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DIFFUSION LIMITS FOR A MARKOV MODULATED BINOMIAL COUNTING PROCESS

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  30 January 2019

Peter Spreij  and
Jaap Storm
Peter Spreij
Affiliation:
Korteweg-de Vries Institute for Mathematics, Universiteit van Amsterdam, Amsterdam, The Netherlands and IMAPP, Radboud University Nijmegen, Nijmegen, The Netherlands E-mail: [email protected]
Jaap Storm
Affiliation:
Department of Mathematics, Vrije Universiteit, Amsterdam, The Netherlands E-mail: [email protected]
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Abstract

In this paper, we study limit behavior for a Markov-modulated binomial counting process, also called a binomial counting process under regime switching. Such a process naturally appears in the context of credit risk when multiple obligors are present. Markov-modulation takes place when the failure/default rate of each individual obligor depends on an underlying Markov chain. The limit behavior under consideration occurs when the number of obligors increases unboundedly, and/or by accelerating the modulating Markov process, called rapid switching. We establish diffusion approximations, obtained by application of (semi)martingale central limit theorems. Depending on the specific circumstances, different approximations are found.

Keywords

central limit theoremscounting processfunctional limit theoremsMarkov-modulated process

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60F05: Central limit and other weak theorems 60G99: None of the above, but in this section
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 34 , Issue 2 , April 2020 , pp. 235 - 257
Copyright
Copyright © Cambridge University Press 2019

