Theory of Probability &Its Applications, Volume 65, Issue 3, Page 497-509, January 2020. We have analyzed some conditions which are essentially involved in deciding whether or not a probability distribution is unique (moment-determinate) or nonunique (moment-indeterminate) by its moments. We suggest new conditions concerning both absolutely continuous and discrete distributions. By using the new conditions

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 482-496, January 2020. For $q\in (0,1)$, $p_1,p_2,p\in \mathbb{R}_+$, we characterize all the solutions of the $q$-functional equations $(1+p_2q^{1/2}x)f(qx)=q^{\beta-1/2}(x+p_1q^{-1/2})f(x)$ and $f(qx)=q^{\beta- 1}(x^2+p^2q^{-1})f(x)$, $x>0$, $\beta\in \mathbb{R}$, and we also show that these solutions solve corresponding indetermined

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 470-481, January 2020. In this article, we show that the centered relay correlation function is a sub-Gaussian random variable. This is done by a careful analysis of its Laplace transform and by estimating the sub-Gaussian standard of the $r$-correlograms.

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 454-469, January 2020. We consider an explosive Cox--Ingersoll--Ross process. We establish a sharp large deviation principle for the maximum likelihood estimator of its drift parameter.

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 418-453, January 2020. It has been understood that the “local” existence of the Markowitz optimal portfolio or the solution to the local-risk minimization problem is guaranteed by some specific mathematical structures on the underlying assets' price processes known in the literature as “structure conditions.” In this paper, we consider

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 405-417, January 2020. In Nevzorov's $F^\alpha$-scheme, one deals with a sequence of independent random variables with distribution functions that are powers of a common continuous distribution function. A key property of the $F^\alpha$-scheme is that the record indicators for such a sequence are independent. This allows one to obtain

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 388-404, January 2020. We present a surprisingly simple version of the fundamental theorem of asset pricing (FTAP) for continuous time large financial markets with two filtrations in an $L^p$-setting for ${1 \leq p < \infty}$. This extends the results of Kabanov and Stricker in [“The Dalang--Morton--Willinger theorem under delayed and

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 375-387, January 2020. We introduce the concept of $\varepsilon$-complexity of an individual continuous finite-dimensional map. This concept is in good accord with the principle of A.N. Kolmogorov's idea of measuring complexity of objects. It is shown that the $\varepsilon$-complexity of an “almost all” Hölder map can be effectively

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 359-374, January 2020. Let $(p_i,q_i)$, $i\in {Z}$, be a sequence of independent identically distributed random vectors such that $p_i,q_i>0$ and $p_i+q_i$ $=1$ a.s. for $i\in {Z}$. We consider a random walk in the random environment $\{(p_i,q_i)$, $i\in {Z}\}$. It is assumed that ${E}\ln (p_0/q_0)=0$ and $0<{E}\ln^{2}(q_0/p_0)<+\infty$

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 341-358, January 2020. This paper is concerned with the optimal stopping problem for Itô diffusion processes over a class of stopping times. Necessary and sufficient optimality conditions are studied for a parametrically specified class of stopping times. A detailed analysis is given for the case of one-dimensional diffusion processes

Theory of Probability &Its Applications, Volume 65, Issue 2, Page 338-339, January 2020. This note takes a retrospective look at the life and accomplishments of mathematician Sagdy Khasanovich Sirazhdinov on the centenary of his birth.

Theory of Probability &Its Applications, Volume 65, Issue 2, Page 330-337, January 2020. We treat a fairly broad class of financial models that includes markets with proportional transaction costs. We consider an investor with cumulative prospect theory preferences and a nonnegativity constraint on portfolio wealth. The existence of an optimal strategy is shown in this context for a class of generalized

Theory of Probability &Its Applications, Volume 65, Issue 2, Page 322-329, January 2020. We provide an equivalent characterization of the absence of arbitrage opportunity for the bid and ask financial market model. This result, which is an analogue of the Dalang--Morton--Willinger theorem formulated for discrete-time financial market models without friction, supplements and improves the Grigoriev theorem

Theory of Probability &Its Applications, Volume 65, Issue 2, Page 292-321, January 2020. We consider the problem of eliciting truthful responses to a survey question when the respondents share a common prior that the survey planner is agnostic about. The planner would therefore like to have a ``universal” mechanism, which would induce honest answers for all possible priors. If the planner also requires

