Home | Articles | Past volumes | About | Login | Notify | Contact | Search
 Stochastic Systems > Vol. 7 (2017) open journal systems 


Asymptotic expansion of stationary distribution for reflected Brownian motion in the quarter plane via analytic approach

Sandro Franceschi, UPMC
Irina Kourkova, UPMC


Abstract
Brownian motion in \(\mathbf{R}_+^2\) with covariance matrix \(\Sigma\) and drift \(\mu\) in the interior and reflection matrix \(R\) from the axes is considered. The asymptotic expansion of the stationary distribution density along all paths in \(\mathbf{R}_+^2\) is found and its main term is identified depending on parameters \((\Sigma, \mu, R)\). For this purpose the analytic approach of Fayolle, Iasnogorodski and Malyshev in in [12] and [36], restricted essentially up to now to discrete random walks in \(\mathbf{Z}_+^2\) with jumps to the nearest-neighbors in the interior is developed in this article for diffusion processes on \(\mathbf{R}_+^2\) with reflections on the axes.

AMS 2000 subject classifications: Primary 60J65, 05A15, 37L40; secondary 60K25, 30F10, 30D05

Keywords: Reflected Brownian motion in the quarter plane, stationary distribution, Laplace transform, asymptotic analysis, saddle-point method, Riemann surface.

Creative Common LOGO

Full Text: PDF


Franceschi, Sandro, Kourkova, Irina, Asymptotic expansion of stationary distribution for reflected Brownian motion in the quarter plane via analytic approach, Stochastic Systems, 7, (2017), 32-94 (electronic). DOI: 10.1214/16-SSY218.

References

[1]     Avram, F., Dai, J. G., and Hasenbein, J. J. (2001). Explicit solutions for variational problems in the quadrant. Queueing Systems. Theory and Applications, 37(1–3):259–289. MR1833666

[2]     Baccelli, F. and Fayolle, G. (1987). Analysis of models reducible to a class of diffusion processes in the positive quarter plane. SIAM Journal on Applied Mathematics, 47(6):1367–1385. MR0916246

[3]     Brychkov, Y., Glaeske, H.-J., Prudnikov, A., and Tuan, V. K. (1992). Multidimensional Integral Transformations. CRC Press. MR1177594

[4]     Dai, J. (1990). Steady-state analysis of reflected Brownian motions: Characterization, numerical methods and queueing applications. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–Stanford University. MR2685786

[5]     Dai, J. G. and Harrison, J. M. (1992). Reflected Brownian motion in an orthant: numerical methods for steady-state analysis. The Annals of Applied Probability, 2(1):65–86. MR1143393

[6]     Dai, J. G. and Miyazawa, M. (2011). Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution. Stochastic Systems, 1(1):146–208. MR2948920

[7]     Dai, J. G. and Miyazawa, M. (2013). Stationary distribution of a two-dimensional SRBM: geometric views and boundary measures. Queueing Systems, 74(2–3):181–217. MR3054655

[8]     Dieker, A. B. and Moriarty, J. (2009). Reflected Brownian motion in a wedge: sum-of-exponential stationary densities. Electronic Communications in Probability, 14:1–16. MR2472171

[9]     Doetsch, G. (1974). Introduction to the Theory and Application of the Laplace Transformation. Springer Berlin Heidelberg, Berlin, Heidelberg. MR0344810

[10]     Dupuis, P. and Williams, R. J. (1994). Lyapunov functions for semimartingale reflecting Brownian motions. The Annals of Probability, 22(2):680–702. MR1288127

[11]     Fayolle, G. and Iasnogorodski, R. (1979). Two coupled processors: The reduction to a Riemann-Hilbert problem. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 47(3):325–351. MR0525314

[12]     Fayolle, G., Iasnogorodski, R., and Malyshev, V. (1999). Random Walks in the Quarter-Plane. Springer Berlin Heidelberg, Berlin, Heidelberg. MR1691900

[13]     Fayolle, G. and Raschel, K. (2015). About a possible analytic approach for walks in the quarter plane with arbitrary big jumps. Comptes Rendus Mathematique, 353(2):89–94. MR3300936

[14]     Fedoryuk, M. V. (1977). The saddle-point method. Izdat. “Nauka”, Moscow. MR0507923

[15]     Fedoryuk, M. V. (1989). Asymptotic methods in analysis. In Analysis I, pages 83–191. Springer. MR0899753

[16]     Foddy, M. E. (1984). Analysis of Brownian motion with drift, confined to a quadrant by oblique reflection (diffusions, Riemann-Hilbert problem). ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–Stanford University. MR2633392

[17]     Franceschi, S. and Raschel, K. (2016). Tutte’s invariant approach for Brownian motion reflected in the quadrant. ESAIM Probab. Stat. to appear.

[18]     Franceschi, S. and Raschel, K. (2017). Explicit expression for the stationary distribution of reflected brownian motion in a wedge. Preprint arXiv:1703.09433.

