An Introduction to Singular Stochastic Control

Bounded variation stochastic control may be defined to include any stochastic control problem in which one restricts the cumulative displacement of the state caused by control to be of bounded variation on finite time intervals. In classical control problems, this cumulative displacement is the integral of the control process (or some function of it), and so is absolutely continuous. In impulse control (see Bensoussan & Lions (1978)), this cumulative displacement has jumps, between which it is either constant or absolutely continuous. Bounded variation control admits both these possibilities and also the possibility that the displacement of the state caused by the optimal control is singularly continuous, at least with positive probability over some interval of time. Problems which exhibit this feature will be called singular, and these are the objects of interest of the present paper.

[1]  Harold J. Kushner,et al.  Optimal stochastic control , 1962 .

[2]  J. Bather.  A continuous time inventory model , 1966, Journal of Applied Probability.

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

[4]  Herman Chernoff,et al.  Sequential decisions in the control of a space-ship (finite fuel) , 1967 .

[5]  J. A. Bather.  A diffusion model for the control of a dam , 1968 .

[6]  Optimum correction under active disturbances , 1968 .

[7]  D. Iglehart,et al.  Multiple channel queues in heavy traffic. I , 1970, Advances in Applied Probability.

[8]  D. Iglehart.  Multiple channel queues in heavy traffic , 1971, Advances in Applied Probability.

[9]  Solution of certain optimal correction problems with error of execution of the control action PMM vol. 38, n≗ 3, 1974, pp. 433-440 , 1974 .

[10]  M. J. Faddy,et al.  Optimal control of finite dams: continuous output procedure , 1974, Advances in Applied Probability.

[11]  Optimal impulse correction under random perturbations , 1975 .

[12]  Stanley R. Pliska,et al.  A diffusion process model for the optimal operation of a reservoir system , 1975, Journal of Applied Probability.

[13]  P. Moerbeke.  On optimal stopping and free boundary problems , 1973, Advances in Applied Probability.

[14]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Optimal stopping rules , 1977 .

[15]  A. Lemoine.  State of the Art---Networks of Queues: A Survey of Weak Convergence Results , 1978 .

[16]  A. J. Taylor,et al.  Optimal control of a Brownian storage system , 1978 .

[17]  Donald Ludwig,et al.  Optimal Harvesting of a Randomly Fluctuating Resource. I: Application of Perturbation Methods , 1979 .

[18]  H. Witsenhausen,et al.  Some solvable stochastic control problemst , 1980 .

[19]  Ioannis Karatzas,et al.  The monotone follower problem in stochastic decision theory , 1981 .

[20]  Robert J. Elliott,et al.  Stochastic calculus and applications , 1984, IEEE Transactions on Automatic Control.

[21]  Maurice Robin,et al.  On some cheap control problems for diffusion processes , 1983 .

[22]  I. Karatzas.  A class of singular stochastic control problems , 1983, Advances in Applied Probability.

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

[24]  Ruth J. Williams.  Brownian motion in a wedge with oblique reflection at the boundary , 1985 .

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

[26]  A finite fuei stochastic control problem , 1983 .

[27]  D. P. Gaver,et al.  Optimal Consumption for General Diffusions with Absorbing and Reflecting Barriers , 1984 .

[28]  Martin I. Reiman,et al.  Open Queueing Networks in Heavy Traffic , 1984, Math. Oper. Res..

[29]  N. L. Lawrie,et al.  Mathematical Computer Performance and Reliability , 1984 .

[30]  S. Shreve,et al.  Connections between Optimal Stopping and Singular Stochastic Control I. Monotone Follower Problems , 1984 .

[31]  S. Shreve,et al.  Connections Between Optimal Stopping and Singular Stochastic Control II. Reflected Follower Problems , 1985 .

[32]  I Karatzas,et al.  Probabilistic aspects of finite-fuel stochastic control. , 1985, Proceedings of the National Academy of Sciences of the United States of America.

[33]  Michael I. Taksar,et al.  Average Optimal Singular Control and a Related Stopping Problem , 1985, Math. Oper. Res..

[34]  J. Harrison,et al.  Brownian motion and stochastic flow systems , 1986 .

[35]  P. Chow,et al.  Additive control of stochastic linear systems with finite horizon , 1985 .

[36]  S. Shreve,et al.  Absolutely continuous and singular stochastic control , 1986 .

[37]  S. Shreve,et al.  Equivalent models for finite-fuel stochastic control , 1986 .

[38]  Friorik mar Baldursson.  Singular stochastic control and optimal stopping , 1987 .