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References

1.Anderson, D., Blom, J., Mandjes, M., Thorsdottir, H., & De Turck, K. (2016). A functional central limit theorem for a Markov-modulated infinite-server queue. Methodology and Computing in Applied Probability 18(1): 153–168.CrossRefGoogle Scholar
2.Andersson, H. & Britton, T. (2012). Stochastic epidemic models and their statistical analysis. Vol. 151. New York: Springer Science & Business Media.Google Scholar
3.Ando, T., Okamura, H., & Dohi, T. (2006). Estimating Markov modulated software reliability models via em algorithm. In Dependable, Autonomic and Secure Computing, 2nd IEEE International Symposium on, pp. 111–118. IEEE.CrossRefGoogle Scholar
4.Ang, A. & Bekaert, G. (2002). Regime switches in interest rates. Journal of Business & Economic Statistics 20(2): 163–182.CrossRefGoogle Scholar
5.Asmussen, S. (2008). Applied probability and queues. Vol. 51. New York: Springer Science & Business Media.Google Scholar
6.Asmussen, S. & Albrecher, H. (2010). Ruin probabilities. Hackensack, NJ: World Scientific Publishing Co Pte Ltd.CrossRefGoogle Scholar
7.Banerjee, T. (2016). Analyzing Credit Risk Models in a Regime Switching Market. Ph.D. thesis, G25537.Google Scholar
8.Banerjee, T., Ghosh, M.K., & Iyer, S.K. (2013). Pricing credit derivatives in a Markov-modulated reduced-form model. International Journal of Theoretical and Applied Finance 16(04): 1350018.CrossRefGoogle Scholar
9.Berchenko, Y., Rosenblatt, J.D., & Frost, S.D. (2017). Modeling and analyzing respondent-driven sampling as a counting process. Biometrics 73(4): 1189–1198.CrossRefGoogle ScholarPubMed
10.Blom, J., De Turck, K., & Mandjes, M. (2016). Functional central limit theorems for Markov-modulated infinite-server systems. Mathematical Methods of Operations Research 83(3): 351–372.CrossRefGoogle Scholar
11.Brémaud, P. (1981). Point processes and queues: martingale dynamics. Vol. 50. New York/Berlin: Springer.CrossRefGoogle Scholar
12.Buffington, J. & Elliott, R.J. (2002). American options with regime switching. International Journal of Theoretical and Applied Finance 5(05): 497–514.CrossRefGoogle Scholar
13.Chen, A. & Delong, Ł (2015). Optimal investment for a defined-contribution pension scheme under a regime switching model. ASTIN Bulletin: The Journal of the IAA 45(2): 397–419.CrossRefGoogle Scholar
14.Choi, S. & Marcozzi, M.D. (2015). A regime switching model for the term structure of credit risk spreads. Journal of Mathematical Finance 5(01): 49.CrossRefGoogle Scholar
15.Coolen-Schrijner, P. & Van Doorn, E.A. (2002). The deviation matrix of a continuous-time Markov chain. Probability in the Engineering and informational Sciences 16(3): 351–366.CrossRefGoogle Scholar
16.Dunbar, K. & Edwards, A.J. (2007). Empirical analysis of credit risk regime switching and temporal conditional default correlation in credit default swap valuation: the market liquidity effect. Working paper 200710, University of Connecticut, Department of Economics.Google Scholar
17.Elliott, R.J. (1993). New finite-dimensional filters and smoothers for noisily observed Markov chains. IEEE Transactions on Information Theory 39(1): 265–271.CrossRefGoogle Scholar
18.Elliott, R. & Hinz, J. (2002). Portfolio optimization, hidden Markov models, and technical analysis of p&f-charts. International Journal of Theoretical and Applied Finance 5(04): 385–399.CrossRefGoogle Scholar
19.Elliott, R.J. & Mamon, R.S. (2002). An interest rate model with a Markovian mean reverting level. Quantitative Finance 2(6): 454–458.CrossRefGoogle Scholar
20.Elliott, R.J. & Siu, T.K. (2009). On Markov-modulated exponential-affine bond price formulae. Applied Mathematical Finance 16(1): 1–15.CrossRefGoogle Scholar
21.Elliott, R.J., Van der Hoek, J. (1997). An application of hidden Markov models to asset allocation problems. Finance and Stochastics 1(3): 229–238.CrossRefGoogle Scholar
22.Elliott, R.J., Kuen Siu, T., Badescu, A. (2011). Bond valuation under a discrete-time regime-switching term-structure model and its continuous-time extension. Managerial Finance 37(11): 1025–1047.CrossRefGoogle Scholar
23.Giampieri, G., Davis, M., & Crowder, M. (2005). Analysis of default data using hidden Markov models. Quantitative Finance 5(1): 27–34.CrossRefGoogle Scholar
24.Glynn, P.W. (1984). Some asymptotic formulas for Markov chains with applications to simulation. Journal of Statistical Computation and Simulation 19(2): 97–112.CrossRefGoogle Scholar
25.Hainaut, D. & Colwell, D.B. (2016). A structural model for credit risk with switching processes and synchronous jumps. The European Journal of Finance 22(11): 1040–1062.CrossRefGoogle Scholar
26.Hamilton, J.D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica: Journal of the Econometric Society 57(2): 357–384.CrossRefGoogle Scholar
27.Hellmich, M. (2016). Statistical inference of a software reliability model by linear filtering. Journal of Statistics and Management Systems 19(2): 163–181.CrossRefGoogle Scholar
28.Huang, G., Mandjes, M., & Spreij, P. (2014). Weak convergence of Markov-modulated diffusion processes with rapid switching. Statistics & Probability Letters 86: 74–79.CrossRefGoogle Scholar
