Multidimensional quality control problems and quasivariational inequalities

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.