Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Multidimensional quality control problems and quasivariational inequalities

Authors: Robert F. Anderson and Avner Friedman
Journal: Trans. Amer. Math. Soc. 246 (1978), 31-76
MSC: Primary 93E20; Secondary 49A29, 62N10
MathSciNet review: 515529
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A machine can manufacture any one of n m-dimensional Brownian motions with drift ${\lambda _j}$, $P_x^{{\lambda _j}}$, defined on the space of all paths $x\left ( t \right ) \in C\left ( {\left [ {0, \infty } \right ); {R^m}} \right )$. It is given that the product is a random evolution dictated by a Markov process $\theta \left ( t \right )$ with n states, and that the product is $P_x^{{\lambda _j}}$ when $\theta \left ( t \right ) = j, 1 \leqslant j \leqslant n$. One observes the $\sigma$-fields of $x\left ( t \right )$, but not of $\theta \left ( t \right )$. With each product $P_x^{{\lambda _j}}$ there is associated a cost ${c_j}$. One inspects $\theta$ at a sequence of times (each inspection entails a certain cost) and stops production when the state $\theta = n$ is reached. The problem is to find an optimal sequence of inspections. This problem is reduced to solving a certain elliptic quasi variational inequality. The latter problem is actually solved in a rather general case.

References [Enhancements On Off] (What's this?)

  • Robert F. Anderson and Avner Friedman, A quality control problem and quasi-variational inequalities, Arch. Rational Mech. Anal. 63 (1976/77), no. 3, 205–252. MR 456857, DOI
  • A. Bensoussan and J.-L. Lions, Problèmes de temps d’arrêt optimal et inéquations variationnelles paraboliques, Applicable Anal. 3 (1973), 267–294 (French). MR 449843, DOI
  • ---, Contrôle impulsionnel et temps d’arrêt inéquations variationnelles et quasi-variationnelles d’évolution, Cahiers Math. de la Décision, no. 7523, Univ. of Paris 9, Dauphine, 1975.
  • E. B. Dynkin, Theory of Markov processes, Prentice-Hall, Inc., Englewood Cliffs, N.J.; Pergamon Press, Oxford-London-Paris, 1961. Translated from the Russian by D. E. Brown; edited by T. Köváry. MR 0131900
  • Avner Friedman, Stochastic differential equations and applications. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Probability and Mathematical Statistics, Vol. 28. MR 0494490
  • R. S. Liptser and A. N. Shiryaev, Statistics of stochastic processes, Moscow, 1974.
  • Paul-A. Meyer, Probability and potentials, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966. MR 0205288
  • Daniel W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients. I, Comm. Pure Appl. Math. 22 (1969), 345–400. MR 253426, DOI
  • A. N. Širjaev, Some new results in the theory of controlled random processes, Selected Transl. in Math. Statist. and Probability, vol. 8, Amer. Math. Soc., Providence, R. I., 1970, pp. 49-130.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 93E20, 49A29, 62N10

Retrieve articles in all journals with MSC: 93E20, 49A29, 62N10

Additional Information

Keywords: Brownian motion, random evolution, Markov process, stopping time, optimal sequence of inspections, quality control, quasi variational inequality
Article copyright: © Copyright 1978 American Mathematical Society