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Probability, Stochastic Processes and Optimization
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Probability, Stochastic Processes and Optimization

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "D1: Probability and Statistics".

Deadline for manuscript submissions: closed (31 May 2023) | Viewed by 41419

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

Special Issue Editors

Prof. Dr. Alexandru Agapie

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Guest Editor
1. Department of Applied Mathematics, The Bucharest University of Economic Studies, 010552 Bucharest, Romania
2. Institute of Mathematical Statistics and Applied Mathematics, 050711 Bucharest, Romania
Interests: stochastic processes; evolutionary algorithms; cellular automata
Prof. Dr. Denis Enachescu

E-Mail Website
Guest Editor
Doctoral School of Computer Science, University of Bucharest, 010014 Bucharest, Romania
Interests: simulation methods; Monte Carlo method; statistical learning; data mining; artificial intelligence; biostatistics
Dr. Vlad Stefan Barbu

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Guest Editor
Laboratory of Mathematics Raphaël Salem, University of Rouen-Normandy, 76801 Saint Étienne du Rouvray, France
Interests: markov and semi-Markov processes; hidden Markov and hidden semi-Markov processes; statistical inference for stochastic processes; parametric and nonparametric estimation; hypotheses testing; stochastic methods in reliability and survival analysis; biostatistics; stochastic methods for DNA modelling; entropy and divergence measures; model selection
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Bogdan Iftimie

E-Mail Website
Guest Editor
Department of Applied Mathematics, Bucharest University of Economic Studies, 010552 Bucharest, Romania
Interests: stochastic processes in finance; stochastic partial differential equations; financial models with advanced or delayed information

Special Issue Information

Dear Colleagues,

In applied research, Probability Theory is usually regarded as a distant, often neglected relative of Statistics. We try to rectify this misconception by publishing papers underlying new applications and original models for real-world problems, from either natural, computer or social sciences that are based on Probability and Stochastic Processes.

The suggested, yet not restrictive, topics include the following:

  • Optimization;
  • Cellular automata;
  • Bayes analysis;
  • Markov processes and related topics;
  • Hidden Markov processes;
  • Applications of stochastic processes in Economics, Finance, Social Sciences, etc.;
  • Stochastic modeling;
  • Measures of divergence and entropy;
  • Clustering;
  • Monte Carlo simulations.

Prof. Dr. Alexandru Agapie
Prof. Dr. Denis Enachescu
Dr. Vlad Stefan Barbu
Prof. Dr. Bogdan Iftimie
Guest Editors

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

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Keywords

  • Probabilistic optimization
  • Evolutionary algorithms
  • Stochastic modeling
  • Probabilistic cellular automata
  • Entropy
  • Markov processes and related topics in social sciences
  • Hidden Markov chains
  • Markov chains for learning human behavior
  • Stochastic processes and their applications in economics
  • Clustering
  • Monte Carlo methods
  • Bayesian analysis
  • Bayesian networks

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Published Papers (15 papers)

