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We considered a method for the determining of the statistical characteristics of the magnitude, location and first-passage time of Markov random process, with piecewise constant drift and diffusion coefficients. We found the closed analytical expressions for distribution functions of the specified random variables. We also analyzed the asymptotic behavior of probability density and ordinary moments of location of the greatest maximum of Markov random process and showed their coincidence with some known results for the particular cases.

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The maximum likelihood algorithm is introduced for measuring the unknown moment of abrupt change and bandwidth jump of a fast-fluctuating Gaussian random process. This algorithm can be technically implemented much simpler than the ones obtained by means of common approaches. The technique for calculating the characteristics of the synthesized measurer is presented and the closed analytical expressions for the conditional biases and variances of the resulting estimates are found using the additive local Markov approximation of the decision statistics. By statistical simulation methods, it is confirmed that the presented measurer is operable, while the theoretical formulas describing its performance well approximate the corresponding experimental data in a wide range of the parameter values of the analyzed random process.