Theory of Probability &Its Applications, Volume 65, Issue 2, Page 270-291, January 2020. The classical Fatou lemma states that the lower limit of a sequence of integrals of functions is greater than or equal to the integral of the lower limit. It is known that Fatou's lemma for a sequence of weakly converging measures states a weaker inequality because the integral of the lower limit is replaced with

Theory of Probability &Its Applications, Volume 65, Issue 2, Page 249-269, January 2020. In this paper, we study the ruin problem with investment in a general framework where the business part $X$ is a Lévy process and the return on investment $R$ is a semimartingale. Under some conditions, we obtain upper and lower bounds on the finite and infinite time ruin probabilities as well as the logarithmic

Theory of Probability &Its Applications, Volume 65, Issue 2, Page 224-248, January 2020. We study a problem of option replication under constant proportional transaction costs in models where stochastic volatility and jumps are combined to capture the market's important features. Assuming some mild condition on the jump size distribution, we show that transaction costs can be approximately compensated

Theory of Probability &Its Applications, Volume 65, Issue 2, Page 191-223, January 2020. For a large financial market (which is a sequence of usual, “small” financial markets), we introduce and study a concept of no asymptotic arbitrage (of the first kind), which is invariant under discounting. We give two dual characterizations of this property in terms of (1) martingale-like properties for each small

Theory of Probability &Its Applications, Volume 65, Issue 2, Page 179-190, January 2020. It is shown that a recently introduced lower cone distribution function, together with the set-valued multivariate quantile, generates a Galois connection between a complete lattice of closed convex sets and the interval [0,1]. This generalizes the corresponding univariate result. It is also shown that an extension

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 175-178, January 2020. Obituary of outstanding mathematician Vladimir Mikhailovich Zolotarev, who passed away November 7, 2019. He is remembered as a supreme master of both modern and classical techniques of analysis who enriched the store of probability theory through his contributions of novel analytical methods and tools.

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 173-174, January 2020. An obituary of Dmitrii Mikhailovich Chibisov, the former deputy editor-in-chief of Theory of Probability and Its Applications, who passed away on November 20, 2019.

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 121-172, January 2020. 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 Gelendzhik) at the Raduga sports and fitness center of the Don State Technical University. Participants included many leading

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 114-120, January 2020. We consider Boolean functions $f$ defined on Boolean cube $\{-1,1\}^n$ of half-spaces, i.e., functions of the form $f(x)=\operatorname{sign}(\omega\cdot x-\theta)$. Half-space functions are often called linear threshold functions. We assume that the Boolean cube $\{-1,1\}^n$ is equipped with a uniform measure.

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 111-113, January 2020. We study the form of excursions of a Gaussian stationary process intersecting a high level $u$. We show that the trajectories fluctuate in this case in a narrow tube around the expected motion. We also find an upper bound for the probability that the trajectory intersects the boundary of this tube as $u$ goes off

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 102-110, January 2020. We consider a stationary linear $\operatorname{AR}(p)$-model with observations subject to gross errors (outliers). The autoregression parameters and the distribution of innovations are unknown. Based on the residuals from the parameter estimators, we construct an analogue of an empirical distribution function and

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 82-101, January 2020. We derive sufficient conditions for the differentiability of all orders for the flow of stochastic differential equations with jumps and prove related $L^p$-integrability results for all orders. Our results extend similar results obtained by H. Kunita [Stochastic differential equations based on Lévy processes and

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 62-81, January 2020. We explore the asymptotic behavior of densities of sums of independent random variables that are convoluted with a small continuous noise.

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 49-61, January 2020. We consider the problem of constructing guarantee procedures of statistical inference with fixed minimal observation number $n^*$ for discrimination of two hypotheses $H_0\colon\theta\in[\theta_1,\theta_2]$ and $H_1\colon\theta\notin[\theta_1,\theta_2]$ with a one-dimensional parameter $\theta$ under the so-called

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 32-48, January 2020. Sample statistics of samples from a fractional Brownian motion with Hurst exponent $H$, and in particular, autocovariance statistics, are considered. Two statistics characterizing the covariate dependence between the increments of this process are studied; in particular, their asymptotic properties and the limit

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 17-31, January 2020. We examine the asymptotic behavior of integral quadratic functionals defined on time-varying Ornstein--Uhlenbeck processes. We find an upper function that majorizes with probability $1$ the deviation of the integral from its expected value as time increases. The results obtained are applied to evaluate the control

