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Explicit Computations for Some Markov Modulated Counting Processes

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Advanced Modelling in Mathematical Finance

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 189))

Abstract

In this paper we present elementary computations for some Markov modulated counting processes, also called counting processes with regime switching. Regime switching has become an increasingly popular concept in many branches of science. In finance, for instance, one could identify the background process with the ‘state of the economy’, to which asset prices react, or as an identification of the varying default rate of an obligor. The key feature of the counting processes in this paper is that their intensity processes are functions of a finite state Markov chain. This kind of processes can be used to model default events of some companies. Many quantities of interest in this paper, like conditional characteristic functions, can all be derived from conditional probabilities, which can, in principle, be analytically computed. We will also study limit results for models with rapid switching, which occur when inflating the intensity matrix of the Markov chain by a factor tending to infinity. The paper is largely expository in nature, with a didactic flavor.

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Notes

  1. 1.

    This argument has been provided by Koen de Turck, University of Ghent.

References

  1. Ang, A., Bekaert, G.: Regime switches in interest rates. J. Bus. Econ. Stat. 20, 163–182 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003)

    MATH  Google Scholar 

  3. Asmussen, S., Albrecher, H.: Ruin Probabilities, 2nd edn. World Scientific, New Jersey (2010)

    Book  MATH  Google Scholar 

  4. Bielecki, T.R., Rutkowski, M.: Credit Risk: Modelling, Valuation and Hedging. Springer, Berlin (2002)

    MATH  Google Scholar 

  5. Blom, J., de Turck, K., Mandjes, M.: Analysis of Markov-modulated infinite-server queues in the central-limit regime. Probab. Eng. Inf. Sci. 29, 433–459 (2015)

    Article  MathSciNet  Google Scholar 

  6. Blom, J., Kella, O., Mandjes, M., Thorsdottir, H.: Markov-modulated infinite-server queues with general service times. Queueing Syst. 76, 403–424 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brémaud, P.: Point Processes and Queues. Springer, New York-Berlin (1980)

    MATH  Google Scholar 

  8. Chen, A., Delong, L.: Optimal Investment for a Defined-Contribution Pension Scheme under a Regime Switching Model, ASTIN Bulletin, 1–23 March 2015

    Google Scholar 

  9. Duffie, D., Filipović, D., Schachermayer, W.: Affine processes and applications in finance. Ann. Appl. Probab. 13, 984–1053 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Elliott, R.J., Aggoun, L., Moore, J.B.: Hidden Markov Models. Springer, New York (1995)

    MATH  Google Scholar 

  11. Elliott, R., Mamon, R.: An interest rate model with a Markovian mean-reverting level. Quant. Financ. 2, 454–458 (2002)

    Article  MathSciNet  Google Scholar 

  12. Elliott, R., Siu, T.K.: On Markov-modulated exponential-affine bond price formulae. Appl. Math. Financ. 16, 1–15 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Filipović, D.: Term-structure Models. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  14. Goel, A.L., Okumoto, K.: Bayesian software prediction models, vol. 1: An imperfect debugging model for reliability and other quantitative measures of software systems. Rome Air Develop. Center, NY, RADC-TR-78-155 (1978)

    Google Scholar 

  15. Di Graziano, G.: Optimal Trading Stops and Algorithmic Trading. SSRN (2014). http://ssrn.com/abstract=2381830

  16. Hamilton, J.: A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Higham, N.J.: Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008)

    Book  MATH  Google Scholar 

  18. Huang, G., Mandjes, M., Spreij, P.: Weak convergence of Markov-modulated diffusion processes with rapid switching. Statist. Probab. Lett. 86, 74–79 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Huang, Gang, Jansen, Marijn, Mandjes, Michel, Spreij, Peter, De Turck, Koen: Markov-modulated Ornstein-Uhlenbeck processes. Adv. Appl. Probab. 48(1), 235–254 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jacod, J., Shiryaev, A.N.: Limit theorems for stochastic processes, 2nd edn. Grundlehren der Mathematischen Wissenschaften 288. Springer, Berlin (2003)

    Google Scholar 

  21. Jelinski, Z., Moranda, P.B.: Software reliability research. In: Freiberger, W. (ed.) Statistical Computer Performance Evaluation, pp. 465–484. Academic, NewYork (1972)

    Chapter  Google Scholar 

  22. Jiang, Z., Pistorius, M.R.: On perpetual American put valuation and first passage in a regime- switching model with jumps. Financ. Stochast. 12, 331–355 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jiang, Z., Pistorius, M.: Optimal dividend distribution under Markov regime switching. Financ. Stochast. 16, 449–476 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kabanov, Y.M., Liptser, R.S., Shiryaev, A.N.: Weak and strong convergence of distributions of counting processes. Teor. Veroyatnost. i Primenen. 28, 288–319 (1983)

    MathSciNet  MATH  Google Scholar 

  25. Koch, G., Spreij, P.J.C.: Software reliability as an application of martingale and filtering theory. IEEE Trans. Reliab. R-32, 342–345 (1983)

    Google Scholar 

  26. Littlewood, B.: Theories of software reliability: how good are they and how can they be improved? IEEE Trans. Softw. Eng. SE-6, 489–500 (1980)

    Google Scholar 

  27. Neuts, M.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Courier Dover Publications, New York (1981)

    MATH  Google Scholar 

  28. van Schuppen, J.H.: Filtering, prediction and smoothing for counting process observations, a martingale approach. SIAM J. Appl. Math. 32, 552–570 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yin, G., Zhu, C.: Hybrid Switching Diffusions: Properties and Applications. Springer, New York (2010)

    Book  MATH  Google Scholar 

  30. Yin, G.: Asymptotic expansions of option price under regime-switching diffusions with a fast-varying switching process. Asymptotic Anal. 65, 203–222 (2009)

    MathSciNet  MATH  Google Scholar 

  31. Zhou, X.Y., Yin, G.: Markowitz mean-variance portfolio selection with regime switching: A continuous-time model. SIAM J. Control Optim. 42, 1466–1482 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Korteweg-de Vries Institute for Mathematics, Universiteit van Amsterdam, PO Box 94248, 1090 GE, Amsterdam, The Netherlands

    Michel Mandjes & Peter Spreij

  2. IMAPP, Radboud Universiteit, Heyendaalseweg 135, 6525AJ, Nijmegen, The Netherlands

    Peter Spreij

Authors
  1. Michel Mandjes
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  2. Peter Spreij
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Correspondence to Peter Spreij .

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Editors and Affiliations

  1. Department of Mathematics, Christian-Albrechts-Universität zu Kiel, Kiel, Germany

    Jan Kallsen

  2. Institute of Mathematics, Technische Universität Berlin, Berlin, Germany

    Antonis Papapantoleon

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Mandjes, M., Spreij, P. (2016). Explicit Computations for Some Markov Modulated Counting Processes. In: Kallsen, J., Papapantoleon, A. (eds) Advanced Modelling in Mathematical Finance. Springer Proceedings in Mathematics & Statistics, vol 189. Springer, Cham. https://doi.org/10.1007/978-3-319-45875-5_3

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