We derive new properties of the sequential parameter estimators for diffusion-type processes X = {Xt, 0 5 t 5 tau}, where tau is a stopping time (this includes the case of fixed sample size estimate). Some earlier theoretical results in this direction can be found in the book [R. S. Liptser and A. N. Shiryaev, Statistics of Random Processes, Springer, 2001]. Under an essentially less restrictive setting, we derive formulas for the moments of the maximum likelihood estimator (MLE) lambda b tau for the parameter lambda of the drift coefficient ft(lambda) = at - lambda bt and prove the exponential boundedness of lambda b tau under a mild condition. In the provided examples we consider the mean-reverting ergodic diffusion process X, where bt = Xt, and the diffusion coefficient sigma t = sigma Xt gamma. In particular, we provide nonasymptotic analytical and numerical results for the bias and mean-square error of lambda b tau for the Ornstein-Uhlenbeck (O-U) and Cox-Ingersoll-Ross (CIR) processes when tau = T is a fixed sample size, and tau = tau H is a specially chosen stopping time that guarantees a prescribed magnitude of 1/H for the variance of lambda b tau H.

This review paper discusses firstly some statistical optimization problems that are known as “changed point” detection based on data given for all possible times of observation. The main text concerns the “quickest detection problems” in the case of sequential observations.

This paper gives summaries of the talks presented during the 2023 fall term and 2024 spring term of the General Seminar of the Department of Probability, Moscow State University. The seminar was held under the direction of A. N. Kolmogorov and B. V. Gnedenko. Current information about the seminar is available from the Department of Probability Theory website.

В работе изучаются свойства последовательных оценок параметров процессов диффузионного типа $\mathbf{X}=\{X_t, 0\leqslant t\leqslant \tau \}$, где $\tau $ - момент остановки (это включает случай оценок с фиксированным размером выборки). Ранее некоторые теоретические результаты в этом направлении были изложены в монографии Р. Ш. Липцера и А. Н. Ширяева "Статистика случайных процессов" (2001 г.). При существенно менее ограничительных условиях мы выводим формулы для моментов оценки максимального правдоподобия (ОМП) $\widehat{\lambda}_{\tau}$ для параметра $\lambda$ коэффициента сноса, рассматриваемого в форме $f_t(\lambda)=a_t-\lambda b_t$, а также указываем условия для экспоненциальной ограниченности распределения $\widehat{\lambda}_{\tau}$. В приводимых примерах мы рассматриваем эргодический диффузионный процесс $\mathbf X$ с коэффициентом диффузии $\sigma_t=\sigma X_t^{\gamma}$. Мы также приводим ряд аналитических и численных результатов для смещения и среднеквадратической ошибки ОМП $\widehat{\lambda}_{\tau}$ в случае процессов Орнштейна-Уленбека (O-U) и Кокса-Ингерсолла-Росса (CIR), когда $\tau =T$ - фиксированный объем выборки и $\tau =\tau_H$ - специально выбранный момент остановки, который гарантирует заданную величину $1/H$ для дисперсии $\widehat{\lambda}_{\tau_H}$.

In the present paper, we first give a survey of the forward and backward Kolmogorov equations for pure jump Markov processes with finite and countable state spaces, and then describe relevant results for the case of Markov processes with values in standard Borel spaces based on results of W. Feller and the authors of the present paper.

В настоящей статье, приуроченной к 120-летию со дня рождения А. Н. Колмогорова, сначала дается обзор прямых и обратных уравнений Колмогорова для чисто скачкообразных марковских процессов с конечным и счетным множеством состояний, а затем описываются соответствующие результаты для случая марковских процессов со значениями в стандартных борелевских пространствах, основанные на работах В. Феллера и авторов.

We give a survey of the results related to extensions of the Kolmogorov inequality for the distribution of the absolute value of the maximum of the sum of centered independent random variables to the case of martingales considered at random stopping times.

Настоящая статья предваряет серию работ, посвященных 120-летию со дня рождения (25 апреля 1903 г.) Андрея Николаевича Колмогорова, основателя нашего журнала "Теория вероятностей и ее применения", носящего теперь его имя.

This article precedes a series of papers dedicated to the 120th anniversary of the birth (on April 25, 1903) of Andrei Nikolaevich Kolmogorov.

