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Mathematics
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Stochastic Processes: Theory and Applications
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Stochastic Processes: Theory and Applications

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 July 2019) | Viewed by 43307

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Prof. Dr. Alexander Zeifman

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Guest Editor
1. Department of Applied Mathematics, Vologda State University, 160000 Vologda, Russia
2. Institute of Informatics Problems of the Federal Research Center “Computer Science and Control”, Russian Academy of Sciences, 119333 Moscow, Russia
3. Vologda Research Center of the Russian Academy of Sciences, 160014 Vologda, Russia
Interests: stochastic models; continuous-time Markov chains; queueing models; biological models
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Victor Korolev

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Guest Editor
1. Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
2. Moscow Center for Fundamental and Applied Mathematics, 119991 Moscow, Russia
3. Federal Research Center “Informatics and Control” of the Russian Academy of Sciences, 119333 Moscow, Russia
Interests: stochastic models; risk processes; queueing theory; limit theorems
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Alexander Sipin

E-Mail Website
Guest Editor
Department of Mathematics, Vologda State University, 160000 Vologda, Russia
Interests: stochastic models; monte carlo methods
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The aim of this Special Issue is to publish original research articles that cover recent advances in the theory and applications of stochastic processes. The focus will especially be on applications of stochastic processes as models of dynamic phenomena in various research areas, such as queuing theory, physics, biology, economics, medicine, reliability theory, and financial mathematics.

Potential topics include, but are not limited to:

  • Markov chains and processes
  • Large deviations and limit theorems
  • Random motions
  • Stochastic biological models
  • Reliability, availability, maintenance, inspection
  • Queueing models
  • Queueing network models
  • Computational methods for stochastic models
  • Applications to risk theory, insurance and mathematical finance

Prof. Dr. Alexander Zeifman
Prof. Dr. Victor Korolev
Prof. Dr. Alexander Sipin
Guest Editors

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Published Papers (15 papers)

