ON A RANDOM NUMBER OF DISORDERS
Abstract: We register a random sequence which has three segments being the homogeneous
Markov processes. Each segment has its own one-step transition probability law and the
length of the segment is unknown and random. It means that at two random moments ,
, where , the source of observation is changed. In effect, the number of
homogeneous segments is random. The transition probabilities of each process are
known and the a priori distribution of the disorder moments is given. The former
research on such a problem has been devoted to various questions concerning the
distribution changes. The random number of distributional segments creates new
problems in solutions with relation to analysis of the model with deterministic number
of segments. Two cases are presented in detail. In the first one the objective is to
stop on or between the disorder moments while in the second one our objective
is to find the strategy which immediately detects the distribution changes. Both
problems are reformulated to optimal stopping of the observed sequences. The detailed
analysis of the problem is presented to show the form of optimal decision function.
2000 AMS Mathematics Subject Classification: Primary: 60G40, 60K99; Secondary:
90D60.
Keywords and phrases: Disorder problem, sequential detection, optimal stopping,
Markov process, change point, double optimal stopping.