29.Huang, G., Jansen, H., Mandjes, M., Spreij, P., & De Turck, K. (2016). Markov-modulated ornstein–uhlenbeck processes. Advances in Applied Probability 48(1): 235–254.CrossRefGoogle Scholar
30.Huang, G., Mandjes, M., & Spreij, P. (2016). Large deviations for Markov-modulated diffusion processes with rapid switching. Stochastic Processes and their Applications 126(6): 1785–1818.CrossRefGoogle Scholar
31.Jacod, J. & Shiryaev, A. (2013). Limit theorems for stochastic processes. Vol. 288. Berlin: Springer Science & Business Media.Google Scholar
32.Jansen, M. (2018). Scaling limits for modulated infinite-server queues and related stochastic processes. Ph.D. thesis, University of Amsterdam and Ghent University.Google Scholar
33.Jelinski, Z. & Moranda, P. (1972). Software reliability research. In Freiberger, W. (ed.), Statistical computer performance evaluation. Providence, RI: Elsevier, pp. 465–484.CrossRefGoogle Scholar
34.Jiang, Z. (2015). Optimal dividend policy when cash reserves follow a jump-diffusion process under Markov-regime switching. Journal of Applied Probability 52(1): 209–223.CrossRefGoogle Scholar
35.Jiang, Z. & Pistorius, M.R. (2008). On perpetual american put valuation and first-passage in a regime-switching model with jumps. Finance and Stochastics 12(3): 331–355.CrossRefGoogle Scholar
36.Jiang, Z. & Pistorius, M. (2012). Optimal dividend distribution under Markov regime switching. Finance and Stochastics 16(3): 449–476.CrossRefGoogle Scholar
37.Koch, G. & Spreij, P. (1983). Software reliability as an application of martingale & filtering theory. IEEE Transactions on Reliability 32(4): 342–345.CrossRefGoogle Scholar
38.Landon, J., Özekici, S., & Soyer, R. (2013). A Markov modulated poisson model for software reliability. European Journal of Operational Research 229(2): 404–410.CrossRefGoogle Scholar
39.Li, J. & Ma, S. (2013). Pricing options with credit risk in Markovian regime-switching markets. Journal of Applied Mathematics 2013: 1–9.Google Scholar
40.Liechty, J. (2013). Regime switching models and risk measurement tools. In Fouque, J.-P. and Langsam, J.A. (eds), Handbook on Systemic Risk. Cambridge: Cambridge University Press, pp. 180.CrossRefGoogle Scholar
41.Liptser, R. & Shiryaev, A.N. (2012). Theory of martingales. Mathematics and its Applications (Soviet Series) Vol. 49. Dordrecht: Kluwer Academic Publishers Group.Google Scholar
42.Littlewood, B. (1975). A reliability model for systems with Markov structure. Applied Statistics 24(2): 172–177.CrossRefGoogle Scholar
43.Mandjes, M. & Spreij, P. (2016). Explicit computations for some Markov modulated counting processes. In Kallsen, J. and Papapantoleon, A. (eds), Advanced Modelling in Mathematical Finance. Cham: Springer, pp. 63–89.CrossRefGoogle Scholar
44.Neuts, M.F. (1981). Matrix-geometric solutions in stochastic models. An algorithmic approach. Johns Hopkins Series in the Mathematical Sciences, 2. Baltimore, MD: Johns Hopkins University Press.Google Scholar
45.Norris, J.R. (1998). Markov chains, Volume 2 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge University Press.Google Scholar
46.Özekici, S. & Soyer, R. (2003). Reliability of software with an operational profile. European Journal of Operational Research 149(2): 459–474.CrossRefGoogle Scholar
47.Özekici, S. & Soyer, R. (2004). Reliability modeling and analysis in random environments. In Soyer, R., Mazzuchi, T.A. and Singpurwalla, N.D. (eds), Mathematical reliability: an expository perspective. Boston, MA: Springer, pp. 249–273.CrossRefGoogle Scholar
48.Ravishanker, N., Liu, Z., & Ray, B.K. (2008). Nhpp models with Markov switching for software reliability. Computational Statistics & Data Analysis 52(8): 3988–3999.CrossRefGoogle Scholar
49.Revuz, D. & Yor, M. (2013). Continuous martingales and Brownian motion. Vol. 293. Berlin: Springer Science & Business Media.Google Scholar
50.Spreij, P. (1990). Self-exciting counting process systems with finite state space. Stochastic processes and their applications 34(2): 275–295.CrossRefGoogle Scholar
51.Spreij, P. (1998). A representation result for finite Markov chains. Statistics & probability letters 38(2): 183–186.CrossRefGoogle Scholar
52.Subrahmaniam, V.T., Dewanji, A., & Roy, B.K. (2015). A semiparametric software reliability model for analysis of a bug-database with multiple defect types. Technometrics 57(4): 576–585.CrossRefGoogle Scholar
53.van Beek, M., Mandjes, M., Spreij, P., & Winands, E. (2014). Markov switching affine processes and applications to pricing. In Actuarial And Financial Mathematics Conference, Brussels, February 6–7, pp. 97–102.Google Scholar
54.Whitt, W. (1980). Some useful functions for functional limit theorems. Mathematics of Operations Research 5(1): 67–85.CrossRefGoogle Scholar
55.Yin, G. (2009). Asymptotic expansions of option price under regime-switching diffusions with a fast-varying switching process. Asymptotic Analysis 65(3–4): 203–222.CrossRefGoogle Scholar
56.Zhou, X.Y. & Yin, G. (2003). Markowitz's mean-variance portfolio selection with regime switching: a continuous-time model. SIAM Journal on Control and Optimization 42(4): 1466–1482.CrossRefGoogle Scholar