatility. Stoch. Anal. Appl. 38, 149–170.10.1080/07362994.2019.1657023CrossRefGoogle Scholar
Hadjiev, D. I. (1985). The first passage problem for generalized Ornstein–Uhlenbeck processes with non-positive jumps. In Séminaire de Probabilités XIX 1983/84, Vol. 1123, ed. J. Azéma and M. Yor, Springer, Berlin, pp. 80–90.10.1007/BFb0075840CrossRefGoogle Scholar
Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. Wiley, New York.Google Scholar
Harrison, J. M. and Lemoine, A. J. (1981). Sticky Brownian motion as the limit of storage processes. J. Appl. Prob. 18, 216–226.10.1017/S0021900200097758CrossRefGoogle Scholar
Jacobsen, M. and Jensen, A. T. (2007). Exit times for a class of piecewise exponential Markov processes with two-sided jumps. Stoch. Process. Appl. 117, 1330–1356.10.1016/j.spa.2007.01.005CrossRefGoogle Scholar
Kella, O. and Stadje, W. (2001). On hitting times for compound Poisson dams with exponential jumps and linear release rate. J. Appl. Prob. 38, 781–786.10.1239/jap/1005091042CrossRefGoogle Scholar
Kobayashi, K. (2011). Stochastic calculus for a time-changed semimartingale and the associated stochastic differential equations. J. Theoret. Prob. 24, 789–820.10.1007/s10959-010-0320-9CrossRefGoogle Scholar
Kou, S. G. and Wang, H. (2003). First passage times of a jump diffusion process. Adv. Appl. Prob. 35, 504–531.10.1239/aap/1051201658CrossRefGoogle Scholar
Magnus, W., Oberhettinger, F. and Soni, R. (1966). Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed., Springer, Berlin.10.1007/978-3-662-11761-3CrossRefGoogle Scholar
Nie, Y. and Linetsky, V. (2018). Diffusions with sticky boundaries, PDEs with Wentzell boundary conditions and interest rates with the zero lower bound. Working paper, Northwestern University.Google Scholar
Nie, Y. and Linetsky, V. (2019). Sticky reflecting Ornstein–Uhlenbeck diffusions and the Vasicek interest rate model with the sticky zero lower bound. Stoch. Models, DOI: 10.1080/15326349.2019.1630287.Google Scholar
Novikov, A. (2004). Martingales and first-passage times for Ornstein–Uhlenbeck processes with a jump component. Theory Prob. Appl. 48, 288–303.10.1137/S0040585X97980403CrossRefGoogle Scholar
Patie, P. (2005). On a martingale associated to generalized Ornstein–Uhlenbeck processes and an application to finance. Stoch. Process. Appl. 115, 593–607.CrossRefGoogle Scholar
Protter, P. (2004). Stochastic Integration and Differential Equations, 2nd ed., Springer, New York.Google Scholar
Salins, M. and Spiliopoulos, K. (2017). Markov processes with spatial delay: path space characterization, occupation time and properties. Stoch. Dynam. 17, 1750042.10.1142/S0219493717500423CrossRefGoogle Scholar
Welch, P. D. (1964). On a generalized M/G/1 queuing process in which the first customer of each busy period receives exceptional service. Operat. Res. 12, 736–752.10.1287/opre.12.5.736CrossRefGoogle Scholar
Yamada, K. (1994). Reflecting or sticky Markov processes with Levy generators as the limit of storage processes. Stoch. Process. Appl. 52, 135–164.10.1016/0304-4149(94)90105-8CrossRefGoogle Scholar
Zhu, J. Y. (2012). Optimal contracts with shirking. Rev. Econom. Stud. 80, 812–839.CrossRefGoogle Scholar