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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Multidimensional quality control problems and quasivariational inequalities
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by Robert F. Anderson and Avner Friedman PDF
Trans. Amer. Math. Soc. 246 (1978), 31-76 Request permission


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.
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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 246 (1978), 31-76
  • MSC: Primary 93E20; Secondary 49A29, 62N10
  • DOI:
  • MathSciNet review: 515529