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Research

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29 pages, 2673 KiB  
Article
Exploring the Predictors of Co-Nationals’ Preference over Immigrants in Accessing Jobs—Evidence from World Values Survey
by Daniel Homocianu
Mathematics 2023, 11(3), 786; https://doi.org/10.3390/math11030786 - 3 Feb 2023
Cited by 1 | Viewed by 2489
Abstract
This paper presents the results of an exploration of the most resilient influences determining the attitude regarding prioritizing co-nationals over immigrants for access to employment. The source data were from the World Values Survey. After many selection and testing steps, a set of [...] Read more.
This paper presents the results of an exploration of the most resilient influences determining the attitude regarding prioritizing co-nationals over immigrants for access to employment. The source data were from the World Values Survey. After many selection and testing steps, a set of the seven most significant determinants was produced (a fair-to-good model as prediction accuracy). These seven determinants (a hepta-core model) correspond to some features, beliefs, and attitudes regarding emancipative values, gender discrimination, immigrant policy, trust in people of another nationality, inverse devoutness or making parents proud as a life goal, attitude towards work, the post-materialist index, and job preferences as more inclined towards self rather than community benefits. Additional controls revealed the significant influence of some socio-demographic variables. They correspond to gender, the number of children, the highest education level attained, employment status, income scale positioning, settlement size, and the interview year. All selection and testing steps considered many principles, methods, and techniques (e.g., triangulation via adaptive boosting (in the Rattle library of R), and pairwise correlation-based data mining—PCDM, LASSO, OLS, binary and ordered logistic regressions (LOGIT, OLOGIT), prediction nomograms, together with tools for reporting default and custom model evaluation metrics, such as ESTOUT and MEM in Stata). Cross-validations relied on random subsamples (CVLASSO) and well-established ones (mixed-effects). In addition, overfitting removal (RLASSO), reverse causality, and collinearity checks succeeded under full conditions for replicating the results. The prediction nomogram corresponding to the most resistant predictors identified in this paper is also a powerful tool for identifying risks. Therefore, it can provide strong support for decision makers in matters related to immigration and access to employment. The paper’s novelty also results from the many robust supporting techniques that allow randomly, and non-randomly cross-validated and fully reproducible results based on a large amount and variety of source data. The findings also represent a step forward in migration and access-to-job research. Full article
(This article belongs to the Special Issue Probability, Stochastic Processes and Optimization)
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22 pages, 423 KiB  
Article
Stabilization of Stochastic Dynamical Systems of a Random Structure with Markov Switches and Poisson Perturbations
by Taras Lukashiv, Yuliia Litvinchuk, Igor V. Malyk, Anna Golebiewska and Petr V. Nazarov
Mathematics 2023, 11(3), 582; https://doi.org/10.3390/math11030582 - 22 Jan 2023
Cited by 5 | Viewed by 1969
Abstract
An optimal control for a dynamical system optimizes a certain objective function. Here, we consider the construction of an optimal control for a stochastic dynamical system with a random structure, Poisson perturbations and random jumps, which makes the system stable in probability. Sufficient [...] Read more.
An optimal control for a dynamical system optimizes a certain objective function. Here, we consider the construction of an optimal control for a stochastic dynamical system with a random structure, Poisson perturbations and random jumps, which makes the system stable in probability. Sufficient conditions of the stability in probability are obtained, using the second Lyapunov method, in which the construction of the corresponding functions plays an important role. Here, we provide a solution to the problem of optimal stabilization in a general case. For a linear system with a quadratic quality function, we give a method of synthesis of optimal control based on the solution of Riccati equations. Finally, in an autonomous case, a system of differential equations was constructed to obtain unknown matrices that are used for the construction of an optimal control. The method using a small parameter is justified for the algorithmic search of an optimal control. This approach brings a novel solution to the problem of optimal stabilization for a stochastic dynamical system with a random structure, Markov switches and Poisson perturbations. Full article
(This article belongs to the Special Issue Probability, Stochastic Processes and Optimization)
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24 pages, 1608 KiB  
Article
Optimal Test Plan of Step Stress Partially Accelerated Life Testing for Alpha Power Inverse Weibull Distribution under Adaptive Progressive Hybrid Censored Data and Different Loss Functions
by Refah Alotaibi, Ehab M. Almetwally, Qiuchen Hai and Hoda Rezk
Mathematics 2022, 10(24), 4652; https://doi.org/10.3390/math10244652 - 8 Dec 2022
Cited by 7 | Viewed by 2111
Abstract
Accelerated life tests are used to explore the lifetime of extremely reliable items by subjecting them to elevated stress levels from stressors to cause early failures, such as temperature, voltage, pressure, and so on. The alpha power inverse Weibull (APIW) distribution is of [...] Read more.
