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  • 1.
    Bauder, David
    et al.
    Department of Mathematics, Humboldt-University of Berlin, Berlin, Germany.
    Bodnar, Taras
    Department of Mathematics, Stockholm University, Stockholm, Sweden.
    Mazur, Stepan
    Örebro University, Örebro University School of Business. Department of Statistics.
    Okhrin, Yarema
    Department of Statistics, University of Augsburg, Augsburg, Germany.
    Bayesian inference for the tangent portfolio2018In: International Journal of Theoretical and Applied Finance, ISSN 0219-0249, Vol. 21, no 8, p. 25article id 1850054Article in journal (Other academic)
    Abstract [en]

    In this paper we consider the estimation of the weights of tangent portfolios from the Bayesian point of view assuming normal conditional distributions of the logarithmic returns. For diffuse and conjugate priors for the mean vector and the covariance matrix, we derive stochastic representations for the posterior distributions of the weights of tangent portfolio and their linear combinations. Separately we provide the mean and variance of the posterior distributions, which are of key importance for portfolio selection. The analytic results are evaluated within a simulation study, where the precision of coverage intervals is assessed. 

  • 2.
    Bodnar, Taras
    et al.
    Department of Mathematics, Stockholm University, Stockholm, Sweden.
    Mazur, Stepan
    Örebro University, Örebro University School of Business.
    Muhinyuza, Stanislas
    Department of Mathematics, Stockholm University, Stockholm, Sweden; Department of Mathematics, College of Science and technology, University of Rwanda, Kigali-Rwanda.
    Parolya, Nestor
    Institute of Statistics, Leibniz University of Hannover, Hannover, Germany.
    On the product of a singular Wishart matrix and a singular Gaussian vector in high dimensions2018In: Theory of Probability and Mathematical Statistics, ISSN 0094-9000, Vol. 99, p. 37-50Article in journal (Refereed)
    Abstract [en]

    In this paper we consider the product of a singular Wishart random matrix and a singular normal random vector. A very useful stochastic representation is derived for this product, in using which the characteristic function of the product and its asymptotic distribution under the double asymptotic regime are established. The application of obtained stochastic representation speeds up the simulation studies where the product of a singular Wishart random matrix and a singular normal random vector is present. We further document a good performance of the derived asymptotic distribution within a numerical illustration. Finally, several important properties of the singular Wishart distribution are provided.

  • 3.
    Bodnar, Taras
    et al.
    Department of Mathematics, Stockholm University, Stockholm, Sweden.
    Mazur, Stepan
    Örebro University, Örebro University School of Business. Unit of Statistics.
    Ngailo, Edward
    Department of Mathematics, Stockholm University, Stockholm, Sweden.
    Parolya, Nestor
    Institute of Empirical Economics, Leibniz University of Hannover, Hannover, Germany.
    Discriminant analysis in small and large dimensions2019In: Theory of Probability and Mathematical Statistics, ISSN 1547-7363, Vol. 100, p. 27p. 24-42Article in journal (Other academic)
    Abstract [en]

    We study the distributional properties of the linear discriminant function under the assumption of normality by comparing two groups with the same covariance matrix but different mean vectors. A stochastic representation for the discriminant function coefficients is derived, which is then used to obtain their asymptotic distribution under the high-dimensional asymptotic regime. We investigate the performance of the classification analysis based on the discriminant function in both small and large dimensions. A stochastic representation is established, which allows to compute the error rate in an efficient way. We further compare the calculated error rate with the optimal one obtained under the assumption that the covariance matrix and the two mean vectors are known. Finally, we present an analytical expression of the error rate calculated in the high-dimensional asymptotic regime. The finite-sample properties of the derived theoretical results are assessed via an extensive Monte Carlo study.

  • 4.
    Bodnar, Taras
    et al.
    Department of Mathematics, Stockholm University, Stockholm, Sweden.
    Mazur, Stepan
    Department of Statistics, Lund University, Lund, Sweden.
    Okhrin, Yarema
    Department of Statistics, University of Augsburg, Augsburg, Germany.
    Bayesian estimation of the global minimum variance portfolio2017In: European Journal of Operational Research, ISSN 0377-2217, E-ISSN 1872-6860, Vol. 256, no 1, p. 292-307Article in journal (Refereed)
    Abstract [en]

    In this paper we consider the estimation of the weights of optimal portfolios from the Bayesian point of view under the assumption that the conditional distributions of the logarithmic returns are normal. Using the standard priors for the mean vector and the covariance matrix, we derive the posterior distributions for the weights of the global minimum variance portfolio. Moreover, we reparameterize the model to allow informative and non-informative priors directly for the weights of the global minimum variance portfolio. The posterior distributions of the portfolio weights are derived in explicit form for almost all models. The models are compared by using the coverage probabilities of credible intervals. In an empirical study we analyze the posterior densities of the weights of an international portfolio. 

