Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T14:48:50.866Z Has data issue: false hasContentIssue false
  • English
  • Français

Nested variational inequalities and related optimal starting–stopping problems

Part of: Stochastic processes Hamilton-Jacobi theories, including dynamic programming Markov processes

Published online by Cambridge University Press:  14 July 2016

M. Sun
M. Sun*
Affiliation:
The University of Alabama
*
∗ Postal address: Department of Mathematics, The University of Alabama, University, AL 35487–0350, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper introduces several versions of starting-stopping problem for the diffusion model defined in terms of a stochastic differential equation. The problem could be regarded as a stochastic differential game in which the player can only decide when to start the game and when to quit the game in order to maximize his fortune. Nested variational inequalities arise in studying such a problem, with which we are able to characterize the value function and to obtain optimal strategies.

Keywords

STARTING-STOPPING GAMENESTED VARIATIONAL INEQUALITYDYNAMIC PROGRAMMING

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60J60: Diffusion processes 49L20: Dynamic programming method 90C39: Dynamic programming
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 1 , March 1992 , pp. 104 - 115
Copyright
Copyright © Applied Probability Trust 1992 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bather, J. A. (1970) Optimal stopping problems for Brownian motion. Adv. Appl. Prob. 2, 259–286.Google Scholar
[2]Bensoussan, A. and Lions, J. L. (1973) Problèmes de temps d'arrêt optimal et inéquations variationelles paraboliques. Appl. Anal. 3, 267–294.Google Scholar
[3]Bensoussan, A. and Lions, J. L. (1982) Applications of Variational Inequalities in Stochastic Control. North-Holland, Amsterdam.Google Scholar
[4]Dynkin, E. B. (1963) Optimal choice of the stopping moment of a Markov process (Russian). Dokl. Akad. Nauk. SSSR 150, 238–240.Google Scholar
[5]Grigelionis, B. I. and Shiryayev, A. N. (1966) On the Stefan problem and optimal stopping rules for Markov processes. Theory Prob. Appl. 11, 541–558.Google Scholar
[6]Mazziotto, G. (1986) Approximations of the optimal stopping problem in partial observation. J. Appl. Prob. 23, 341–354.Google Scholar
[7]Menaldi, J. L. (1980) On the optimal stopping time problem for degenerate diffusions. SIAM J. Control Optim. 18, 697–721.Google Scholar
[8]Menaldi, J. L. (1982) On a degenerate variational inequality with Neumann boundary conditions. J. Optim. Theory Appl. 36, 535–563.Google Scholar
[9]van Moerbeke, P. (1974) Optimal stopping and free boundary problems. Rocky Mountain J. Math. 4, 539–578.Google Scholar
[10]Nagai, H. (1978) On an optimal stopping problem and a variational inequality. J. Math. Soc. Japan 30, 303–312.Google Scholar
[11]Robin, M. (1982) On the semigroup approach for ergodic problems of optimal stopping. In Lecture Notes in Control and Information Science 38, pp. 384–392, Springer-Verlag, Berlin.Google Scholar
[12]Shiryayev, A. N. (1969) Statistical Sequential Analysis: Optimal Stopping Rules (Russian). Nauka, Moscow.Google Scholar
[13]Snell, J. L. (1952) Applications of martingale system theorems. Trans. Amer. Math. Soc. 73, 293–312.Google Scholar
[14]Sun, M. (1986) An optimal stopping time problem with time average cost in a bounded interval. Syst. Control Lett. 8, 173–180.Google Scholar
[15]Sun, M. (1990) A multi-dimensional optimal stopping time problem with time average criterion. Opt. Control Appl. Meth. 11, 85–93.Google Scholar
[16]Thomas, R. (1987) Optimal stopping and impulsive control of one-dimensional diffusion processes. Czechoslovak Math. J. 37, 271–292.Google Scholar