[19]     Harrison, J. M. (1978). The diffusion approximation for tandem queues in heavy traffic. Advances in Applied Probability, 10(4):886–905. MR0509222

[20]     Harrison, J. M. and Hasenbein, J. J. (2009). Reflected Brownian motion in the quadrant: tail behavior of the stationary distribution. Queueing Systems, 61(2–3):113–138. MR2485885

[21]     Harrison, J. M. and Nguyen, V. (1993). Brownian models of multiclass queueing networks: current status and open problems. Queueing Systems. Theory and Applications, 13(1–3):5–40. MR1218842

[22]     Harrison, J. M. and Williams, R. J. (1987a). Brownian models of open queueing networks with homogeneous customer populations. Stochastics, 22(2):77–115. MR0912049

[23]     Harrison, J. M. and Williams, R. J. (1987b). Multidimensional reflected Brownian motions having exponential stationary distributions. The Annals of Probability, 15(1):115–137. MR0877593

[24]     Hobson, D. G. and Rogers, L. C. G. (1993). Recurrence and transience of reflecting Brownian motion in the quadrant. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 113, pages 387–399. Cambridge Univ Press. MR1198420

[25]     Ignatyuk, I. A., Malyshev, V. A., and Shcherbakov, V. V. (1994). The influence of boundaries in problems on large deviations. Rossiĭskaya Akademiya Nauk. Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, 49(2(296)):43–102. MR1283135

[26]     Kourkova, I. and Raschel, K. (2011). Random walks in Z+2 with non-zero drift absorbed at the axes. Bulletin de la Société Mathématique de France, 139:341–387. MR2869310

[27]     Kourkova, I. and Raschel, K. (2012). On the functions counting walks with small steps in the quarter plane. Publications mathématiques de l’IHES, 116(1):69–114. MR3090255

[28]     Kurkova, I. A. and Malyshev, V. A. (1998). Martin boundary and elliptic curves. Markov Processes and Related Fields, 4(2):203–272. MR1641546

[29]     Kurkova, I. A. and Suhov, Y. M. (2003). Malyshev’s Theory and JS-Queues. Asymptotics of Stationary Probabilities. The Annals of Applied Probability, 13(4):1313–1354. MR2023879

[30]     Latouche, G. and Miyazawa, M. (2013). Product-form characterization for a two-dimensional reflecting random walk. Queueing Systems, 77(4):373–391. MR3225816

[31]     Lieshout, P. and Mandjes, M. (2007). Tandem Brownian queues. Mathematical Methods of Operations Research, 66(2):275–298. MR2342215

[32]     Lieshout, P. and Mandjes, M. (2008). Asymptotic analysis of Lévy-driven tandem queues. Queueing Systems. Theory and Applications, 60(3–4):203–226. MR2461616

[33]     Majewski, K. (1996). Large Deviations of Stationary Reflected Brownian Motions. Stochastic Networks: Theory and Applications.

[34]     Majewski, K. (1998). Large deviations of the steady-state distribution of reflected processes with applications to queueing systems. Queueing Systems, 29(2–4):351–381. MR1654452

[35]     Malyshev, V. A. (1970). Sluchainye bluzhdaniya Uravneniya. Vinera-Khopfa v chetverti ploskosti. Avtomorfizmy Galua. Izdat. Moskov. Univ., Moscow. MR0428464

[36]     Malyshev, V. A. (1973). Asymptotic behavior of the stationary probabilities for two-dimensional positive random walks. Siberian Mathematical Journal, 14(1):109–118. MR0433604

[37]     Pemantle, R. and Wilson, M. C. (2013). Analytic combinatorics in several variables, volume 140 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge. MR3088495

[38]     Raschel, K. (2012). Counting walks in a quadrant: a unified approach via boundary value problems. Journal of the European Mathematical Society, pages 749–777. MR2911883

[39]     Reiman, M. I. and Williams, R. J. (1988). A boundary property of semimartingale reflecting Brownian motions. Probability Theory and Related Fields, 77(1):87–97. MR0921820

[40]     Taylor, L. M. and Williams, R. J. (1993). Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant. Probability Theory and Related Fields, 96(3):283–317. MR1231926

[41]     Williams, R. J. (1985). Recurrence classification and invariant measure for reflected Brownian motion in a wedge. Annals of Probabability, 13:758–778. MR0799421

[42]     Williams, R. J. (1995). Semimartingale reflecting Brownian motions in the orthant. Stochastic Networks, 13. MR1381009

[43]     Williams, R. J. (1996). On the Approximation of Queuing Networks in Heavy Traffic. In Stochastic Networks: Theory and Applications, Royal Statistical Society Series. F. P. Kelly, S. Zachary, and I. Ziedins, oxford university press edition.




Home | Articles | Past volumes | About | Login | Notify | Contact | Search

Stochastic Systems. ISSN: 1946-5238