On Singular Control Problems, the Time-Stretching Method, and the Weak-M1 Topology

We consider a general class of singular control problems with state constraints. Budhiraja and Ross (2006) established the existence of optimal controls for a relaxed version of this class of problems by using the so-called `time-stretching' method and the J1-topology. We show that the weak-M1 topology is better suited for establishing existence, since by using it, one bypasses the need for time-transformations, without any additional effort. Furthermore, we reveal how the time-scaling feature in the definition of the weak-M1 distance embeds the time-stretching method's scheme. This case study suggests that one can benefit from working with the weak-M1 topology in other singular control frameworks, such as queueing control problems under heavy traffic.

[1]  A. Skorokhod.  Limit Theorems for Stochastic Processes , 1956 .

[2]  L. F. Martins,et al.  Routing and singular control for queueing networks in heavy traffic , 1990 .

[3]  A. R. Norman,et al.  Portfolio Selection with Transaction Costs , 1990, Math. Oper. Res..

[4]  J. Michael Harrison,et al.  Instantaneous Control of Brownian Motion , 1983, Math. Oper. Res..

[5]  Herman Chernoff,et al.  Sequential decisions in the control of a spaceship , 1967 .

[6]  Amarjit Budhiraja,et al.  Convergent Numerical Scheme for Singular Stochastic Control with State Constraints in a Portfolio Selection Problem , 2007, SIAM J. Control. Optim..

[7]  Bojan Basrak,et al.  A Multivariate Functional Limit Theorem in Weak M1 Topology , 2013 .

[8]  H. Soner,et al.  Regularity of the value function for a two-dimensional singular stochastic control problem , 1989 .

[9]  F. Delarue,et al.  Particle systems with a singular mean-field self-excitation. Application to neuronal networks , 2014, 1406.1151.

[10]  Asaf Cohen,et al.  Asymptotic Analysis of a Multiclass Queueing Control Problem Under Heavy Traffic with Model Uncertainty , 2017, Stochastic Systems.

[11]  Ruth J. Williams,et al.  Existence and Uniqueness of Semimartingale Reflecting Brownian Motions in Convex Polyhedrons , 1996 .

[12]  Steven E. Shreve,et al.  A free boundary problem related to singular stochastic control , 1990 .

[13]  Ward Whitt,et al.  Functional large deviation principles for first-passage-time processes , 1997 .

[14]  Bojan Basrak,et al.  A Multivariate Functional Limit Theorem in Weak $$M_{1}$$M1 Topology , 2013, 1308.3624.

[15]  Mark Shifrin,et al.  An asymptotic optimality result for the multiclass queue with finite buffers in heavy traffic , 2014, 1412.6775.

[16]  Guodong Pang,et al.  Continuity of a queueing integral representation in the ${M}_{\mathbf{1}}$ topology , 2010, 1001.2381.

[17]  R. Atar,et al.  Singular control with state constraints on unbounded domain , 2005, math/0501016.

[18]  A. Budhiraja,et al.  Existence of optimal controls for singular control problems with state constraints , 2006, math/0702418.

[19]  R. J. Williams,et al.  Workload reduction of a generalized Brownian network , 2005, math/0602495.

[20]  Ward Whitt,et al.  Functional Large Deviation Principles for Waiting and Departure Processes , 1998, Probability in the Engineering and Informational Sciences.

[21]  W. Whitt.  Weak convergence of first passage time processes , 1971, Journal of Applied Probability.

[22]  Mykhaylo Shkolnikov,et al.  Particle systems with singular interaction through hitting times: Application in systemic risk modeling , 2017, The Annals of Applied Probability.

[23]  Avishai Mandelbaum,et al.  Strong Approximations for Time-Dependent Queues , 1995, Math. Oper. Res..

[24]  L. F. Martins,et al.  Numerical Methods for Stochastic Singular Control Problems , 1991 .

[25]  Søren Asmussen,et al.  Controlled diffusion models for optimal dividend pay-out , 1997 .

[26]  Weian Zheng,et al.  Tightness criteria for laws of semimartingales , 1984 .

[27]  Ward Whitt,et al.  LIMITS FOR CUMULATIVE INPUT PROCESSES TO QUEUES , 2000, Probability in the Engineering and Informational Sciences.

[28]  Thomas G. Kurtz,et al.  Random Time Changes and Convergence in Distribution Under the Meyer-Zheng Conditions , 1991 .

[29]  W. Whitt,et al.  Diffusion approximations for queues with server vacations , 1990, Advances in Applied Probability.

[30]  J. Michael Harrison,et al.  Brownian Models of Queueing Networks with Heterogeneous Customer Populations , 1988 .

[31]  Sidney I. Resnick,et al.  Weak Convergence of high-speed network traffic models , 2000 .

[32]  Amarjit Budhiraja,et al.  Diffusion approximations for controlled stochastic networks: An asymptotic bound for the value function , 2006 .

[33]  Ward Whitt,et al.  An Introduction to Stochastic-Process Limits and their Application to Queues , 2002 .

[34]  M. J. Wichura.  Functional laws of the Iterated Logarithm for the Partial Sums of I. I. D. Random Variables in the Domain of Attraction of a Completely Asymmetric Stable Law , 1974 .

[35]  Ruth J. Williams,et al.  A multiclass closed queueing network with unconventional heavy traffic behavior , 1996 .

[36]  Jan A. Van Mieghem,et al.  Dynamic Control of Brownian Networks: State Space Collapse and Equivalent Workload Formulations , 1997 .

[37]  Ulrich Horst,et al.  Mean Field Games with Singular Controls , 2016, SIAM J. Control. Optim..