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Research

12 pages, 1042 KiB  
Article
A Note on a Generalized Gerber–Shiu Discounted Penalty Function for a Compound Poisson Risk Model
by Jiechang Ruan, Wenguang Yu, Ke Song, Yihan Sun, Yujuan Huang and Xinliang Yu
Mathematics 2019, 7(10), 891; https://doi.org/10.3390/math7100891 - 24 Sep 2019
Cited by 7 | Viewed by 2351
Abstract
In this paper, we propose a new generalized Gerber–Shiu discounted penalty function for a compound Poisson risk model, which can be used to study the moments of the ruin time. First, by taking derivatives with respect to the original Gerber–Shiu discounted penalty function, [...] Read more.
In this paper, we propose a new generalized Gerber–Shiu discounted penalty function for a compound Poisson risk model, which can be used to study the moments of the ruin time. First, by taking derivatives with respect to the original Gerber–Shiu discounted penalty function, we construct a relation between the original Gerber–Shiu discounted penalty function and our new generalized Gerber–Shiu discounted penalty function. Next, we use Laplace transform to derive a defective renewal equation for the generalized Gerber–Shiu discounted penalty function, and give a recursive method for solving the equation. Finally, when the claim amounts obey the exponential distribution, we give some explicit expressions for the generalized Gerber–Shiu discounted penalty function. Numerical illustrations are also given to study the effect of the parameters on the generalized Gerber–Shiu discounted penalty function. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
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15 pages, 342 KiB  
Article
Valuing Guaranteed Minimum Death Benefits by Cosine Series Expansion
by Wenguang Yu, Yaodi Yong, Guofeng Guan, Yujuan Huang, Wen Su and Chaoran Cui
Mathematics 2019, 7(9), 835; https://doi.org/10.3390/math7090835 - 10 Sep 2019
Cited by 42 | Viewed by 3103
Abstract
Recently, the valuation of variable annuity products has become a hot topic in actuarial science. In this paper, we use the Fourier cosine series expansion (COS) method to value the guaranteed minimum death benefit (GMDB) products. We first express the value of GMDB [...] Read more.
Recently, the valuation of variable annuity products has become a hot topic in actuarial science. In this paper, we use the Fourier cosine series expansion (COS) method to value the guaranteed minimum death benefit (GMDB) products. We first express the value of GMDB by the discounted density function approach, then we use the COS method to approximate the valuation Equations. When the distribution of the time-until-death random variable is approximated by a combination of exponential distributions and the price of the fund is modeled by an exponential Lévy process, explicit equations for the cosine coefficients are given. Some numerical experiments are also made to illustrate the efficiency of our method. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
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12 pages, 644 KiB  
Article
On Two Interacting Markovian Queueing Systems
by Valeriy A. Naumov, Yuliya V. Gaidamaka and Konstantin E. Samouylov
Mathematics 2019, 7(9), 799; https://doi.org/10.3390/math7090799 - 1 Sep 2019
Cited by 1 | Viewed by 2235
Abstract
In this paper, we study a Markovian queuing system consisting of two subsystems of an arbitrary structure. Each subsystem generates a multi-class Markovian arrival process of customers arriving to the other subsystem. We derive the necessary and sufficient conditions for the stationary distribution [...] Read more.
In this paper, we study a Markovian queuing system consisting of two subsystems of an arbitrary structure. Each subsystem generates a multi-class Markovian arrival process of customers arriving to the other subsystem. We derive the necessary and sufficient conditions for the stationary distribution to be of product form and consider some particular cases of the subsystem interaction for which these conditions can be easily verified. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
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10 pages, 438 KiB  
Article
On Truncation of the Matrix-Geometric Stationary Distributions
by Valeriy A. Naumov, Yuliya V. Gaidamaka and Konstantin E. Samouylov
Mathematics 2019, 7(9), 798; https://doi.org/10.3390/math7090798 - 1 Sep 2019
Cited by 1 | Viewed by 2146
Abstract
In this paper, we study queueing systems with an infinite and finite number of waiting places that can be modeled by a Quasi-Birth-and-Death process. We derive the conditions under which the stationary distribution for a loss system is a truncation of the stationary [...] Read more.
In this paper, we study queueing systems with an infinite and finite number of waiting places that can be modeled by a Quasi-Birth-and-Death process. We derive the conditions under which the stationary distribution for a loss system is a truncation of the stationary distribution of the Quasi-Birth-and-Death process and obtain the stationary distributions of both processes. We apply the obtained results to the analysis of a semi-open network in which a customer from an external queue replaces a customer leaving the system at the node from which the latter departed. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
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14 pages, 620 KiB  
Article
Optimization of Queueing Model with Server Heating and Cooling
by Olga Dudina and Alexander Dudin
Mathematics 2019, 7(9), 768; https://doi.org/10.3390/math7090768 - 21 Aug 2019
Cited by 2 | Viewed by 4518
Abstract
The operation of many real-world systems, e.g., servers of data centers, is accompanied by the heating of a server. Correspondingly, certain cooling mechanisms are used. If the server becomes overheated, it interrupts processing of customers and needs to be cooled. A customer is [...] Read more.
The operation of many real-world systems, e.g., servers of data centers, is accompanied by the heating of a server. Correspondingly, certain cooling mechanisms are used. If the server becomes overheated, it interrupts processing of customers and needs to be cooled. A customer is lost when its service is interrupted. To prevent overheating and reduce the customer loss probability, we suggest temporal termination of service of new customers when the temperature of the server reaches the predefined threshold value. Service is resumed after the temperature drops below another threshold value. The problem of optimal choice of the thresholds (with respect to the chosen economical criterion) is numerically solved under quite general assumptions about the parameters of the system (Markovian arrival process, phase-type distribution of service time, and accounting for customers impatience). Numerical examples are presented. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