Accelerated life tests are used to explore the lifetime of extremely reliable items by subjecting them to elevated stress levels from stressors to cause early failures, such as temperature, voltage, pressure, and so on. The alpha power inverse Weibull (APIW) distribution is of great significance and practical applications due to its appealing characteristics, such as its flexibilities in the probability density function and the hazard rate function. We analyze the step stress partially accelerated life testing model with samples from the APIW distribution under adaptive type II progressively hybrid censoring. We first obtain the maximum likelihood estimates and two types of approximate confidence intervals of the distributional parameters and then derive Bayes estimates of the unknown parameters under different loss functions. Furthermore, we analyze three probable optimum test techniques for identifying the best censoring under different optimality criteria methods. We conduct simulation studies to assess the finite sample performance of the proposed methodology. Finally, we provide a real data example to further demonstrate the proposed technique. Full article
(This article belongs to the Special Issue Probability, Stochastic Processes and Optimization)
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26 pages, 806 KiB  
Article
Fisher-like Metrics Associated with ϕ-Deformed (Naudts) Entropies
by Cristina-Liliana Pripoae, Iulia-Elena Hirica, Gabriel-Teodor Pripoae and Vasile Preda
Mathematics 2022, 10(22), 4311; https://doi.org/10.3390/math10224311 - 17 Nov 2022
Viewed by 1511
Abstract
The paper defines and studies new semi-Riemannian generalized Fisher metrics and Fisher-like metrics, associated with entropies and divergences. Examples of seven such families are provided, based on exponential PDFs. The particular case when the basic entropy is a ϕ-deformed one, in the [...] Read more.
The paper defines and studies new semi-Riemannian generalized Fisher metrics and Fisher-like metrics, associated with entropies and divergences. Examples of seven such families are provided, based on exponential PDFs. The particular case when the basic entropy is a ϕ-deformed one, in the sense of Naudts, is investigated in detail, with emphasis on the variation of the emergent scalar curvatures. Moreover, the paper highlights the impact on these geometries determined by the addition of some group logarithms. Full article
(This article belongs to the Special Issue Probability, Stochastic Processes and Optimization)
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16 pages, 9416 KiB  
Article
Approximation of the Statistical Characteristics of Piecewise Linear Systems with Asymmetric Damping and Stiffness under Stationary Random Excitation
by Tudor Sireteanu, Ana-Maria Mitu, Ovidiu Solomon and Marius Giuclea
Mathematics 2022, 10(22), 4275; https://doi.org/10.3390/math10224275 - 15 Nov 2022
Cited by 4 | Viewed by 1634
Abstract
In this paper, the dynamic response of piecewise linear systems with asymmetric damping and stiffness for random excitation is studied. In order to approximate the statistical characteristics for each significant output of piecewise linear system, a method based on transmissibility factors is applied. [...] Read more.
In this paper, the dynamic response of piecewise linear systems with asymmetric damping and stiffness for random excitation is studied. In order to approximate the statistical characteristics for each significant output of piecewise linear system, a method based on transmissibility factors is applied. A stochastic linear system with the same transmissibility factor is attached, and the statistical parameters of the studied output corresponding to random excitation having rational spectral densities are determined by solving the associated Lyapunov equation. Using the attached linear systems for root mean square and for standard deviation of displacement, the shift of the sprung mass average position in a dynamic regime, due to damping or stiffness asymmetry, can be predicted with a good accuracy for stationary random input. The obtained results are compared with those determined by the Gaussian equivalent linearization method and by the numerical integration of asymmetric piecewise linear system equations. It is shown that the piecewise linear systems with asymmetrical damping and stiffness characteristics can provide a better vibration isolation (lower force transmissibility) than the linear system. Full article
(This article belongs to the Special Issue Probability, Stochastic Processes and Optimization)
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19 pages, 586 KiB  
Article
A More Flexible Asymmetric Exponential Modification of the Laplace Distribution with Applications for Chemical Concentration and Environment Data
by Jimmy Reyes, Mario A. Rojas, Pedro L. Cortés and Jaime Arrué
Mathematics 2022, 10(19), 3515; https://doi.org/10.3390/math10193515 - 26 Sep 2022
Cited by 1 | Viewed by 1560
Abstract
In this work, a new family of distributions based on the Laplace distribution is introduced. We define this new family by its stochastic representation as the sum of two independent random variables, one with a Laplace distribution and the other with an exponential [...] Read more.
In this work, a new family of distributions based on the Laplace distribution is introduced. We define this new family by its stochastic representation as the sum of two independent random variables, one with a Laplace distribution and the other with an exponential distribution. Using a Monte Carlo simulation study, the statistical performance of the estimators obtained by the moments and maximum likelihood methods were empirically evaluated. We studied the coverage probabilities and mean length of the confidence intervals of the corresponding parameters based on the asymptotic normality of these estimators. This simulation study reported a good statistical performance of these estimators. Fits were made to three real data sets with the new distribution, two related to chemical concentrations and one to the environment, comparing it with three similar distributions given in the literature. We have used information criteria for the selection of models. These results showed that the exponentially modified Laplace model can be an alternative distribution to model skewed data with high kurtosis. The new approach is a contribution to the tools of statisticians and various professionals interested in modeling data with high kurtosis. Full article
(This article belongs to the Special Issue Probability, Stochastic Processes and Optimization)
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23 pages, 2947 KiB  
Article
Influence of Binomial Crossover on Approximation Error of Evolutionary Algorithms
by Cong Wang, Jun He, Yu Chen and Xiufen Zou
Mathematics 2022, 10(16), 2850; https://doi.org/10.3390/math10162850 - 10 Aug 2022
