On a random number of disorders

We register a random sequence which has the following properties: it has three segments being the homogeneous Markov processes. Each segment has his own one step transition probability law and the length of the segment is unknown and random. It means that at two random successive moments (they can be equal also and equal zero too) the source of observations is changed and the first observation in new segment is chosen according to new transition probability starting from the last state of the previous segment. In effect the number of homogeneous segments is random. The transition probabilities of each process are known and a priori distribution of the disorder moments is given. The former research on such 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 details. In the first one the objectives 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.

💡 Research Summary

The paper studies a stochastic observation sequence that consists of three homogeneous Markov segments whose lengths are random and unknown. Each segment i (i = 1, 2, 3) is governed by a known one‑step transition matrix (P_i). The change‑points (disorders) (\tau_1) and (\tau_2) are random variables with a given prior distribution; they may be equal, may be zero, and therefore the number of effective homogeneous segments is itself random. The observation process (X_0,X_1,\dots) starts from a known initial state and evolves according to the transition matrix of the current segment; at a disorder moment the first observation of the new segment is drawn using the transition law of the new matrix, starting from the last state of the previous segment, which guarantees continuity of the path.

Two optimal‑stopping problems are formulated.

1. Stopping near the disorders. A loss function
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