DOI QR코드

DOI QR Code

Application of GTH-like algorithm to Markov modulated Brownian motion with jumps

  • Received : 2021.02.15
  • Accepted : 2021.04.13
  • Published : 2021.09.30

Abstract

The Markov modulated Brownian motion is a substantial generalization of the classical Brownian Motion. On the other hand, the Markovian arrival process (MAP) is a point process whose family is dense for any stochastic point process and is used to approximate complex stochastic counting processes. In this paper, we consider a superposition of the Markov modulated Brownian motion (MMBM) and the Markovian arrival process of jumps which are distributed as the bilateral ph-type distribution, the class of which is also dense in the space of distribution functions defined on the whole real line. In the model, we assume that the inter-arrival times of the MAP depend on the underlying Markov process of the MMBM. One of the subjects of this paper is introducing how to obtain the first passage probabilities of the superposed process using a stochastic doubling algorithm designed for getting the minimal solution of a nonsymmetric algebraic Riccatti equation. The other is to provide eigenvalue and eigenvector results on the superposed process to make it possible to apply the GTH-like algorithm, which improves the accuracy of the doubling algorithm.

Keywords

Acknowledgement

This work was supported by the 2018 Research Fund of the University of Seoul.

References

  1. Ahn S and Ramaswami V (2005). Bilateral phase type distributions. Stochastic Models, 21 239-259. https://doi.org/10.1081/STM-200056029
  2. Ahn S (2016). Total shift during the first passages of Markov modulated Brownian motion with bilateral ph-type jumps: Formulas driven by the minimal solution matrix of a Riccati equation, Stochastic Models, 32, 433-459. https://doi.org/10.1080/15326349.2016.1165621
  3. Ahn S (2017). Time-dependent and stationary analyses of the two-sided reflected Markov modulated Brownian motion with bilateral ph-type jumps, Journal of the Korean Statistical Society, 46, 45-69. https://doi.org/10.1016/j.jkss.2016.06.002
  4. Alfa AS, Xue J, and Ye Q (2001) Accurate computation of the smallest eigenvalue of a diagonally dominant M-matrix, Math. Comp., 71, 217-236. https://doi.org/10.1090/S0025-5718-01-01325-4
  5. Asmussen S (1995). Stationary distributions for fluid flow models with or without Brownian noise. Stochastic Models, 11, 1-20. https://doi.org/10.1080/15326349508807329
  6. Asmussen S (2003). Applied Probability and Queues(2nd ed.), Springer, Verlag.
  7. Belhaj M (2010). Optimal dividend payments when risk reserves follow a jump diffusion process. Mathematical Finance, 20, 313-325. https://doi.org/10.1111/j.1467-9965.2010.00399.x
  8. Bini DA, Iannazzo B, and Meini B (2012). Numerical Solution of Algebraic Riccati Equations, Siam, Philadelphia.
  9. Guo XX, Lin WW, and Xu SF (2006). A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation, Numer. Math., 103, 393-412. https://doi.org/10.1007/s00211-005-0673-7
  10. Jiang Z and Pistorius MR (2008). On perpetual American put valuation and first passage in a regimeswitching model with jumps. Finance Stoch. 12, 331-355. https://doi.org/10.1007/s00780-008-0065-9
  11. Liu C and Xue J (2012). Complex nonsymmetric algebraic Riccati equations arising in Markov modulated fluid flows, SIAM J. Matrix Anal. Appl., 33, 569-596. https://doi.org/10.1137/110847731
  12. Wang W, Wang W, and Li R (2012). Alternating-directional doubling algorithm for M-matrix algebraic Riccati equations, SIAM. J. Matrix Anal. & Appl., 33, 170-194. https://doi.org/10.1137/110835463
  13. Whitt W (1980). Some useful functions for functional limit theorems, Mathematics of Operations Research, 5, 67-85. https://doi.org/10.1287/moor.5.1.67
  14. Xue J and Li R (2017) Highly accurate doubling algorithm for M-matrix algebraic Riccati equations, Numer. Math. 135, 733-767. https://doi.org/10.1007/s00211-016-0815-0