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Regular version of the site
Of all publications in the section: 12
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Article
Smirnov S. N. Mathematics. 2019. Vol. 7. No. 12. P. 1246.

This paper considers super-replication in a guaranteed deterministic problem setting with discrete time. The aim of hedging a contingent claim is to ensure the coverage of possible payoffs under the option contract for all admissible scenarios. These scenarios are given by means of a priori given compacts that depend on the history of prices. The increments of the price at each moment in time must lie in the corresponding compacts. The absence of transaction costs is assumed. The game–theoretic interpretation of pricing American options implies that the corresponding Bellman–Isaacs equations hold for both pure and mixed strategies. In the present paper, we study some properties of the least favorable (for the “hedger”) mixed strategies of the “market” and of their supports in the special case of convex payoff functions.

Added: Dec 26, 2019
Article
Lapshin V. A. Mathematics. 2019. Vol. 7. No. 11. P. 1121.

We consider the issue of short term immunization of a bond-like obligation with respect to changes in interest rates using a portfolio of bonds. In the case that the zero-coupon yield curve belongs to a fixed low-dimensional manifold, the problem is widely known as parametric immunization. Parametric immunization aims to make the price sensitivity of the hedged portfolio  for all parameters of the model zero. However, within a popular approach to estimating non-parametric (smoothing splines) structural terms, parametric hedging is not applied immediately. We present a non-parametric approach to hedging a bond-like obligation, allowing for a common form of assessment of the timing structure with possible smoothing. We show that our approach gives standard immunization on the basis of the maximum duration, when the degree of smoothing goes to infinity. We also reinstating the industry's best hedging approach, based on the length of the key rate, as another specific case. The hedging portfolio is easy to calculate using only the basic operations of linear algebra.

Added: Nov 21, 2019
Article
Averboukh Y. Mathematics. 2018. Vol. 6. No. 9.

The paper is devoted to inverse Stackelberg games with many players. We consider both static and differential games. The main assumption of the paper is the compactness of the strategy sets. We obtain the characterization of inverse Stackelberg solutions and under additional concavity conditions, establish the existence theorem.

Added: Apr 17, 2020
Article
Popkov Y., Dubnov Y. A., Popkov A. Mathematics. 2016. Vol. 4. No. 1. P. 1-16.

We propose a new method of randomized forecasting (RF-method), which operates with models described by systems of linear ordinary differential equations with random parameters. The RF-method is based on entropy-robust estimation of the probability density functions (PDFs) of model parameters and measurement noises. The entropy-optimal estimator uses a limited amount of data. The method of randomized forecasting is applied to World population prediction. Ensembles of entropy-optimal prognostic trajectories of World population and their probability characteristics are generated. We show potential preferences of the proposed method in comparison with existing methods.

Added: Feb 28, 2017
Article
Galatenko A., Nersisyan S., Zhuk D. Mathematics. 2019. Vol. 7. No. 8. P. 711.

We consider the problem of finding a position of a d-dimensional box with given edge lengths that maximizes the number of enclosed points of the given finite set P⊂Rd , i.e., the problem of optimal box positioning. We prove that while this problem is polynomial for fixed values of d, it is NP-hard in the general case. The proof is based on a polynomial reduction technique applied to the considered problem and the 3-CNF satisfiability problem.

Added: Apr 28, 2020
Article
Galatenko A. V., Nersisyan S. A., Zhuk D. Mathematics. 2019. Vol. 7. No. 8.

We consider the problem of finding a position of a d-dimensional box with given edge lengths that maximizes the number of enclosed points of the given finite set P⊂Rd , i.e., the problem of optimal box positioning. We prove that while this problem is polynomial for fixed values of d, it is NP-hard in the general case. The proof is based on a polynomial reduction technique applied to the considered problem and the 3-CNF satisfiability problem.

Added: Jun 15, 2020
Article
Nazarov A., Nikitin Ya. Yu. Mathematics. 2018. Vol. 6(4). No. 55. P. 1-9.

We study the exact small deviation asymptotics with respect to the Hilbert norm for some mixed Gaussian processes. The simplest example here is the linear combination of the Wiener process and the Brownian bridge. We get the precise final result in this case and in some examples of more complicated processes of similar structure. The proof is based on Karhunen–Loève expansion together with spectral asymptotics of differential operators and complex analysis methods.

Added: May 28, 2018
Article
Lazarev A. A., Nikolay Pravdivets, Werner F. Mathematics. 2020. Vol. 8. No. 7. P. 1131.

In this paper, we consider the single-machine scheduling problem with given release dates and the objective to minimize the maximum penalty which is NP-hard in the strong sense. For this problem, we introduce a dual and an inverse problem and show that both these problems can be solved in polynomial time. Since the dual problem gives a lower bound on the optimal objective function value of the original problem, we use the optimal function value of a sub-problem of the dual problem in a branch and bound algorithm for the original single-machine scheduling problem. We present some initial computational results for instances with up to 20 jobs.

Added: Sep 2, 2020
Article
Gromova E., Malakhova A., Palestini A. Mathematics. 2018. Vol. 6. No. 9. P. 1-17.

A nonrenewable resource extraction game model is analyzed in a differential game theory framework with random duration. If the cumulative distribution function (c.d.f.) of the final time is discontinuous, the related subgames are differentiated based on the position of the initial instant with respect to the jump. We investigate properties of optimal trajectories and of imputation distribution procedures if the game is played cooperatively.

Added: Mar 11, 2020
Article
Andreev N. A. Mathematics. 2019. Vol. 7. No. 12. P. 1147.

We present a robust dynamic programming approach to the general portfolio selection problem in the presence of transaction costs and trading limits. We formulate the problem as a dynamic infinite game against nature and obtain the corresponding Bellman-Isaacs equation. Under~several additional assumptions, we get an alternative form of the equation, which is more feasible for a numerical solution. The framework covers a wide range of control problems, such as the estimation of the portfolio liquidation value, or portfolio selection in an adverse market. The~results can be used in the presence of model errors, non-linear transaction costs and a price impact.

Added: Oct 30, 2019
Article
Ulyanov V. V., Christoph G. Mathematics. 2020. Vol. 8. No. 7. P. 1-28.

We consider high-dimension low-sample-size data taken from the standard multivariate normal distribution under assumption that dimension is a random variable. The second order Chebyshev–Edgeworth expansions for distributions of an angle between two sample observations and corresponding sample correlation coefficient are constructed with error bounds. Depending on the type of normalization, we get three different limit distributions: Normal, Student’s t-, or Laplace distributions. The paper continues studies of the authors on approximation of statistics for random size samples.

Added: Jul 15, 2020
Article
Kelbert M., Sazonov I., Bocharov G. et al. Mathematics. 2020. Vol. 8. No. 8. P. 1207.

Many human virus infections including those with the human immunodeficiency virus type 1 (HIV) are initiated by low numbers of founder viruses. Therefore, random effects have a strong influence on the initial infection dynamics, e.g., extinction versus spread. In this study, we considered the simplest (so-called, ‘consensus’) virus dynamics model and incorporated a delay between infection of a cell and virus progeny release from the infected cell. We then developed an equivalent stochastic virus dynamics model that accounts for this delay in the description of the random interactions between the model components. The new model is used to study the statistical characteristics of virus and target cell populations. It predicts the probability of infection spread as a function of the number of transmitted viruses. A hybrid algorithm is suggested to compute efficiently the system dynamics in state space domain characterized by the mix of small and large species densities.  

Added: Jul 31, 2020