Viewed by 1735
Abstract
Although differential evolution (DE) algorithms perform well on a large variety of complicated optimization problems, only a few theoretical studies are focused on the working principle of DE algorithms. To make the first attempt to reveal the function of binomial crossover, this paper [...] Read more.
Although differential evolution (DE) algorithms perform well on a large variety of complicated optimization problems, only a few theoretical studies are focused on the working principle of DE algorithms. To make the first attempt to reveal the function of binomial crossover, this paper aims to answer whether it can reduce the approximation error of evolutionary algorithms. By investigating the expected approximation error and the probability of not finding the optimum, we conduct a case study comparing two evolutionary algorithms with and without binomial crossover on two classical benchmark problems: OneMax and Deceptive. It is proven that using binomial crossover leads to the dominance of transition matrices. As a result, the algorithm with binomial crossover asymptotically outperforms that without crossover on both OneMax and Deceptive, and outperforms on OneMax, however, not on Deceptive. Furthermore, an adaptive parameter strategy is proposed which can strengthen the superiority of binomial crossover on Deceptive. Full article
(This article belongs to the Special Issue Probability, Stochastic Processes and Optimization)
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21 pages, 410 KiB  
Article
Decomposition of Finitely Additive Markov Chains in Discrete Space
by Alexander Zhdanok and Anna Khuruma
Mathematics 2022, 10(12), 2083; https://doi.org/10.3390/math10122083 - 15 Jun 2022
Cited by 3 | Viewed by 2001
Abstract
In this study, we consider general Markov chains (MC) defined by a transition probability (kernel) that is finitely additive. These Markov chains were constructed by S. Ramakrishnan within the concepts and symbolism of game theory. Here, we study these MCs by using the [...] Read more.
In this study, we consider general Markov chains (MC) defined by a transition probability (kernel) that is finitely additive. These Markov chains were constructed by S. Ramakrishnan within the concepts and symbolism of game theory. Here, we study these MCs by using the operator approach. In our work, the state space (phase space) of the MC has any cardinality and the sigma-algebra is discrete. The construction of a phase space allows us to decompose the Markov kernel (and the Markov operators that it generates) into the sum of two components: countably additive and purely finitely additive kernels. We show that the countably additive kernel is atomic. Some properties of Markov operators with a purely finitely additive kernel and their invariant measures are also studied. A class of combined finitely additive MC and two of its subclasses are introduced, and the properties of their invariant measures are proven. Some asymptotic regularities of such MCs were revealed. Full article
(This article belongs to the Special Issue Probability, Stochastic Processes and Optimization)
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10 pages, 258 KiB  
Article
On Geometric Mean and Cumulative Residual Entropy for Two Random Variables with Lindley Type Distribution
by Marius Giuclea and Costin-Ciprian Popescu
Mathematics 2022, 10(9), 1499; https://doi.org/10.3390/math10091499 - 30 Apr 2022
Cited by 3 | Viewed by 1894
Abstract
In this paper, we focus on two generalizations of the Lindley distribution and investigate, for each one separately, some special properties related to the geometric mean (GM) and the cumulative residual entropy (CRE), both of them [...] Read more.
In this paper, we focus on two generalizations of the Lindley distribution and investigate, for each one separately, some special properties related to the geometric mean (GM) and the cumulative residual entropy (CRE), both of them being of great importance from the theoretical as well as from the practical point of view. Full article
(This article belongs to the Special Issue Probability, Stochastic Processes and Optimization)
17 pages, 337 KiB  
Article
Fourth Cumulant Bound of Multivariate Normal Approximation on General Functionals of Gaussian Fields
by Yoon-Tae Kim and Hyun-Suk Park
Mathematics 2022, 10(8), 1352; https://doi.org/10.3390/math10081352 - 18 Apr 2022
Cited by 1 | Viewed by 1935
Abstract
We develop a technique for obtaining the fourth moment bound on the normal approximation of F, where F is an Rd-valued random vector whose components are functionals of Gaussian fields. This study transcends the case of vectors of multiple stochastic [...] Read more.
We develop a technique for obtaining the fourth moment bound on the normal approximation of F, where F is an Rd-valued random vector whose components are functionals of Gaussian fields. This study transcends the case of vectors of multiple stochastic integrals, which has been the subject of research so far. We perform this task by investigating the relationship between the expectations of two operators Γ and Γ*. Here, the operator Γ was introduced in Noreddine and Nourdin (2011) [On the Gaussian approximation of vector-valued multiple integrals. J. Multi. Anal.], and Γ* is a muilti-dimensional version of the operator used in Kim and Park (2018) [An Edgeworth expansion for functionals of Gaussian fields and its applications, stoch. proc. their Appl.]. In the specific case where F is a random variable belonging to the vector-valued multiple integrals, the conditions in the general case of F for the fourth moment bound are naturally satisfied and our method yields a better estimate than that obtained by the previous methods. In the case of d=1, the method developed here shows that, even in the case of general functionals of Gaussian fields, the fourth moment theorem holds without conditions for the multi-dimensional case. Full article
(This article belongs to the Special Issue Probability, Stochastic Processes and Optimization)
17 pages, 450 KiB  
Article
Testing for the Rayleigh Distribution: A New Test with Comparisons to Tests for Exponentiality Based on Transformed Data
by Gerrit Lodewicus Grobler, Elzanie Bothma and James Samuel Allison
Mathematics 2022, 10(8), 1316; https://doi.org/10.3390/math10081316 - 15 Apr 2022
