On a formulation of the multiple “disorder” problem

This paper deals with a formulation of the multiple "disorder" problem. Let $\tau_1,\ldots, \tau_n$ be “disorder” emergence times. Available for observation is a process $x$ with differential $dx_t = \theta_t \,dt +\sigma \,dW_t$, where $\theta_t = \sum_{i=1}^{n}a_i I\{t \ge \tau_i\}$ is a Markov process. Having the realization of the process $x$, it is required to estimate $\tau_i$, $i=1,\ldots,n$. The estimation of time $\tau_i$ is based on verifying the hypothesis about reaching the level $A_i = \sum_{k=1}^{i}a_k$ by the process $\theta$.