### A Continuous-Time Sequential Testing Problem.

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Suppose that the process $X=\{{X}_{n},n\in \mathbb{N}\}$ is observed sequentially. There are two random moments of time ${\theta}_{1}$ and ${\theta}_{2}$, independent of X, and X is a Markov process given ${\theta}_{1}$ and ${\theta}_{2}$. The transition probabilities of X change for the first time at time ${\theta}_{1}$ and for the second time at time ${\theta}_{2}$. Our objective is to find a strategy which immediately detects the distribution changes with maximal probability based on observation of X. The corresponding problem of double optimal stopping is constructed. The optimal strategy is found...

We offer the quantitative estimation of stability of risk-sensitive cost optimization in the problem of optimal stopping of Markov chain on a Borel space $X$. It is supposed that the transition probability $p(\xb7|x)$, $x\in X$ is approximated by the transition probability $\tilde{p}(\xb7|x)$, $x\in X$, and that the stopping rule ${\tilde{f}}_{*}$ , which is optimal for the process with the transition probability $\tilde{p}$ is applied to the process with the transition probability $p$. We give an upper bound (expressed in term of the total variation distance: ${sup}_{x\in X}\parallel p(\xb7|x)-\tilde{p}(\xb7|x)\parallel )$ for...

A homogeneous Poisson process (N(t),t ≥ 0) with the intensity function m(t)=θ is observed on the interval [0,T]. The problem consists in estimating θ with balancing the LINEX loss due to an error of estimation and the cost of sampling which depends linearly on T. The optimal T is given when the prior distribution of θ is not uniquely specified.

The Bayesian sequential estimation problem for an exponential family of processes is considered. Using a weighted square error loss and observing cost involving a linear function of the process, the Bayes sequential procedures are derived.

The paper presents the stopping rule for random search for Bayesian model-structure estimation by maximising the likelihood function. The inspected maximisation uses random restarts to cope with local maxima in discrete space. The stopping rule, suitable for any maximisation of this type, exploits the probability of finding global maximum implied by the number of local maxima already found. It stops the search when this probability crosses a given threshold. The inspected case represents an important...

The principle of smooth fit is probably the most used tool to find solutions to optimal stopping problems of one-dimensional diffusions. It is important, e.g., in financial mathematical applications to understand in which kind of models and problems smooth fit can fail. In this paper we connect-in case of one-dimensional diffusions-the validity of smooth fit and the differentiability of excessive functions. The basic tool to derive the results is the representation theory of excessive functions;...