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7 pages, 254 KiB  
Article
Monte Carlo Methods and the Koksma-Hlawka Inequality
by Sergey Ermakov and Svetlana Leora
Mathematics 2019, 7(8), 725; https://doi.org/10.3390/math7080725 - 9 Aug 2019
Cited by 4 | Viewed by 3320
Abstract
The solution of a wide class of applied problems can be represented as an integral over the trajectories of a random process. The process is usually modeled with the Monte Carlo method and the integral is estimated as the average value of a [...] Read more.
The solution of a wide class of applied problems can be represented as an integral over the trajectories of a random process. The process is usually modeled with the Monte Carlo method and the integral is estimated as the average value of a certain function on the trajectories of this process. Solving this problem with acceptable accuracy usually requires modeling a very large number of trajectories; therefore development of methods to improve the accuracy of such algorithms is extremely important. The paper discusses Monte Carlo method modifications that use some classical results of the theory of cubature formulas (quasi-random methods). A new approach to the derivation of the well known Koksma-Hlawka inequality is pointed out. It is shown that for high ( s > 5 ) dimensions of the integral, the asymptotic decrease of the error comparable to the asymptotic behavior of the Monte Carlo method, can be achieved only for a very large number of nodes N. It is shown that a special criterion can serve as a correct characteristic of the error decrease (average order of the error decrease). Using this criterion, it is possible to analyze the error for reasonable values of N and to compare various quasi-random sequences. Several numerical examples are given. Obtained results make it possible to formulate recommendations on the correct use of the quasi-random numbers when calculating integrals over the trajectories of random processes. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
8 pages, 598 KiB  
Article
Exact Time-Dependent Queue-Length Solution to a Discrete-Time Geo/D/1 Queue
by Jung Woo Baek and Yun Han Bae
Mathematics 2019, 7(8), 717; https://doi.org/10.3390/math7080717 - 7 Aug 2019
Viewed by 2464
Abstract
Time-dependent solutions to queuing models are beneficial for evaluating the performance of real-world systems such as communication, transportation, and service systems. However, restricted results have been reported due to mathematical complexity. In this study, we present a time-dependent queue-length formula for a discrete-time [...] Read more.
Time-dependent solutions to queuing models are beneficial for evaluating the performance of real-world systems such as communication, transportation, and service systems. However, restricted results have been reported due to mathematical complexity. In this study, we present a time-dependent queue-length formula for a discrete-time G e o / D / 1 queue starting with a positive number of initial customers. We derive the time-dependent formula in closed form. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
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19 pages, 930 KiB  
Article
Analysis of a Semi-Open Queuing Network with a State Dependent Marked Markovian Arrival Process, Customers Retrials and Impatience
by Chesoong Kim, Sergey Dudin, Alexander Dudin and Konstantin Samouylov
Mathematics 2019, 7(8), 715; https://doi.org/10.3390/math7080715 - 7 Aug 2019
Cited by 9 | Viewed by 2936
Abstract
We consider a queuing network with single-server nodes and heterogeneous customers. The number of customers, which can obtain service simultaneously, is restricted. Customers that cannot be admitted to the network upon arrival make repeated attempts to obtain service. The service time at the [...] Read more.
We consider a queuing network with single-server nodes and heterogeneous customers. The number of customers, which can obtain service simultaneously, is restricted. Customers that cannot be admitted to the network upon arrival make repeated attempts to obtain service. The service time at the nodes is exponentially distributed. After service completion at a node, the serviced customer can transit to another node or leave the network forever. The main features of the model are the mutual dependence of processes of customer arrivals and retrials and the impatience and non-persistence of customers. Dynamics of the network are described by a multidimensional Markov chain with infinite state space, state inhomogeneous behavior and special structure of the infinitesimal generator. The explicit form of the generator is derived. An effective algorithm for computing the stationary distribution of this chain is recommended. The expressions for computation of the key performance measures of the network are given. Numerical results illustrating the importance of the account of the mentioned features of the model are presented. The model can be useful for capacity planning, performance evaluation and optimization of various wireless telecommunication networks, transportation and manufacturing systems. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
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11 pages, 344 KiB  
Article
On the Rate of Convergence and Limiting Characteristics for a Nonstationary Queueing Model
by Yacov Satin, Alexander Zeifman and Anastasia Kryukova
Mathematics 2019, 7(8), 678; https://doi.org/10.3390/math7080678 - 30 Jul 2019
Cited by 11 | Viewed by 2707
Abstract
Consideration is given to the nonstationary analogue of M / M / 1 queueing model in which the service happens only in batches of size 2, with the arrival rate λ ( t ) and the service rate μ ( t ) . [...] Read more.
Consideration is given to the nonstationary analogue of M / M / 1 queueing model in which the service happens only in batches of size 2, with the arrival rate λ ( t ) and the service rate μ ( t ) . One proposes a new and simple method for the study of the queue-length process. The main probability characteristics of the queue-length process are computed. A numerical example is provided. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
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20 pages, 5048 KiB  
Article
Statistical Tests for Extreme Precipitation Volumes
by Victor Korolev, Andrey Gorshenin and Konstatin Belyaev
Mathematics 2019, 7(7), 648; https://doi.org/10.3390/math7070648 - 19 Jul 2019
Cited by 4 | Viewed by 3253
Abstract
The analysis of the real observations of precipitation based on the novel statistical approach using the negative binomial distribution as a model for describing the random duration of a wet period is considered and discussed. The study shows that this distribution fits very [...] Read more.