  • 5.
    Bodnar, Taras
    et al.
    Department of Mathematics, Humboldt-University of Berlin, Berlin, Germany .
    Mazur, Stepan
    Department of Statistics, Lund University, Lund, Sweden.
    Okhrin, Yarema
    Department of Statistics, University of Augsburg, Augsburg, Germany.
    Distribution of the product of a singular Wishart matrix and a normal vector2014In: Theory of Probability and Mathematical Statistics, ISSN 0094-9000, no 91, p. 1-15Article in journal (Refereed)
    Abstract [en]

    In this paper we derive a very useful formula for the stochastic representation of the product of a singular Wishart matrix with a normal vector. Using this result, the expressions of the density function as well as of the characteristic function are established. Moreover, the derived stochastic representation is used to generate random samples from the product which leads to a considerable improvement in the computation efficiency. Finally, we present several important properties of the singular Wishart distribution, like its characteristic function and distributional properties of the partitioned singular Wishart matrix. 

  • 6.
    Bodnar, Taras
    et al.
    Department of Mathematics, Humboldt University of Berlin, Berlin, Germany.
    Mazur, Stepan
    Department of Statistics, European University Viadrina, Frankfurt an der Oder, Germany.
    Okhrin, Yarema
    Department of Statistics, University of Augsburg, Augsburg, Germany.
    On the exact and approximate distributions of the product of a Wishart matrix with a normal vector2013In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 122, p. 70-81Article in journal (Refereed)
    Abstract [en]

    In this paper we consider the distribution of the product of a Wishart random matrix and a Gaussian random vector. We derive a stochastic representation for the elements of the product. Using this result, the exact joint density for an arbitrary linear combination of the elements of the product is obtained. Furthermore, the derived stochastic representation allows us to simulate samples of arbitrary size by generating independently distributed chi-squared random variables and standard multivariate normal random vectors for each element of the sample. Additionally to the Monte Carlo approach, we suggest another approximation of the density function, which is based on the Gaussian integral and the third order Taylor expansion. We investigate, with a numerical study, the properties of the suggested approximations. A good performance is documented for both methods. 

  • 7.
    Bodnar, Taras
    et al.
    Department of Mathematics, Stockholm University, Stockholm, Sweden.
    Mazur, Stepan
    Örebro University, Örebro University School of Business. Department of Statistics.
    Parolya, Nestor
    Institute of Statistics, Leibniz University of Hannover, Hannover, Germany.
    Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix-variate location mixture of normal distributions2019In: Scandinavian Journal of Statistics, ISSN 0303-6898, E-ISSN 1467-9469, Vol. 46, no 2, p. 636-660Article in journal (Refereed)
    Abstract [en]

    In this paper we consider the asymptotic distributions of functionals of the sample covariance matrix and the sample mean vector obtained under the assumption that the matrix of observations has a matrix-variate location mixture of normal distributions. The central limit theorem is derived for the product of the sample covariance matrix and the sample mean vector. Moreover, we consider the product of the inverse sample covariance matrix and the mean vector for which the central limit theorem is established as well. All results are obtained under the large-dimensional asymptotic regime where the dimension p and the sample size n approach to infinity such that p/n → c ∈ [0, +∞) when the sample covariance matrix does not need to be invertible and p/n → c ∈ [0, 1) otherwise.

  • 8.
    Bodnar, Taras
    et al.
    Department of Mathematics, Stockholm University, Stockholm, Sweden.
    Mazur, Stepan
    Department of Mathematics, Aarhus University, Aarhus, Denmark.
    Podgorski, Krzysztof
    Department of Statistics, Lund University, Lund, Sweden.
    A test for the global minimum variance portfolio for small sample and singular covariance2017In: AStA Advances in Statistical Analysis, ISSN 1863-8171, E-ISSN 1863-818X, Vol. 101, no 3, p. 253-265Article in journal (Refereed)
    Abstract [en]

    Recently, a test dealing with the linear hypothesis for the global minimum variance portfolio weights was obtained under the assumption of non-singular covariance matrix. However, the problem of potential multicollinearity and correlations of assets constitutes a limitation of the classical portfolio theory. Therefore, there is an interest in developing theory in the presence of singularities in the covariance matrix. In this paper, we extend the test by analyzing the portfolio weights in the small sample case with a singular population covariance matrix. The results are illustrated using actual stock returns and a discussion of practical relevance of the model is presented. 

