# Publications

In the paper, we study a new rate of convergence estimate for homogeneous discrete-time nonlinear Markov chains based on the Markov-Dobrushin condition. This result generalizes the convergence estimates for any positive number of transition steps. An example of a class such a process provided indicates that such types of estimates considering several transition steps may be applicable when one transition can not guarantee any convergence. Moreover, a better estimate can be obtained for a higher number of transitions steps. A law of large numbers is presented for a class of ergodic nonlinear Markov chains with finite state space that may serve as a basis for nonparametric estimation and other statistics.

This paper studies the bailout optimal dividend problem with regime switching under the constraint that dividend payments can be made only at the arrival times of an independent Poisson process while capital can be injected continuously in time. We show the optimality of the regime-modulated Parisian-classical reflection strategy when the underlying risk model follows a general spectrally negative Markov additive process. In order to verify the optimality, first we study an auxiliary problem driven by a single spectrally negative Lévy process with a final payoff at an exponential terminal time and characterise the optimal dividend strategy. Then, we use the dynamic programming principle to transform the global regime-switching problem into an equivalent local optimization problem with a final payoff up to the first regime switching time. The optimality of the regime modulated Parisian-classical barrier strategy can be proven by using the results from the auxiliary problem and approximations via recursive iterations.

In the article [Theory of Probability & Its Applications 62(2) (2018), 216–235], a class W of terminal joint distributions of integrable increasing processes and their compensators was introduced. In this paper, it is shown that the discrete distributions lying in W form a dense subset in the set W for *ψ*-weak topology with a gauge function *ψ* of linear growth.

We produce a series of results extending information-theoretical inequalities (dis- cussed by Dembo and CoverThomas in 1988-1991) to a weighted version of entropy. Most of the resulting inequalities involve the Gaussian weighted entropy; they imply a number of new relations for determinants of positive-definite matrices. Unlike the Shannon entropy where the contribution of an outcome depends only upon its prob- ability, the weighted (or context-dependent) entropy takes into account a `value' of an outcome determined by a given weight function '. An example of a new result is a weighted version of the strong Hadamard inequality (SHI) between the determinants of a positive-definitenite d by d matrix and its square blocks (sub-matrices) of different size. When the weight equals 1, the weighted inequality becomes a `standard' SHI; in general, the weighted version requires some assumptions upon '. The SHI and its weighted version generalize a widely known `usual' Hadamard inequality.

We present a number of upper and low bounds for the total variation distances between the most popular probability distributions. In particular, some estimates of the total variation distances between one-dimensional Gaussian distributions, between two Poisson distributions, between two binomial distributions, between a binomial and a Poisson distribution, and also between two negative binomial distributions are given. The Kolmogorov – Smirnov distance is also presented.

We continue the work of improving the rate of convergence of ergodic homogeneous Markov chains. The setting is more general than in previous papers: we are able to get rid of the assumption about a common dominating measure and consider the case of inhomogeneous Markov chains as well as more general state spaces. We give examples where the new bound for the rate of convergence is the same as (resp. better than) the classical Markov-Dobrushin inequality.

We consider the set $\Lambda$ of all edge joint distributions $\Law ([X_a, A_a], [X_b, A_b])$ at the moments $t =a$ and $t = b$ of integrable increasing processes $(X_t)_{t\in [a; b]}$ and their compensators $(A_t)_{t\in [a; b]}$, which start from an arbitrary integrable initial condition $[X_a, A_a]$. The convexity and closure of the set $\Lambda$ in $\psi$-weak topology with a gauge function $\psi$ of linear growth are established. Necessary and sufficient conditions are obtained that a given probability measure $\lambda$ on $\mathcal{B}(\mathbb{R}^2\times\mathbb{R}^2)$ belongs to the class of measures $\Lambda$. The main result of the work is the following: for two measures $\mu_a$ and $\mu_b$ given on $\mathcal{B}(\mathbb{R}^2)$ necessary and sufficient conditions are obtained that the set $\Lambda$ contains the measure $\lambda$, for which $\mu_a$ and $\mu_b$ are marginal distributions.

We consider non degenerate Brownian SDEs with Hölder continuous in space diffusion coefficient and unbounded drift with linear growth. We derive two sided bounds for the associated density and pointwise controls of its derivatives up to order two under some additional spatial Hölder continuity assumptions on the drift. Importantly, the estimates reflect the transport of the initial condition by the unbounded drift through an auxiliary, possibly regularized, flow.

We investigate when a mean field-type control system can fulfill a given constraint. Namely, given a closed set of probability measures on the torus, starting from any initial probability measure belonging to this set, does there exist a solution to the mean field control system remaining in it for any time? This property—the so-called viability property—is equivalently characterized through a property involving normals to the given set of probability measures. We prove that, if the Hamiltonian is nonpositive at any normal distribution to the given set, then the feedback strategy realizing the extremal shift rule provides the approximate viability. This implies the usual viability property. Conversely, the Hamiltonian is nonpositive at any normal distribution if the given set is viable. Our approach enables us to derive generalized feedback laws which ensure the trajectory to fulfill the constraint. This generalized feedback called here extremely shift rule is inspired by constructive motions developed by Krasovskii and Subbotin for differential games.

In this paper, we consider the distribution of the supremum of non-stationary Gaus- sian processes, and present a new theoretical result on the asymptotic behaviour of this distribution. We focus on the case when the processes have finite number of points attaining their maximal variance, but, unlike previously known facts in this field, our main theorem yields the asymptotic representation of the corresponding distribution function with exponentially decaying remainder term. This result can be efficiently used for studying the projection density estimates, based, for instance, on Legendre polynomials. More precisely, we construct the sequence of accompanying laws, which approximates the distribution of maximal deviation of the considered estimates with polynomial rate. Moreover, we construct the confidence bands for densities, which are honest at polynomial rate to a broad class of densities.

This paper deals with the extreme value analysis for the triangular arrays which appear when some parameters of the mixture model vary as the number of observations grows. When the mixing parameter is small, it is natural to associate one of the components with “an impurity” (in the case of regularly varying distribution, “heavy-tailed impurity”), which “pollutes” another component. We show that the set of possible limit distributions is much more diverse than in the classical Fisher–Tippett–Gnedenko theorem, and provide the numerical examples showing the efficiency of the proposed model for studying the maximal values of the stock returns.

In this paper, we aim to determine an optimal insurance premium rate for health-care in deterministic and stochastic SEIR models. The studied models consider two standard SEIR centres characterised by migration fluxes and vaccination of population. The premium is calculated using the basic equivalence principle. Even in this simple set-up, there are non-intuitive results that illustrate how the premium depends on migration rates, the severity of a disease and the initial distribution of healthy and infected individuals through the centres. We investigate how the vaccination program affects the insurance costs by comparing the savings in benefits with the expenses for vaccination. We compare the results of deterministic and stochastic models.

Games of inspection and corruption are well developed in the game-theoretic literature. However, there are only a few publications that approach these problems from the evolutionary point of view. In previous papers of this author, a generalization of the replicator dynamics of the evolutionary game theory was suggested for inspection modeling, namely the pressure and resistance framework, where a large pool of small players plays against a distinguished major player and evolves according to certain myopic rules. In this paper, we develop this approach further in a setting of the two-level hierarchy, where a local inspector can be corrupted and is further controlled by the higher authority (thus combining the modeling of inspection and corruption in a unifying setting). Mathematical novelty arising in this investigation involves the analysis of the generalized replicator dynamics (or kinetic equation) with switching, which occurs on the “efficient frontier of corruption”. We try to avoid parameters that are difficult to observe or measure, leading to some clear practical consequences. We prove a result that can be called the “principle of quadratic fines”: We show that if the fine for violations (both for criminal businesses and corrupted inspectors) is proportional to the level of violations, the stable rest points of the dynamics support the maximal possible level of both corruption and violation. The situation changes if a convex fine is introduced. In particular, starting from the quadratic growth of the fine function, one can effectively control the level of violations. Concrete settings that we have in mind are illegal logging, the sales of products with substandard quality, and tax evasion

The BIS indicated in July 2020 an unprecedented rise in default risk correlation as a result of pandemics-induced credit risks’ accumulation. A third of the world banking assets credit risk measurement depends on the Basel internal-ratings-based (IRB) models. To ensure financial stability, we wish IRB models to be accurate in default probability (PD) forecasting. There naturally arises a question of which model may be deemed accurate if the data demonstrates the presence of the default correlation. The existing prudential IRB validation guidelines suggest a confidence interval of up to 100 percentage points’ length for such a case. Such an interval is useless as any model and any PD forecast seem accurate. The novelty of this paper is the justification for the use of twin confidence intervals to validate PD model accuracy. Those intervals more concentrate around the two extremes (default and its absence), the higher the default correlation is.

In this paper, we derive a stability result for L1 and L∞ perturbations of diffusions under weak regularity conditions on the coefficients. In particular, the drift terms we consider can be unbounded with at most linear growth, and we do not require uniform convergence of perturbed diffusions. Instead, we require a weaker convergence condition in a special metric introduced in this paper, related to the Holder norm of the diffusion matrix differences. Our approach is based on a special version of the McKean-Singer parametrix expansion.

The theory of first-order mean field type differential games examines the systems of infinitely many identical agents interacting via some external media under assumption that each agent is controlled by two players. We study the approximations of the value function of the first-order mean field type differential game using solutions of model finite-dimensional differential games. The model game appears as a mean field type continuous-time Markov game, i.e., the game theoretical problem with the infinitely many agents and dynamics of each agent determined by a controlled finite state nonlinear Markov chain. Given a supersolution (resp. subsolution) of the Hamilton-Jacobi equation for the model game, we construct a suboptimal strategy of the first (resp. second) player and evaluate the approximation accuracy using the modulus of continuity of the reward function and the distance between the original and model games. This gives the approximations of the value function of the mean field type differential game by values of the finite-dimensional differential games. Furthermore, we present the way to build a finite-dimensional differential game that approximates the original game with a given accuracy.

Objective. Our objectives were to (1) compare different regimens of hormonal therapy (HT) in young women with atypical endometrial hyperplasia (AEH) and early endometrial cancer (EC), (2) assess reproductive and on- cologic outcomes and (3) explore possible predictors of complete response (CR) and disease free survival (DFS).

Methods. Reproductive age women with AEH and Grade 1–2 endometrioid EC with no or minimal myometrial invasion on MRI treated with different regimens of HT were prospectively analyzed. Treatment pro- tocols included levonorgestrel intrauterine device (LNG IUD), gonadotropin-releasing hormone agonist (aGnRH) or high-dose oral medroxyprogesteron acetate (MPA) separately and in combinations.

Results. Total of 418 patients with AEH (n = 228) and EC (n = 190) aged 19–46 years received HT. Overall CR rate was 96% in AEH and 88% in EC patients (р < 0.001). None of the regimens used in AEH (LNG IUD + 2 D&C vs. LNG IUD + aGnRH vs. LNG IUD + 3 D&C) was found inferior to the others (CR of 98%, 95%, 100%, respectively, p > 0.05) except for MPA alone (CR 87%, р = 0.009). Out of four HT regimens used in EC LNG IUD + aGnRH+3 D&C was superior to all others (CR 96%, р = 0.026) where 2 D&Cs were performed or oral MPA was prescribed. The median follow-up for 339 patients was 33 months (range: 3–136), 68% of patients (n = 232) attempted con- ception, 38% (n = 89) of them used ART. The birth rate was 42% (n = 97). The rate of recurrence was 26% (50/ 196) in AEH group and 36% (51/143) in EC group (p = 0.05). Birth after treatment (HR = 0.24) or LNG IUD main- tenance (HR = 0.18) were associated with superior DFS (p < 0.001 for both). ART use did not influence DFS.

Conclusion. Hormonal therapy of AEH and early EC with LNG IUD is superior to MPA-containing regimens, however still carries high risk of recurrence. Post-treatment pregnancy rates are satisfactory and can be further improved by broader ART use which was proven safe. Initial diagnosis of AEH, post-treatment child birth and LNG IUD maintenance were associated with decreased rates of recurrence.

The paper is devoted to the numerical solutions of fractional PDEs based on its probabilistic interpretation, that is, we construct approximate solutions via certain Monte Carlo simulations. The main results represent the upper bound of errors between the exact solution and the Monte Carlo approximation, the estimate of the fluctuation via the appropriate central limit theorem (CLT) and the construction of confidence intervals. Moreover, we provide rates of convergence in the CLT via Berry-Esseen type bounds. Concrete numerical computations and illustrations are included.

Several methods of establishing coupling for stochastic differential equations are presented.