One of the basic facts known for discrete-time Markov decision processes is that, if the probability distribution of an initial state is fixed, then for every policy it is easy to construct a (randomized) Markov policy with the same marginal distributions of state-action pairs as for the original policy. This equality of marginal distributions implies that the values of major objective criteria, including expected discounted total costs and average rewards per unit time, are equal for these two policies. This paper investigates the validity of the similar fact for continuous-time jump Markov decision processes (CTJMDPs). It is shown in this paper that the equality of marginal distributions takes place for a CTJMDP if the corresponding Markov policy defines a nonexplosive jump Markov process. If this Markov process is explosive, then at each time instance, the marginal probability, that a state-action pair belongs to a measurable set of state-action pairs, is not greater for the described Markov policy than the same probability for the original policy. These results are applied in this paper to CTJMDPs with expected discounted total costs and average costs per unit time. It is shown for these criteria that, if the initial state distribution is fixed, then for every policy, there exists a Markov policy with the same or better value of the objective function.

A celebration of the life and accomplishments of internationally known mathematician Doctor Hans-Jürgen Engelbert, who passed away on May 23, 2021.

Recognition of the 75th birthday of Andrei Mikhailovich Zubkov and the 70th birthday of Vladimir Alekseevich Vatutin.

Настоящая статья является расширенным текстом вступительного слова на Международной конференции "П. Л. Чебышeв - 200" (Москва, 17-22 мая 2021 г.).

The prominent mathematician Vladimir Igorevich Bogachev, Professor at the Department of the Theory of Functions and Functional Analysis of the Faculty of Mechanics and Mathematics at Lomonosov Moscow State University, Professor at the Faculty of Mathematics of the HSE University, and Professor at the Department of Mathematics of the Faculty of Informatics and Applied Mathematics at St Tikhon’s Orthodox University, celebrated his sixtieth birthday on 14 February 2021. He was born in Moscow. His parents worked for defence industry and were involved directly in launching Earth satellites and ballistic missiles. After graduating from Moscow secondary school no. 19 with a gold medal, where B. L. Geidman was his mathematics teacher, Bogachev enrolled at the Faculty of Mechanics and Mathematics at Moscow State University, and later started postgraduate studies there with O. G. Smolyanov as his scientific advisor. He completed his postgraduate studies ahead of time, and in 1986, after defending his PhD thesis, begun to work at the same Faculty. Bogachev is a major expert in measure theory, the theory of probability, infinitedimensional analysis, and partial differential equations. He has solved a number of difficult problems stated by well-known mathematicians, and has obtained fundamental results in the theory of Gaussian distributions, investigated the differentiability properties of measures, and developed a new line of research in the theory of Fokker–Planck–Kolmogorov equations. His first papers, published in the early 1980s, concerned measure theory in infinite-dimensional spaces and the theory of differentiable measures, where he continued the research of his advisor Smolyanov. Bogachev gained recognition by successfully solving three problems posed by Aronszajn in the theory of infinite-dimensional probability distributions. Aronszajn proposed the following definition as an infinite-dimensional analogue of a set with Lebesgue measure zero.

Background and Aims: High lipoprotein (a) [Lp(a)] level is an independent predictor of atherosclerosis. Diffuse coronary artery disease (CAD) is considered unfavorable for myocardial revascularization. The objective: To compare serum lipid and Lp (a) levels and one-year clinical results of patients after coronary artery bypass grafting (CABG) between two groups of patients with deferent type of lesions.

This paper describes the structure of solutions to Kolmogorov’s equations for nonhomogeneous jump Markov processes and applications of these results to control of jump stochastic systems. These equations were studied by Feller (1940), who clarified in 1945 in the errata to that paper that some of its results covered only nonexplosive Markov processes. In this work, which is largely of a survey nature, the case of explosive processes is also considered. This paper is based on the invited talk presented by the authors at the conference “Chebyshev-200”, and it describes the results of their joined studies with Manasa Mandava (1984-2019).

This article celebrates the life and professional accomplishments of outstanding mathematician Vladimir Vasil'evich Senatov, who passed away on June 22, 2021. During his long career, he made essential contributions toward the solution of several central problems in probability theory.