Quickest Change-point Detection Problems for Multidimensional Wiener Processes

We study the quickest change-point (disorder) detection problems for an observable multidimensional Wiener process with the constantly correlated components changing their drift rates at certain unobservable random (change-point) times. These problems seek to determine the times of alarms which should be as close as possible to the unknown change-point times at which some of the components have changed their drift rates. The optimal stopping times of alarm are shown to be the first times at which the appropriate posterior probability processes exit certain regions restricted by the stopping boundaries. We characterise the value functions and optimal boundaries as unique solutions to the associated free-boundary problems for partial differential equations. It is observed that the optimal stopping boundaries can also be uniquely specified by means of the equivalent nonlinear Fredholm integral equations in the class of continuous functions of bounded variation. We also provide estimates for the value functions and boundaries which are solutions to the appropriately constructed ordinary differential free-boundary problems.