Skip to main content
SpringerLink
Log in

Exit Times, Undershoots and Overshoots for Reflected CIR Process with Two-Sided Jumps

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

In this paper, we investigate the reflected CIR process with two-sided jumps to capture the jump behavior and its non-negativeness. Applying the method of (complex) contour integrals, the closed-form solution to the joint Laplace transform of the first passage time crossing a lower level and the corresponding undershoot is derived. We further extend our arguments to the exit problem from a finite interval and obtain joint Laplace transforms. Our results are expressed in terms of the real and imaginary parts of complex functions by complex matrix. Numerical results are included.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Asmussen S, Pihlsgard M (2007) Loss rates for Lévy process with two reflecting barriers. Math Oper Res 32:308–321

    Article  MathSciNet  Google Scholar 

  • Avram F, Usabel M (2008) The Gerber-shiu expected discounted penalty-reward function under an affine jump-diffusion model. Astin Bull 38(2):461–481

    Article  MathSciNet  Google Scholar 

  • Avram F, Kyprianou AE, Pistorius M R (2004) Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann Appl Probab 14:215–238

    Article  MathSciNet  Google Scholar 

  • Bo L (2013) First passage times of reflected Ornstein–Uhlenbeck processes with two-sided jumps. Queueing Syst 73:105–118

    Article  MathSciNet  Google Scholar 

  • Bo L, Wang Y, Yang X (2011) Some integral functionals of reflected SDEs and their applications in finance. Quant Finance 11:343–348

    Article  MathSciNet  Google Scholar 

  • Brigo D, El-Bachir N (2010) An exact formula for default Swaptions’ pricing in the SSRJD stochastic intensity model. Math Financ 20(3):365–382

    Article  MathSciNet  Google Scholar 

  • Cai N (2009) On first passage times of a hyper-exponential jump diffusion process. Oper Res Lett 37(2):127–134

    Article  MathSciNet  Google Scholar 

  • Chou C, Lin H (2006) Some properties of CIR processes. Stoch Anal Appl 24:901–912

    Article  MathSciNet  Google Scholar 

  • Cox JC, Ingersoll JE, Ross SA (1985) A theory of the term structure of interest rates. Econometrica 53:385–407

    Article  MathSciNet  Google Scholar 

  • Duffie D, Pan J, Singleton K (2000) Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68(6):1343–1376

    Article  MathSciNet  Google Scholar 

  • Feller W (1951) Two singular diffusion problems. Ann Math 54(1):173–182

    Article  MathSciNet  Google Scholar 

  • Fu Z, Li Z (2010) Stochastic equations of non-negative process with jumps. Stoch Process Appl 120:306–330

    Article  Google Scholar 

  • Heston S (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6:327–343

    Article  MathSciNet  Google Scholar 

  • Göing-Jaeschke A, Yor M (2003) A survey and some generalizations of bessel processes. Bernoulli 9(2):313–349

    Article  MathSciNet  Google Scholar 

  • Harrison M (1986) Brownian motion and stochastic flow systems. Wiley, New York

    Google Scholar 

  • Jacobson M, Jenson A T (2007) Exit times for a class of piecewise exponential Markov process with two-sided jumps. Stoch Process Appl 117:1330–1356

    Article  Google Scholar 

  • Jarrow R A, Lando D, Turnbull S M (1997) A Markov model for the term structure of credit risk spreads. Rev Financ Stud 10(2):481–523

    Article  Google Scholar 

  • Jiao Y, Ma C, Scotti S (2017) Alpha-CIR model with branching processes in sovereign interest rate modeling. Finance Stochast 21(3):1–25

    Article  MathSciNet  Google Scholar 

  • Kawazu K, Watanabe S (1971) Branching processes with immigration and related limit theorems. Theory Probab Appl 16:36–54

    Article  MathSciNet  Google Scholar 

  • Kou S G, Wang H (2003) First passage times of a jump diffusion process. Add Appl Probab 35:504–531

    Article  MathSciNet  Google Scholar 

  • Mendoza-Arriaga R, Linetsky V (2014) Time-changed CIR default intensities with two-sided mean-reverting jumps. Ann Appl Probab 24(2):811–856

    Article  MathSciNet  Google Scholar 

  • Novikov A A (2003) Martingales and first-exit times for the Ornstein-Uhlenbeck process with jumps. Theory Probab Appl 48:340–358

    MathSciNet  Google Scholar 

  • Pihlsgard M, Glynn P (2013) On the dynamics of semimartingales with two reflecting barrier. J Appl Probab 50:671–685

    Article  MathSciNet  Google Scholar 

  • Pitman J, Yor M (2004) A decomposition of Bessel Bridges. Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete 59:425–457

    Article  MathSciNet  Google Scholar 

  • Protter P (2004) Stochastic integration and differential equations. Springer, Berlin

    MATH  Google Scholar 

  • Revuz D, Yor M (1999) Continuous martingales and Brownian motion. Springer, Berlin

    Book  Google Scholar 

  • Scott LO (1997) Pricing stock options in a jump-diffusion model with stochastic volatility and interest rates: applications of Fourier inversion methods. Math Financ 7 (4):413–424

    Article  MathSciNet  Google Scholar 

  • Ward A, Glynn P (2003) Properties of the reflected Ornstein-Uhlenbeck process. Queueing Syst 43:103–128

    Article  MathSciNet  Google Scholar 

  • Zhu L (2013) Limit theorems for a Cox-Ingersoll-Ross process with Hawkes jumps. J Appl Probab 51(3):699–712

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

We are deeply grateful to Professor Renming Song at Department of Mathematics, University of Illinois at Urbana-Champaign for his guidance and detailed suggestions. This work is partially supported by the National Natural Science Foundation of China (No. 11631004, 71532001) and the LPMC at Nankai University.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Nankai University, Tianjin, 300071, People’s Republic of China

    Pingping Jiang & Bo Li

  2. Business School, Nankai University, Tianjin, 300071, People’s Republic of China

    Yongjin Wang

Authors
  1. Pingping Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bo Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yongjin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pingping Jiang.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiang, P., Li, B. & Wang, Y. Exit Times, Undershoots and Overshoots for Reflected CIR Process with Two-Sided Jumps. Methodol Comput Appl Probab 22, 693–710 (2020). https://doi.org/10.1007/s11009-019-09730-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-019-09730-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation