# Publications

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.

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.

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.

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.

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

New sufficient conditions are established for a diffusion with switching to be positive recurrent.

Non-uniform exponential bounds of convergece to the stationary rejime are established for one-dimensional Wright - Fisher's diffusion with mitations in the continuous time are established.

We consider the problem of designing robust numerical integration scheme of the solution of a one-dimensional SDE with non-globally Lipschitz drift and diffusion coefficients behaving as x^α, with α>1. We propose an (semi-explicit) exponential-Euler scheme for which we obtain a theoretical convergence rate for the weak error. To this aim, we analyze the C1,4 regularity of the solution of the associated backward Kolmogorov PDE using its Feynman–Kac representation and the flow derivative of the involved processes. Under some suitable hypotheses on the parameters of the model, we prove a rate of weak convergence of order one for the proposed exponential Euler scheme, and illustrate it with some numerical experiments.

We consider a linear-quadratic control problem where a time parameter evolves according to a stochastic time scale. The stochastic time scale is defined via a stochastic process with continuously differentiable paths. We obtain an optimal infinite-time control law under criteria similar to the long-run averages. Some examples of stochastic time scales from various applications have been examined.

We consider quotients of the group algebra of the 3-string braid group B3 by p-th order generic polynomial relations on the elementary braids. If p=2,3,4,5, these quotient algebras are finite dimensional. We give semisimplicity criteria for these algebras and present explicit formulas for all their irreducible representations.

In many rare disease Phase II clinical trials, two objectives are of interest to an investigator: maximising the statistical power and maximising the number of patients responding to the treatment. These two objectives are competing, therefore, clinical trial designs offering a balance between them are needed. Recently, it was argued that response-adaptive designs such as families of multi-arm bandit (MAB) methods could provide the means for achieving this balance. Furthermore, response-adaptive designs based on a concept of context-dependent (weighted) information criteria were recently proposed with a focus on Shannon’s differential entropy. The information-theoretic designs based on the weighted Renyi, Tsallis and Fisher informations are also proposed. Due to built-in parameters of these novel designs, the balance between the statistical power and the number of patients that respond to the treatment can be tuned explicitly. The asymptotic properties of these measures are studied in order to construct intuitive criteria for arm selection. A comprehensive simulation study shows that using the exact criteria over asymptotic ones or using information measures with more parameters, namely Renyi and Tsallis entropies, brings no sufficient gain in terms of the power or proportion of patients allocated to superior treatments. The proposed designs based on information-theoretical criteria are compared to several alternative approaches. For example, via tuning of the built-in parameter, one can find designs with power comparable to the fixed equal randomisation’s but a greater number of patients responded in the trials.

In many rare disease Phase II clinical trials, two objectives are of interest to an investigator: maximising the statistical power and maximising the number of patients responding to the treatment. These two objectives are competing, therefore, clinical trial designs offering a balance between them are needed. Recently, it was argued that response-adaptive designs such as families of multi-arm bandit (MAB) methods could provide the means for achieving this balance. Furthermore, response-adaptive designs based on a concept of context-dependent (weighted) information criteria were recently proposed with a focus on Shannon’s differential entropy. The information-theoretic designs based on the weighted Renyi, Tsallis and Fisher informations are also proposed. Due to built-in parameters of these novel designs, the balance between the statistical power and the number of patients that respond to the treatment can be tuned explicitly. The asymptotic properties of these measures are studied in order to construct intuitive criteria for arm selection. A comprehensive simulation study shows that using the exact criteria over asymptotic ones or using information measures with more parameters, namely Renyi and Tsallis entropies, brings no sufficient gain in terms of the power or proportion of patients allocated to superior treatments. The proposed designs based on information-theoretical criteria are compared to several alternative approaches. For example, via tuning of the built-in parameter, one can find designs with powercomparable to the fixed equal randomisation’s but a greater number of patients responded in the trials.

We prove the well-posedness of some non-linear stochastic differential equations in the sense of McKean-Vlasov driven by non-degenerate symmetric αα-stable Lévy processes with values in RdRd under some mild Hölder regularity assumptions on the drift and diffusion coefficients with respect to both space and measure variables. The methodology developed here allows to consider unbounded drift terms even in the so-called super-critical case, i.e. when the stability index α∈(0,1)α∈(0,1). New strong well-posedness results are also derived from the previous analysis.