Cited by 1 | Viewed by 2853
Abstract
We propose a new goodness-of-fit test for the Rayleigh distribution which is based on a distributional fixed-point property of the Stein characterization. The limiting null distribution of the test is derived and the consistency against fixed alternatives is also shown. The results of [...] Read more.
We propose a new goodness-of-fit test for the Rayleigh distribution which is based on a distributional fixed-point property of the Stein characterization. The limiting null distribution of the test is derived and the consistency against fixed alternatives is also shown. The results of a finite-sample comparison is presented, where we compare the power performance of the new test to a variety of other tests. In addition to existing tests for the Rayleigh distribution we also exploit the link between the exponential and Rayleigh distributions. This allows us to include some powerful tests developed specifically for the exponential distribution in the comparison. It is found that the new test outperforms competing tests for many of the alternative distributions. Interestingly, the highest estimated power, against all alternative distributions considered, is obtained by one of the tests specifically developed for the Rayleigh distribution and not by any of the exponentiality tests based on the transformed data. The use of the new test is illustrated on a real-world COVID-19 data set. Full article
(This article belongs to the Special Issue Probability, Stochastic Processes and Optimization)
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12 pages, 1123 KiB  
Article
Algorithmic Strategies for Precious Metals Price Forecasting
by Gil Cohen
Mathematics 2022, 10(7), 1134; https://doi.org/10.3390/math10071134 - 1 Apr 2022
Cited by 13 | Viewed by 5391
Abstract
This research is the first attempt to create machine learning (ML) algorithmic systems that would be able to automatically trade precious metals. The algorithm uses three forecast methodologies: linear regression (LR), Darvas boxes (DB), and Bollinger bands (BB). Our data consists of 20 [...] Read more.
This research is the first attempt to create machine learning (ML) algorithmic systems that would be able to automatically trade precious metals. The algorithm uses three forecast methodologies: linear regression (LR), Darvas boxes (DB), and Bollinger bands (BB). Our data consists of 20 years of daily price data concerning five precious metals futures: gold, silver, copper, platinum, and palladium. We found that all of the examined precious metals’ current daily returns are negatively autocorrelated to their former day’s returns and identified lagged interdependencies among the examined metals. Silver futures prices were found to be best forecasted by our systems, and platinum the worst. Moreover, our system better forecasts price-up trends than downtrends for all examined techniques and commodities. Linear regression was found to be the best technique to forecast silver and gold prices trends, while the Bollinger band technique best fits palladium forecasting. Full article
(This article belongs to the Special Issue Probability, Stochastic Processes and Optimization)
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9 pages, 255 KiB  
Article
Estimating the Coefficients of a System of Ordinary Differential Equations Based on Inaccurate Observations
by Gurami Tsitsiashvili, Marina Osipova and Yury Kharchenko
Mathematics 2022, 10(3), 502; https://doi.org/10.3390/math10030502 - 4 Feb 2022
Cited by 5 | Viewed by 1803
Abstract
In this paper, we solve the problem of estimating the parameters of a system of ordinary differential equations from observations on a short interval of argument values. By analogy with linear regression analysis, a sufficiently large number of observations are selected on this [...] Read more.
In this paper, we solve the problem of estimating the parameters of a system of ordinary differential equations from observations on a short interval of argument values. By analogy with linear regression analysis, a sufficiently large number of observations are selected on this segment and the values of the functions on the right side of the system and the values of the derivatives are estimated. According to the obtained estimates, unknown parameters are determined, using the differential equations system. The consistency of the estimates obtained in this way is proved with an increase in the number of observations over a short period of argument values. Here, an algorithm for estimating parameters acts as a system. The error of the obtained estimate is an indicator of its quality. A sequence of inaccurate measurements is a random process. The method of linear regression analysis applied to an almost linear regression function is used as an optimization procedure. Full article
(This article belongs to the Special Issue Probability, Stochastic Processes and Optimization)
15 pages, 318 KiB  
Article
Spherical Distributions Used in Evolutionary Algorithms
by Alexandru Agapie
Mathematics 2021, 9(23), 3098; https://doi.org/10.3390/math9233098 - 30 Nov 2021
Cited by 4 | Viewed by 2344
Abstract
Performance of evolutionary algorithms in real space is evaluated by local measures such as success probability and expected progress. In high-dimensional landscapes, most algorithms rely on the normal multi-variate, easy to assemble from independent, identically distributed components. This paper analyzes a different distribution, [...] Read more.
Performance of evolutionary algorithms in real space is evaluated by local measures such as success probability and expected progress. In high-dimensional landscapes, most algorithms rely on the normal multi-variate, easy to assemble from independent, identically distributed components. This paper analyzes a different distribution, also spherical, yet with dependent components and compact support: uniform in the sphere. Under a simple setting of the parameters, two algorithms are compared on a quadratic fitness function. The success probability and the expected progress of the algorithm with uniform distribution are proved to dominate their normal mutation counterparts by order n!!. Full article
(This article belongs to the Special Issue Probability, Stochastic Processes and Optimization)
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Review

