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В статье описывается структура решений уравнений Колмогорова для неоднородных скачкообразных марковских процессов и обсуждаются применения этих результатов в задачах управления стохастическими системами со скачками. Такие уравнения рассматривались В. Феллером (1940 г.), который позднее (в 1945 г.) уточнил, что некоторые его результаты верны лишь для невзрывающихся процессов. В настоящей работе, во многом носящей обзорный характер, рассматривается и случай взрывающихся процессов. Статья основана на результатах, представленных авторами на конференции "П. Л. Чебышeв - 200", и опирается на их совместные исследования с Манасой Мандава (1984-2019).

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

This paper presents abstracts of the talks presented at The Fourth International Conference on Stochastic Methods (ICSM-4), which was held June 2--9, 2019 at Divnomorskoe (near the town of Gelendzh...

The present paper is concerned with the conditionally extremal settings of the sequential testing problem of two simple hypotheses about a parameter responsible for the local return rate of a stati...

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.

1. We will denote by θ the time of “disorder” onset (disorder time for short), assuming that \(0 \leqslant \theta \leqslant \infty \), and by \(X=(X_t)_{t \geqslant 0}\) the observed process with the stochastic differential $$\displaystyle {} dX_t=\mu I (t \geqslant \theta ) \,dt+\sigma \,dB_t,\quad X_0=0. $$

The parameter θ is the unknown disorder time, at which the drift coefficient of the process X changes its positive valueμ1 to a negative valueμ2. Accordingly, it is natural to call X a Brownian motion with change-of-trend disorder. (Related, but different from the one treated here, is the model where the volatility (σ) undergoes a jump. Such models are known as bubble models.) Another appropriate name for the model (10.1) is that of Bachelier model with disorder (with initial condition X0 = 0; compare with [94, 95]).

1. At the outset we will assume that with the original probabilistic-statistical experiment \((\Omega ,\mathcal {F}, (\mathcal {F}_t)_{t \geqslant 0};\mathrm {P}^{\kern 1pt 0}, \mathrm {P}^{\infty })\), where $$\displaystyle \{\varnothing ,\Omega \} =\mathcal {F}_0 \subseteq \mathcal {F}_1 \subseteq \cdots , $$ there are associated “θ-models” and “G-models” (Sects. 1.2 and 1.3).

1. Suppose that we observe the stochastic process \(X=(X_t)_{t \geqslant 0}\), $$\displaystyle X_t=\theta \mu t+B_t, \quad X_0=0, $$ on the filtered probability space Open image in new window , where \(B=(B_t)_{t \geqslant 0}\) is the standard Brownian motion (as a martingale with respect to the filtration flow \((\mathcal {F}_t)_{t \geqslant 0}\), EBt = 0 and \(\mathrm {E} B_t^2=t\)).

1. A specific peculiarity of the quickest detection problems considered here is that in them one is required to determine a stopping time that is close, in some sense, to the “regime-failure” time in the observed process. For quite some time individual problems have emerged that effectively reduce to optimal stopping time problems and eventually lead to the construction of a theory of optimal stopping rules, the results of which are essential in dealing with the quickest detection problems we are considering.

Prove the Corollary to Theorem 1 in [P}§4.1]. Hint. Use the fact that the distribution function of the variable η takes only the values 0 and 1.

1. Recall that the problems concerning optimal stopping and optimal stopping rules considered above were stated as follows.

We assume again that at the basis of our considerations lies a filtered probabilistic-statistical experiment.

1. From the considerations presented in the introduction it follows that many probabilistic-statistical schemes like, for example, those used in production quality control, are such that in the behavior of the observable random sequences a “failure” occurs in the character of the probability distributions. In this section we will describe a sufficiently general scheme, called the “stochastic θ-model”, which also covers the cases considered above, in which θ is understood as the time at which the probability characteristics change (i.e., “disorder”, “disruption”, or “chaos” occurs).

1. In the second chapter the A, B, C, and D variants of quickest detection problems were considered in the case of discrete time and arbitrary random sequences (on filtered probability spaces).

Known conditions of existence of solutions of optimal stopping problems for Markov processes assume that payoff functions are bounded in some sense. In this paper we prove weaker conditions which are applicable to unbounded payoff functions. The results obtained are applied to the optimal stopping problem for a Brownian motion with the payoff function G(τ, Bτ )= |Bτ |− c/(1 − τ ).

As is well-known, transition probabilities of jump Markov processes satisfy Kolmogorov’s backward and forward equations. In the seminal 1940 paper, William Feller investigated solutions of Kolmogorov’s equations for jump Markov processes. Recently the authors solved the problem studied by Feller and showed that the minimal solution of Kolmogorov’s backward and forward equations is the transition probability of the corresponding jump Markov process if the transition rate at each state is bounded. This paper presents more general results. For Kolmogorov’s backward equation, the sufficient condition for the described property of the minimal solution is that the transition rate at each state is locally integrable, and for Kolmogorov’s forward equation the corresponding sufficient condition is that the transition rate at each state is locally bounded.

This paper studies three ways to construct a nonhomogeneous jump Markov process: (i) via a compensator of the random measure of a multivariate point process, (ii) as a minimal solution of the backward Kolmogorov equation, and (iii) as a minimal solution of the forward Kolmogorov equation. The main conclusion of this paper is that, for a given measurable transition intensity, commonly called a Q-function, all these constructions define the same transition function. If this transition function is regular, that is, the probability of accumulation of jumps is zero, then this transition function is the unique solution of the backward and forward Kolmogorov equations. For continuous Q-functions, Kolmogorov equations were studied in Fellerʼs seminal paper. In particular, this paper extends Fellerʼs results for continuous Q-functions to measurable Q-functions and provides additional results.

In this paper, the authors apply a continuous-time stochastic process model developed by Shiryaev and Zhutlukhin for optimally stopping random price processes that appear to be bubbles, defined as price increases that are largely based on the expectation of higher and higher future prices. Futures traders, such as George Soros, attempt to trade such markets, trying to exit near the peak from a starting long position. The model applies equally well to the question of when to enter and exit a short position. In this article, the authors test the model in two technology markets. These include the price of Apple computer stock from various times in 2009–2012 after the local low of March 6, 2009, plus a market in which the generally very successful bubble trader George Soros lost money by shorting the NASDAQ-100 stock index too soon in 2000. The model provides good exit points in both situations; these would have been profitable to speculators who employed the model.

An obituary of Robert Shevilevich Liptser, who passed away on January 2, 2019. He was a prominent scientist and author in the field of random processes and their applications. His books are widely ...

We formulate a general Bayesian disorder detection problem, which generalizes models considered in the literature. We study properties of basic statistics, which allow us to reduce problems of quickest detection of disorder moments to optimal stopping problems. Using general results, we consider in detail a disorder problem for Brownian motion on a finite time segment.