  • 9.
    Bodnar, Taras
    et al.
    Department of Mathematics, Stockholm University, Stockholm, Sweden.
    Mazur, Stepan
    Örebro University, Örebro University School of Business. Department of Statistics.
    Podgorski, Krzysztof
    Department of Statistics, Lund University, Lund, Sweden.
    Tyrcha, Joanna
    Department of Mathematics, Stockholm University, Stockholm, Sweden.
    Tangency portfolio weights for singular covariance matrix in small and large dimensions: estimation and test theory2019In: Journal of Statistical Planning and Inference, ISSN 0378-3758, E-ISSN 1873-1171, Vol. 201, p. 28p. 40-57Article in journal (Refereed)
    Abstract [en]

    In this paper we derive the finite-sample distribution of the estimated weights of the tangency portfolio when both the population and the sample covariance matrices are singular. These results are used in the derivation of a statistical test on the weights of the tangency portfolio where the distribution of the test statistic is obtained under both the null and the alternative hypotheses. Moreover, we establish the high-dimensional asymptotic distribution of the estimated weights of the tangency portfolio when both the portfolio dimension and the sample size increase to infinity. The theoretical findings are implemented in an empirical application dealing with the returns on the stocks included into the S&P 500 index. 

  • 10.
    Bodnar, Taras
    et al.
    Department of Mathematics, Stockholm University, Stockholm, Sweden.
    Mazur, Stepan
    Department of Statistics, Lund University, Lund, Sweden.
    Podgórski, Krzysztof
    Department of Statistics, Lund University, Lund, Sweden.
    Singular inverse Wishart distribution and its application to portfolio theory2016In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 143, p. 314-326Article in journal (Refereed)
    Abstract [en]

    The inverse of the standard estimate of covariance matrix is frequently used in the portfolio theory to estimate the optimal portfolio weights. For this problem, the distribution of the linear transformation of the inverse is needed. We obtain this distribution in the case when the sample size is smaller than the dimension, the underlying covariance matrix is singular, and the vectors of returns are independent and normally distributed. For the result, the distribution of the inverse of covariance estimate is needed and it is derived and referred to as the singular inverse Wishart distribution. We use these results to provide an explicit stochastic representation of an estimate of the mean-variance portfolio weights as well as to derive its characteristic function and the moments of higher order. The results are illustrated using actual stock returns and a discussion of practical relevance of the model is presented. 

  • 11.
    Gulliksson, Mårten
    et al.
    Örebro University, School of Science and Technology.
    Mazur, Stepan
    Örebro University, Örebro University School of Business.
    An Iterative Approach to Ill-Conditioned Optimal Portfolio Selection2019In: Computational Economics, ISSN 0927-7099, , p. 21Article in journal (Refereed)
    Abstract [en]

    Covariance matrix of the asset returns plays an important role in the portfolioselection. A number of papers is focused on the case when the covariance matrixis positive definite. In this paper, we consider portfolio selection with a singu-lar covariance matrix. We describe an iterative method based on a second orderdamped dynamical systems that solves the linear rank-deficient problem approxi-mately. Since the solution is not unique, we suggest one numerical solution that canbe chosen from the iterates that balances the size of portfolio and the risk. The nu-merical study confirms that the method has good convergence properties and givesa solution as good as or better than the constrained least norm Moore-Penrose solu-tion. Finally, we complement our result with an empirical study where we analyzea portfolio with actual returns listed in S&P 500 index.

  • 12.
    Javed, Farrukh
    et al.
    Örebro University, Örebro University School of Business.
    Loperfido, Nicola
    Dipartimento di Economia, Societ`a e Politica, Universit`a degli Studi di Urbino ”Carlo Bo”, Urbino (PU), Italy.
    Mazur, Stepan
    Örebro University, Örebro University School of Business.
    Fourth Cumulant of Multivariate Aggregate Claim ModelsManuscript (preprint) (Other academic)
    Abstract [en]

    The fourth cumulant for the aggregated multivariate claims is considered. A formula is presented for the general case when the aggregating variable is independent of the multivariate claims. Two important special cases are considered. In the first one, multivariate skewed normal claims are considered and aggregated by a Poisson variable. The second case is dealing with multivariate asymmetric generalized Laplace and aggregation is made by a negative binomial variable. Due to the invariance property the latter case can be derived directly, leading to the identity involving the cumulant of the claims and the aggregated claims. There is a well established relation between asymmetric Laplace motion and negative binomial process that corresponds to the invariance principle of the aggregating claims for the generalized asymmetric Laplace distribution. We explore this relation and provide multivariate continuous time version of the results. It is discussed how these results that deals only with dependence in the claim sizes can be used to obtain a formula for the fourth cumulant for more complex aggregate models of multivariate claims in which the dependence is also in the aggregating variables.

