Splitting Algorithms for Rare Events of Semimartingale Reflecting Brownian Motions

References

• Asmussen S, Glynn P (2007) Stochastic Simulation: Algorithms and Analysis (Springer-Verlag, New York).Google Scholar
• Avram F, Dai J, Hasenbein JJ (2001) Explicit solutions for variational problems in the quadrant. Queueing Systems 37(1–3):259–289.Google Scholar
• Blanchet J, Leder K, Shi Y (2011) Analysis of a splitting estimator for rare event probabilities in Jackson networks. Stochastic Systems 1(2):306–339.Link, Google Scholar
• Bramson M, Dai JG, Harrison JM (2010) Positive recurrence of reflecting Brownian motion in three dimensions. Ann. Appl. Probab. 20(2):753–783.Google Scholar
• Budhiraja A, Dupuis P (1999) Simple necessary and sufficient conditions for the stability of constrained processes. SIAM J. Appl. Math. 59(5):1686–1700.Google Scholar
• Budhiraja A, Dupuis P (2019) Analysis and Approximation of Rare Events: Representations and Weak Convergence Methods, vol. 94 (Springer US, New York).Google Scholar
• Budhiraja A, Lee C (2007) Long time asymptotics for constrained diffusions in polyhedral domains. Stochastic Processes Their Appl. 117(8):1014–1036.Google Scholar
• Dai JG, Miyazawa M (2011) Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution. Stochastic Systems 1(1):146–208.Link, Google Scholar
• Dai J, Miyazawa M, Wu J (2014) A multi-dimensional SRBM: Geometric views of its product form stationary distribution. Queueing Systems 78(4):313–335.Google Scholar
• Dean T, Dupuis P (2009) Splitting for rare event simulation: A large deviation approach to design and analysis. Stochastic Processes Their Appl. 119(2):562–587.Google Scholar
• Dean T, Dupuis P (2011) The design and analysis of a generalized RESTART/DPR algorithm for rare event simulation. Ann. Oper. Res. 189(1):63–102.Google Scholar
• Dembo A, Zeitouni O (2010) Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability, vol. 38. (Springer-Verlag, Berlin, Germany)Google Scholar
• Dupuis P, Ishii H (1991) On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications. Stochastics 35(1):31–62.Google Scholar
• Dupuis P, Ramanan K (1999) Convex duality and the Skorokhod problem. I. Probab. Theory Related Fields 115(2):153–195.Google Scholar
• Dupuis P, Ramanan K (2002) A time-reversed representation for the tail probabilities of stationary reflected brownian motion. Stochastic Process. Appl. 98(2):253–287.Google Scholar
• Dupuis P, Williams RJ (1994) Lyapunov functions for semimartingale reflecting Brownian motions. Ann. Probab. 22(2):680–702.Google Scholar
• El Kharroubi A, Yaacoubi A, Tahar AB, Bichard K (2012) Variational problem in the non-negative orthant of R3: Reflective faces and boundary influence cones. Queueing Systems 70(4):299–337.Google Scholar
• Farlow KG (2013) The reflected quasipotential: Characterization and exploration. Unpublished PhD thesis, Virginia Tech, Blacksburg.Google Scholar
• Farlow KG, Day MV (2015) A characterization of the reflected quasipotential. Appl. Math. Optim. 72(3):435–468.Google Scholar
• Harrison JM, Hasenbein JJ (2009) Reflected Brownian motion in the quadrant: Tail behavior of the stationary distribution. Queueing Systems 61(2–3):113–138.Google Scholar
• Harrison JM, Nguyen V (1993) Brownian models of multiclass queueing networks: Current status and open problems. Queueing Systems 13(1–3):5–40.Google Scholar
• Harrison JM, Reiman MI (1981) Reflected Brownian motion on an orthant. Ann. Probab. 9(2):302–308.Google Scholar
• Harrison JM, Williams RJ (1987) Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22(2):77–115.Google Scholar
• Hobson DG, Rogers LC (1993) Recurrence and transience of reflecting Brownian motion in the quadrant. Math. Proc. Cambridge Philos. Soc. 113(2):387–399.Google Scholar
• Karatzas I, Shreve S (2012) Brownian Motion and Stochastic Calculus, vol. 113 (Springer-Verlag, New York).Google Scholar
• Kingman J (1962) On queues in heavy traffic. J. Roy. Statist. Soc. B. 24(2):383–392.Google Scholar
• Liang Z, Hasenbein JJ (2013) Optimal paths in large deviations of symmetric reflected Brownian motion in the octant. Stochastic Systems 3(1):187–229.Link, Google Scholar
• Plemmons R (1977) M-characterizations I—Nonsingular m-matrices. Linear Algebra Appl. 18(2):175–188.Google Scholar
• Reiman MI (1984) Open queueing networks in heavy traffic. Math. Oper. Res. 9(3):441–458.Link, Google Scholar
• Skorohod AV (1961) Stochastic differential equations for a bounded region. Theory Probab. Appl. 6:264–274.Google Scholar
• Stroock DW, Varadhan SS (1971) Diffusion processes with boundary conditions. Comm. Pure Appl. Math. 24(2):147–225.Google Scholar
• Taylor LM, Williams RJ (1993) Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant. Probab. Theory Related Fields 96(3):283–317.Google Scholar
• Watanabe S (1971) On stochastic differential equations for multi-dimensional diffusion processes with boundary conditions. J. Math. Kyoto Univ. 11(1):169–180.Google Scholar
• Williams R (1985) Recurrence classification and invariant measure for reflected Brownian motion in a wedge. Ann. Probab. 13(3):758–778.Google Scholar
• Williams RJ (1995) Semimartingale reflecting Brownian motions in the orthant. Kelly FP, Williams RJ, eds. IMA Volumes in Mathmatics and its Applications, vol. 71 (Springer-Verlag, New York), 125–137.Google Scholar
• Williams R (1996) On the approximation of queueing networks in heavy traffic. Edinburgh, Kelly FP, Zachary S, Ziedins I, eds. Stochastic Networks: Theory Appl., Proc. Royal Soc. Workshop, (Oxford University Press, Oxford, UK), 35–56.Google Scholar
• Williams RJ (2017) Skorokhod problems (lecture notes). Accessed June 2020, http://www.math.ucsd.edu/williams/courses/m28917/skorokhod32017.pdf.Google Scholar