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 1-16, January 2020. We consider statistics of the form $T =\sum_{j=1}^n \xi_{j} f_{j}+ \mathcal R $, where $\xi_j, f_j$, $j=1, \dots, n$, and $\mathcal R$ are $\mathfrak M$-measurable random variables for some $\sigma$-algebra $ \mathfrak M$. Assume that there exist $\sigma$-algebras $\mathfrak M^{(1)}, \dots, \mathfrak M^{(n)}$, $ \mathfrak

Theory of Probability &Its Applications, Volume 64, Issue 4, Page 656-656, January 2020. A remembrance of the life and accomplishments of Vadim Rolandovich Fatalov, who passed away on June 4, 2019.

Theory of Probability &Its Applications, Volume 64, Issue 4, Page 646-655, January 2020. In this paper a two-dimensional Brownian motion (modeling the endowment of two companies), absorbed at the boundary of the positive quadrant, with controlled drift, is considered. We allow that both drifts add up to the maximal value of one. Our target is to choose the strategy in a way s.t. the expected value

Theory of Probability &Its Applications, Volume 64, Issue 4, Page 636-645, January 2020. When collecting samples, sometimes failures of observations occur and consequently missing data. This can have an impact on the analysis and subsequent inference, especially if the study focuses on the extreme values where the data is more scarce. In this work, we analyze the effect of different types of failures

Theory of Probability &Its Applications, Volume 64, Issue 4, Page 631-635, January 2020. It is known that the values of functions that depend on a large number of similar variables are almost constant from the point of view of an observer evaluating their values at random points of the domains of their definition (the “nonlinear law of large numbers''). We show that, under certain normalization conditions

Theory of Probability &Its Applications, Volume 64, Issue 4, Page 615-630, January 2020. This paper describes Fatou's lemma for a sequence of measures converging weakly to a finite measure and for a sequence of functions whose negative parts are uniformly integrable with respect to these measures. The paper also provides new formulations of uniform Fatou's lemma, uniform Lebesgue's convergence theorem

Theory of Probability &Its Applications, Volume 64, Issue 4, Page 595-614, January 2020. Slicing a Voronoi tessellation in ${R}^n$ with a $k$-plane gives a $k$-dimensional weighted Voronoi tessellation, also known as a power diagram or Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay mosaic to the radius of the smallest empty circumscribed sphere whose center lies in the $k$-plane

Theory of Probability &Its Applications, Volume 64, Issue 4, Page 579-594, January 2020. We study the relationship between the well-known Carleman's condition guaranteeing that a probability distribution is uniquely determined by its moments, and a recent, easily checkable condition on the rate of growth of the moments. We use asymptotic methods in the theory of integrals and involve properties of

Theory of Probability &Its Applications, Volume 64, Issue 4, Page 564-578, January 2020. The remainder term in the integro-local version of the multidimensional central limit theorem for a sum of independent random vectors is studied with due account of asymptotic expansions. It is assumed that the distribution of this sum can be absolutely continuous and/or lattice in some coordinates.

Theory of Probability &Its Applications, Volume 64, Issue 4, Page 553-563, January 2020. We consider a problem of utility maximization for multiperiod asset trading in a general model of connected financial markets represented by a graph. The main result of the paper is a theorem providing conditions for the existence of a system of supporting prices in this model. Using the general result, a specific

Theory of Probability &Its Applications, Volume 64, Issue 4, Page 535-552, January 2020. A conditional limit theorem is proved describing the distribution of the population size in the initial evolution stage of a weakly subcritical branching process in a random environment given its survival for a long time.

Theory of Probability &Its Applications, Volume 64, Issue 4, Page 513-534, January 2020. We consider a supercritical catalytic branching random walk (CBRW) with finite number of catalysts on a multidimensional lattice $\mathbb{Z}^d$, $d\in{N}$. The behavior of a cloud of particles in space and time is studied. When estimating the rate of the population propagation for the front of a multidimensional

Theory of Probability &Its Applications, Volume 64, Issue 4, Page 499-512, January 2020. We find the asymptotics for the logarithm of the Laplace transform of the distribution of a compound renewal process as time increases unboundedly. It is assumed that the elements of the governing sequences of the renewal process satisfy Cramér's moment condition. Representations for the deviation rate function