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20 pages, 441 KiB  
Review
Evolution Strategies under the 1/5 Success Rule
by Alexandru Agapie
Mathematics 2023, 11(1), 201; https://doi.org/10.3390/math11010201 - 30 Dec 2022
Cited by 2 | Viewed by 6127
Abstract
For large space dimensions, the log-linear convergence of the elitist evolution strategy with a 1/5 success rule on the sphere fitness function has been observed, experimentally, from the very beginning. Finding a mathematical proof took considerably more time. This paper presents a review [...] Read more.
For large space dimensions, the log-linear convergence of the elitist evolution strategy with a 1/5 success rule on the sphere fitness function has been observed, experimentally, from the very beginning. Finding a mathematical proof took considerably more time. This paper presents a review and comparison of the most consistent theories developed so far, in the critical interpretation of the author, concerning both global convergence and the estimation of convergence rates. I discuss the local theory of the one-step expected progress and success probability for the (1+1) ES with a normal/uniform distribution inside the sphere mutation, thereby minimizing the SPHERE function, but also the adjacent global convergence and convergence rate theory, essentially based on the 1/5 rule. Small digressions into complementary theories (martingale, irreducible Markov chain, drift analysis) and different types of algorithms (population based, recombination, covariance matrix adaptation and self-adaptive ES) complete the review. Full article
(This article belongs to the Special Issue Probability, Stochastic Processes and Optimization)
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