  • 13.
    Javed, Farrukh
    et al.
    Örebro University, Örebro University School of Business.
    Mazur, Stepan
    Örebro University, Örebro University School of Business.
    Ngailo, Edward
    Department of Mathematics, Linköping University, Linköping, Sweden.
    Higher order moments of the estimated tangency portfolio weights2020In: Journal of Applied Statistics, ISSN 0266-4763, E-ISSN 1360-0532, , p. 18Article in journal (Other academic)
    Abstract [en]

    In this paper, we consider the estimated weights of the tangency portfolio. We derive analytical expressions for the higher order non-central and central moments of these weights when the returns are assumed to be independently and multivariate normally distributed. Moreover, the expressions for mean, variance, skewness and kurtosis of the estimated weights are obtained in closed forms. Later, we complement our results with a simulation study where data from the multivariate normal and t-distributions are simulated, and the first four moments of estimated weights are computed by using the Monte Carlo experiment. It is noteworthy to mention that the distributional assumption of returns is found to be important, especially for the first two moments. Finally, through an empirical illustration utilizing returns of four financial indices listed in NASDAQ stock exchange, we observe the presence of time dynamics in higher moments.

  • 14.
    Karlsson, Sune
    et al.
    Örebro University, Örebro University School of Business.
    Mazur, Stepan
    Örebro University, Örebro University School of Business.
    Flexible Fat-tailed BVARs2019Conference paper (Refereed)
    Abstract [en]

    We propose a general class of fat-tailed distributions which includes the t,Cauchy, Laplace and slash distributions as well as the normal distribution as spe-cial cases. Full conditional posterior distributions for the Bayesian VAR-model arederived and used to construct a MCMC-sampler for the joint posterior distribution.The framework allows for selection of a specic special case as the distribution forthe error terms in the VAR if the evidence in the data is strong while at the sametime allowing for considerable exibility and more general distributions than oeredby any of the special cases.

  • 15.
    Kotsiuba, Ihor
    et al.
    Ivan Franko National University of Lviv, Lviv, Ukraine.
    Mazur, Stepan
    School of Economics and Management, Lund University, Lund, Sweden.
    Conditions of equilibrium for european option2014In: Mathematical and Computer Modelling, Series: Physical & Mathematical Sciences, ISSN 2308-5878, Vol. 11, p. 114-121Article in journal (Refereed)
    Abstract [en]

    The article deals with the Black-Scholes model where parameters depend on the time and the environmental state, conditions under which the fair price of an option before and after averaging coincide are considered. Furthermore, the main mathematical characteristics for the fair price of the European call option under the finite discrete-time homogenous Markov chain process are given.

  • 16.
    Kotsiuba, Ihor
    et al.
    van Franko National University, Lviv, Ukraine.
    Mazur, Stepan
    Department of Statistics, Lund University, Lund, Sweden.
    On the asymptotic and approximate distributions of the product of an inverse Wishart matrix and a Gaussian random vector2015In: Theory of Probability and Mathematical Statistics, ISSN 0868-6904, Vol. 93, p. 95-104Article in journal (Refereed)
    Abstract [en]

    In this paper we study the distribution of the product of an inverse Wishart random matrix and a Gaussian random vector. We derive its asymptotic distribution as well as its approximate density function formula which is based on the Gaussian integral and the third order Taylor expansion. Furthermore, we compare obtained asymptotic and approximate density functions with the exact density which is obtained by Bodnar and Okhrin (2011). A good performance of obtained results is documented in the numerical study. 

  • 17.
    Loperfido, Nicola
    et al.
    Dipartimento di Economia, Società e Politica, Università degli Studi di Urbino “Carlo Bo”, Urbino PU, Italy.
    Mazur, Stepan
    Örebro University, Örebro University School of Business. Unit of Statistics.
    Podgorski, Krzysztof
    Department of Statistics, Lund University, Lund, Sweden.
    Third cumulant for multivariate aggregate claim models2018In: Scandinavian Actuarial Journal, ISSN 0346-1238, E-ISSN 1651-2030, no 2, p. 109-128Article in journal (Refereed)
    Abstract [en]

