Matrix equations in Markov modulated Brownian motion: theoretical properties and numerical solution. (English) Zbl 1451.60095
Summary: A Markov modulated Brownian motion (MMBM) is a substantial generalization of the classical Brownian motion and is obtained by allowing the Brownian parameters to be modulated by an underlying Markov chain of environments. As in Brownian motion, the stationary analysis of the MMBM becomes easy once the distributions of the first passage time between levels are determined. S. Asmussen [Commun. Stat., Stochastic Models 11, No. 1, 21–49 (1995; Zbl 0817.60086)] proved that such distributions can be obtained by solving a suitable quadratic matrix equation (QME), while, more recently, S. Ahn and V. Ramaswami [Stoch. Models 33, No. 1, 59–96 (2017; Zbl 1361.60070)] derived the distributions from the solution of a suitable algebraic Riccati equation (NARE). In this paper we provide an explicit algebraic relation between the QME and the NARE, based on a linearization of a matrix polynomial. Moreover, we discuss the doubling algorithms such as the structure-preserving doubling algorithm (SDA) and alternating-directional doubling algorithm (ADDA), with shifting technique, which are used for finding the sought of the NARE.
MSC:
| 60J65 | Brownian motion |
| 60-08 | Computational methods for problems pertaining to probability theory |
| 60J60 | Diffusion processes |
Keywords:
doubling algorithm; first passage time distribution; Markov modulated Brownian motion; matrix polynomials; nonsymmetric algebraic Riccati equation; quadratic convergence; quadratic matrix equationReferences:
| [1] | Agapie, M.; Sohraby, K., 1261-1270, Anchorage, Alaska, USA |
| [2] | Ahn, S.; Badescu, A.; Cheung, E., An IBNR-RBNS Insurance Risk Model with Marked Poisson Arrivals, Insurance: Math. Econ., 79, 26-42 (2018) · Zbl 1400.91238 · doi:10.1016/j.insmatheco.2017.12.004 |
| [3] | Ahn, S.; Ramaswami, V., A Quadratically Convergent Algorithm for First Passage Time Distributions in the Markov Modulated Brownian Motion, Stoch. Models, 33, 59-96 (2017) · Zbl 1361.60070 · doi:10.1080/15326349.2016.1211018 |
| [4] | Asmussen, S., Stationary Distributions for Fluid Flow Models with or without Brownian Noise, Stoch. Models., 11, 1-20 (1995) |
| [5] | Berman, A.; Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences (1994), SIAM: SIAM, Philadelphia, PA · Zbl 0815.15016 |
| [6] | Bini, D. A.; Iannazzo, B.; Meini, B., Numerical Solution of Algebraic Riccati Equations (2012), SIAM: SIAM, Philadelphia, PA · Zbl 1244.65058 |
| [7] | Borodin, A. N.; Salminen, P., Handbook of Brownian Motion—Facts and Formulae (1996), Birkhauser: Birkhauser, Berlin · Zbl 0859.60001 |
| [8] | Dong, L.; Li, J.; Li, G., The Double Deflating Technique for Irreducible Singular M-Matrix Algebraic Riccati Equations in the Critical Case, Linear Multilinear Algebra, 8, 1653-1684 (2019) · Zbl 1417.65121 · doi:10.1080/03081087.2018.1466862 |
| [9] | Gohberg, I.; Lancaster, P.; Rodman, L., Matrix Polynomials. Reprint of the 1982 Original. Classics in Applied Mathematics (2009), SIAM: SIAM, Philadelphia, PA · Zbl 1170.15300 |
| [10] | Guo, C. H.; Iannazzo, B.; Meini, B., On the Doubling Algorithm for a Shifted Nonsymmetric Algebraic Riccati Equation, SIAM J. Matrix Anal. Appl., 63, 109-129 (2007) |
| [11] | Guo, C. H.; Liu, C.; Xue, J., Performance Enhancement of Doubling Algorithms for a Class of Complex Nonsymmetric Algebraic Riccati Equations, IMA J. Numer. Anal., 35, 270-288 (2015) · Zbl 1320.65063 · doi:10.1093/imanum/dru007 |
| [12] | Guo, C. H.; Lu, D., On Algebraic Riccati Equations with Regular Singular M-Matrices, Linear Algebra Appl., 493, 108-119 (2016) · Zbl 1329.15040 · doi:10.1016/j.laa.2015.11.024 |
| [13] | Guo, X. X.; Lin, W. W.; Xu, S. F., A Structure-Preserving Doubling Algorithm for Nonsymmetric Algebraic Riccati Equation, Numer. Math., 103, 393-412 (2006) · Zbl 1097.65055 · doi:10.1007/s00211-005-0673-7 |
| [14] | Karandikar, R. L.; Kulkarni, V. G., Second-Order Fluid Flow Models: Reflected Brownian Motion in a Random Environment, Oper. Res., 43, 77-88 (1995) · Zbl 0821.60087 · doi:10.1287/opre.43.1.77 |
| [15] | Liu, C.; Xue, J., Complex Nonsymmetric Algebraic Riccati Equations Arising in Markov Modulated Fluid Flows, SIAM J. Matrix Anal. Appl., 33, 569-596 (2012) · Zbl 1343.65042 · doi:10.1137/110847731 |
| [16] | Nguyen, G. T.; Latouche, G., The Morphing of Fluid Queues into Markov Modulated Brownian Motion, Stoch. Syst., 5, 62-86 (2015) · Zbl 1336.60151 · doi:10.1214/13-SSY133 |
| [17] | Nguyen, G. T.; Poloni, F., Componentwise Accurate Brownian Motion Computations Using Cyclic Reduction (2015) |
| [18] | Poloni, F.; Reis, T., A Structure-Preserving Doubling Algorithm for Lur’e Equations, Numer. Linear Algebra Appl., 23, 169-186 (2016) · Zbl 1413.65102 · doi:10.1002/nla.2019 |
| [19] | Rogers, L. C. G., Fluid Models in Queueing Theory and Wiener-Hopf Factorization of Markov Chains, Ann. Appl. Probab., 4, 390-413 (1994) · Zbl 0806.60052 · doi:10.1214/aoap/1177005065 |
| [20] | Wang, W.; Wang, W.; Li, R., Alternating-Directional Doubling Algorithm for M-Matrix Algebraic Riccati Equations, SIAM J. Matrix Anal. Appl., 33, 170-194 (2012) · Zbl 1258.65045 · doi:10.1137/110835463 |
| [21] | Xue, J.; Li, R., Highly Accurate Doubling Algorithm for M-Matrix Algebraic Riccati Equations, Numer. Math., 135, 733-767 (2017) · Zbl 1362.65049 · doi:10.1007/s00211-016-0815-0 |
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