The analysis of the real observations of precipitation based on the novel statistical approach using the negative binomial distribution as a model for describing the random duration of a wet period is considered and discussed. The study shows that this distribution fits very well to the real observations and generalized standard methods used in meteorology to detect an extreme volume of precipitation. It also provides a theoretical base for the determination of asymptotic approximations to the distributions of the maximum daily precipitation volume within a wet period, as well as the total precipitation volume over a wet period. The paper demonstrates that the relation of the unique precipitation volume, having the gamma distribution, divided by the total precipitation volume taken over the wet period is given by the Snedecor–Fisher or beta distributions. It allows us to construct statistical tests to determine the extreme precipitations. Within this approach, it is possible to introduce the notions of relatively and absolutely extreme precipitation volumes. An alternative method to determine an extreme daily precipitation volume based on a certain quantile of the tempered Snedecor–Fisher distribution is also suggested. The results of the application of these methods to real data are presented. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
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11 pages, 268 KiB  
Article
Non-Parametric Threshold Estimation for the Wiener–Poisson Risk Model
by Honglong You and Yuan Gao
Mathematics 2019, 7(6), 506; https://doi.org/10.3390/math7060506 - 3 Jun 2019
Cited by 2 | Viewed by 2196
Abstract
In this paper, we consider the Wiener–Poisson risk model, which consists of a Wiener process and a compound Poisson process. Given the discrete record of observations, we use a threshold method and a regularized Laplace inversion technique to estimate the survival probability. In [...] Read more.
In this paper, we consider the Wiener–Poisson risk model, which consists of a Wiener process and a compound Poisson process. Given the discrete record of observations, we use a threshold method and a regularized Laplace inversion technique to estimate the survival probability. In addition, we also construct an estimator for the distribution function of jump size and study its consistency and asymptotic normality. Finally, we give some simulations to verify our results. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
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10 pages, 254 KiB  
Article
On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process
by Alexander Zeifman, Yacov Satin, Ksenia Kiseleva and Victor Korolev
Mathematics 2019, 7(5), 477; https://doi.org/10.3390/math7050477 - 26 May 2019
Cited by 4 | Viewed by 2786
Abstract
We consider a multidimensional inhomogeneous birth-death process. In this paper, a general situation is studied in which the intensity of birth and death for each coordinate (“each type of particle”) depends on the state vector of the whole process. A one-dimensional projection of [...] Read more.
We consider a multidimensional inhomogeneous birth-death process. In this paper, a general situation is studied in which the intensity of birth and death for each coordinate (“each type of particle”) depends on the state vector of the whole process. A one-dimensional projection of this process on one of the coordinate axes is considered. In this case, a non-Markov process is obtained, in which the transitions to neighboring states are possible in small periods of time. For this one-dimensional process, by modifying the method previously developed by the authors of the note, estimates of the rate of convergence in weakly ergodic and null-ergodic cases are obtained. The simplest example of a two-dimensional process of this type is considered. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
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25 pages, 1363 KiB  
Article
Estimating the Expected Discounted Penalty Function in a Compound Poisson Insurance Risk Model with Mixed Premium Income
by Yunyun Wang, Wenguang Yu, Yujuan Huang, Xinliang Yu and Hongli Fan
Mathematics 2019, 7(3), 305; https://doi.org/10.3390/math7030305 - 26 Mar 2019
Cited by 7 | Viewed by 3158
Abstract
In this paper, we consider an insurance risk model with mixed premium income, in which both constant premium income and stochastic premium income are considered. We assume that the stochastic premium income process follows a compound Poisson process and the premium sizes are [...] Read more.
In this paper, we consider an insurance risk model with mixed premium income, in which both constant premium income and stochastic premium income are considered. We assume that the stochastic premium income process follows a compound Poisson process and the premium sizes are exponentially distributed. A new method for estimating the expected discounted penalty function by Fourier-cosine series expansion is proposed. We show that the estimation is easily computed, and it has a fast convergence rate. Some numerical examples are also provided to show the good properties of the estimation when the sample size is finite. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
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10 pages, 233 KiB  
Article
Monte Carlo Algorithms for the Parabolic Cauchy Problem
by Alexander Sipin
Mathematics 2019, 7(2), 177; https://doi.org/10.3390/math7020177 - 15 Feb 2019
Viewed by 1990
Abstract
New Monte Carlo algorithms for solving the Cauchy problem for the second order parabolic equation with smooth coefficients are considered. Unbiased estimators for the solutions of this problem are constructed. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
19 pages, 415 KiB  
Article
Cumulative Measure of Inaccuracy and Mutual Information in k-th Lower Record Values
by Maryam Eskandarzadeh, Antonio Di Crescenzo and Saeid Tahmasebi
Mathematics 2019, 7(2), 175; https://doi.org/10.3390/math7020175 - 14 Feb 2019
Cited by 2 | Viewed by 2801
Abstract
In this paper, we discuss the cumulative measure of inaccuracy in k-lower record values and study characterization results of dynamic cumulative inaccuracy. We also present some properties of the proposed measures, and the empirical cumulative measure of inaccuracy in k-lower record [...] Read more.
In this paper, we discuss the cumulative measure of inaccuracy in k-lower record values and study characterization results of dynamic cumulative inaccuracy. We also present some properties of the proposed measures, and the empirical cumulative measure of inaccuracy in k-lower record values. We prove a central limit theorem for the empirical cumulative measure of inaccuracy under exponentially distributed populations. Finally, we analyze the mutual information for measuring the degree of dependency between lower record values, and we show that it is distribution-free. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
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