    The third cumulant for the aggregated multivariate claims is considered. A formula is presented for the general case when the aggregating variable is independent of the multivariate claims. Two important special cases are considered. In the first one, multivariate skewed normal claims are considered and aggregated by a Poisson variable. The second case is dealing with multivariate asymmetric generalized Laplace and aggregation is made by a negative binomial variable. Due to the invariance property the latter case can be derived directly, leading to the identity involving the cumulant of the claims and the aggregated claims. There is a well-established relation between asymmetric Laplace motion and negative binomial process that corresponds to the invariance principle of the aggregating claims for the generalized asymmetric Laplace distribution. We explore this relation and provide multivariate continuous time version of the results. It is discussed how these results that deal only with dependence in the claim sizes can be used to obtain a formula for the third cumulant for more complex aggregate models of multivariate claims in which the dependence is also in the aggregating variables.

  • 18.
    Mazur, Stepan
    et al.
    Örebro University, Örebro University School of Business.
    Otryakhin, Dmitry
    Department of Mathematics, Aarhus University, Aarhus, Denmark.
    Linear Fractional Stable Motion with the RLFSM R Package2019Report (Other academic)
    Abstract [en]

    Linear fractional stable motion is a type of a stochastic integral driven by symmetric alpha-stable Levy motion. The integral could be considered as a non-Gaussian analogue of the fractional Brownian motion. The present paper discusses R package rlfsm created for numerical procedures with the linear fractional stable motion. It is a set of tools for simulation of these processes as well as performing statistical inference and simulation studies on them. We introduce: tools that we developed to work with that type of motions as well as methods and ideas underlying them. Also we perform numerical experiments to show finite-sample behavior of certain estimators of the integral, and give an idea of how to envelope workflow related to the linear fractional stable motion in S4 classes and methods. Supplementary materials, including codes for numerical experiments, are available online. rlfsm could be found on CRAN and gitlab.

  • 19.
    Mazur, Stepan
    et al.
    Örebro University, Örebro University School of Business.
    Otryakhin, Dmitry
    Department of Mathematics, Aarhus University, Aarhus, Denmark.
    Podolskij, Mark
    Department of Mathematics, Aarhus University, Aarhus, Denmark.
    Estimation of the linear fractional stable motion2020In: Bernoulli, ISSN 1350-7265, E-ISSN 1573-9759, Vol. 26, no 1, p. 226-252Article in journal (Other academic)
    Abstract [en]

    In this paper, we investigate the parametric inference for the linear fractional stable motion in high and low frequency setting. The symmetric linear fractional stable motion is a three-parameter family, which constitutes a natural non-Gaussian analogue of the scaled fractional Brownian motion. It is fully characterised by the scaling parameter σ>0, the self-similarity parameter H∈(0,1) and the stability index α∈(0,2) of the driving stable motion. The parametric estimation of the model is inspired by the limit theory for stationary increments Lévy moving average processes that has been recently studied in (Ann. Probab. 45 (2017) 4477–4528). More specifically, we combine (negative) power variation statistics and empirical characteristic functions to obtain consistent estimates of (σ,α,H). We present the law of large numbers and some fully feasible weak limit theorems.

  • 20. Yeleyko, Yaroslav
    et al.
    Lazariv, Taras
    Mazur, Stepan
    Lund University, Lund, Sweden.
    Main mathematical characteristics of payment functional2012In: Bulletin of the Lviv University, Series in Mathematics & Informatics, ISSN 2078-3744, no 18, p. 157-164Article in journal (Refereed)
    Abstract [en]

    The aim of our work was to calculate basic mathematical characteristics of payment functional, such as mathematical expectation, variation and expected risk in discrete and continuous market models. In our paper we present a model of financial market in which the entire time interval in the continuous model is divided into steps with exponential distribution, and in the discrete model into steps of length 1. Using the appropriate ergodic theorems for continuous and discrete Markov chains we have found the mathematical expectation, variation and expected risk. The results have theoretical and practical application in verifying the accuracy of modeling prices of derivative securities in the economy and finance.

  • 21. Yeleyko, Yaroslav
    et al.
    Lazariv, Taras
    Mazur, Stepan
    Lund University, Lund, Sweden.
    Multifractal products of diffusion processes and randomized scenario2011In: Bulletin of the Lviv University, Series in Mechanics & Mathematics, ISSN 2078-3744, no 74, p. 83-88Article in journal (Refereed)
    Abstract [en]

    We investigate the properties of multifractal products of the exponential of stationary diffusion processes defined by stochastic differential equations with linear drift and certain form of the diffusion coefficient corresponding to a variety of marginal distribution.

1 - 21 of 21
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