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We study the preservation of impulse response non-negativity, also known as external positivity, for general discretization methods. The results yield new subsets in the space of transfer function coefficients which ensure external positivity. Consequently, linear programming approaches for designing controllers ensuring monotonic step responses are derived for both continuous and discrete-time systems. [ABSTRACT FROM AUTHOR]

We study the problem of detecting as quickly as possible the disorder time at which a purely jump Lévy process changes its probabilistic features. Assuming that its jumps are completely monotone, the monitored process is approximated by a sequence of hyperexponential processes. Then, the solution to the disorder problem for a hyperexponential process is used to approximate the one of the original problem. The efficiency of the proposed approximation scheme is investigated for some popular Lévy processes, such as the gamma, inverse Gaussian, variance-gamma and CGMY processes. [ABSTRACT FROM AUTHOR]

Abstract In this paper we introduce the new class of generalized Volterra functions. We prove some integral representations for them via Fox–Wright H-functions and Meijer G-functions. From positivity conditions on the weight in these representations, we found sufficient conditions on parameters of the generalized Volterra function to prove its complete monotonicity. As applications, a Turán type inequality for generalized Volterra functions is derived, infinite integral of some special functions are expressed in terms of the generalized Volterra functions and closed-form integral representations for a family of convergent Mathieu-type series defined in terms of generalized Volterra functions are established. [ABSTRACT FROM AUTHOR]

We prove complete monotonicity of sums of squares of generalized Baskakov basis functions by deriving the corresponding results for hypergeometric functions. Moreover, in the central Baskakov case we study the distribution of the complex zeros for large values of a parameter. We finally discuss the extension of some results for sums of higher powers. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present new approximations for the gamma function in terms of x x - 1 . The monotonicity and convexity of some involved functions are studied. As applications, sharp bounds are presented. It is also solved an open problem posed by Mortici (2013). [ABSTRACT FROM AUTHOR]

There exist only a few known examples of subordinators for which the transition probability density can be computed explicitly along side an expression for its Lévy measure and Laplace exponent. Such examples are useful in several areas of applied probability. For example, they are used in mathematical finance for modeling stochastic time change. They appear in combinatorial probability to construct sampling formulae, which in turn is related to a variety of issues in the theory of coalescence models. Moreover, they have also been extensively used in the potential analysis of subordinated Brownian motion in dimension d≥2. In this paper, we show that Kendall's classic identity for spectrally negative Lévy processes can be used to construct new families of subordinators with explicit transition probability semigroups. We describe the properties of these new subordinators and emphasize some interesting connections with explicit and previously unknown Laplace transform identities and with complete monotonicity properties of certain special functions. [ABSTRACT FROM AUTHOR]

Abstract: In the paper, the authors establish integral representations of some functions related to the remainder of Burnside’s formula for the gamma function and find the (logarithmically) complete monotonicity of these and related functions. These results extend and generalize some known conclusions. [Copyright &y& Elsevier]

Abstract: Motivated by the work of Chen and Qi (2005) [3] we study the products and . We prove that some functions associated to the previous products are completely monotonic and we establish some sharp inequalities. [Copyright &y& Elsevier]

Abstract: We consider the optimal dividend problem for the insurance risk process in a general Lévy process setting. The objective is to find a strategy which maximizes the expected total discounted dividends until the time of ruin. We give sufficient conditions under which the optimal strategy is of barrier type. In particular, we show that if the Lévy density is a completely monotone function, then the optimal dividend strategy is a barrier strategy. This approach was inspired by the work of Avram et al. [F. Avram, Z. Palmowski, M.R. Pistorius, On the optimal dividend problem for a spectrally negative Lévy process, The Annals of Applied Probability 17 (2007) 156–180], Loeffen [R. Loeffen, On optimality of the barrier strategy in De Finetti’s dividend problem for spectrally negative Lévy processes, The Annals of Applied Probability 18 (2008) 1669–1680] and Kyprianou et al. [A.E. Kyprianou, V. Rivero, R. Song, Convexity and smoothness of scale functions with applications to De Finetti’s control problem, Journal of Theoretical Probability 23 (2010) 547–564] in which the same problem was considered under the spectrally negative Lévy processes setting. [Copyright &y& Elsevier]

Abstract: In this paper we prove new complete monotonicity properties of some functions involving the gamma function. As consequences of them we establish various new upper and lower bounds for the gamma function and the harmonic numbers. [Copyright &y& Elsevier]

Abstract: We establish new sharp bounds for the digamma and trigamma functions, using the complete monotonicity of some class of functions. [ABSTRACT FROM AUTHOR]

Abstract: This paper introduces an exchangeable negative binomial distribution resulting from relaxing the independence of the Bernoulli sequence associated with a negative binomial distribution to exchangeability. It is demonstrated that the introduced distribution is a mixture of negative binomial distributions and can be characterized by infinitely many parameters that form a completely monotone sequence. The moments of the distribution are derived and a small simulation is conducted to illustrate the distribution. For data analytic purposes, two methods, truncation and completely-monotone links, are given for converting the saturated distribution of infinitely many parameters to parsimonious distributions of finitely many parameters. A full likelihood procedure is described which can be used to investigate correlated and overdispersed count data common in biomedical sciences and teratology. In the end, the introduced distribution is applied to analyze a real clinical data of burn wounds on patients. [Copyright &y& Elsevier]

Abstract: The optimal dividend problem proposed in de Finetti  is to find the dividend-payment strategy that maximizes the expected discounted value of dividends which are paid to the shareholders until the company is ruined. Avram et al.  studied the case when the risk process is modelled by a general spectrally negative Lévy process and Loeffen  gave sufficient conditions under which the optimal strategy is of the barrier type. Recently Kyprianou et al. strengthened the result of Loeffen  which established a larger class of Lévy processes for which the barrier strategy is optimal among all admissible ones. In this paper we use an analytical argument to re-investigate the optimality of barrier dividend strategies considered in the three recent papers. [Copyright &y& Elsevier]

Abstract: Let We prove that is completely monotonic on if and only if . Also, is completely monotonic on if and only if . An application of this result reveals that the best possible nonnegative constants in are given by and . [Copyright &y& Elsevier]

Abstract: Shi, Liu and Hu [X. Shi, F. Liu, M. Hu, A new asymptotic series for the Gamma function, J. Comput. Appl. Math. 195 (2006) 134–154] proved that the function defined by is strictly increasing for . The aim of our work is to prove that is strictly completely monotonic on . As direct consequences, we show that is strictly convex on , and then we prove that is strictly decreasing on , and strictly increasing on , where . [Copyright &y& Elsevier]