Optimality in Accelerated Life Tests

We formulate a stochastic control problem arising from the optimal design of accelerated life tests. The model is obtained, in a natural way, in the setup of point processes. The cost functional depends on the total number of items under test at time t, the observed number of items failed up to time t, and the stress level applied to the items under test at time t. The optimal policy minimizing the cost functional is characterized via the dynamic programming equation. After a further specification of the model, we prove that the solution is of bang-bang type. In some specific cases, we also prove that the control problem is equivalent to an optimal stopping problem. This amounts to simply running the test, with the number of items and the stress level at their highest available value, up to the exit time of the state process from a given region. In this special case we exhibit an explicit solution and give a criterion in order to decide whether to let the experiment run or to stop it.