Search ResultsShowing 1-17 of 17
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arXiv:2206.02137 (Published 2022-06-05)
Approximating the first passage time density from data using generalized Laguerre polynomials
Comments: 22 pages, 6 figuresThis paper analyzes a method to approximate the first passage time probability density function which turns to be particularly useful if only sample data are available. The method relies on a Laguerre-Gamma polynomial approximation and iteratively looks for the best degree of the polynomial such that the fitting function is a probability density function. The proposed iterative algorithm relies on simple and new recursion formulae involving first passage time moments. These moments can be computed recursively from cumulants, if they are known. In such a case, the approximated density can be used also for the maximum likelihood estimates of the parameters of the underlying stochastic process. If cumulants are not known, suitable unbiased estimators relying on k-statistics are employed. To check the feasibility of the method both in fitting the density and in estimating the parameters, the first passage time problem of a geometric Brownian motion is considered.
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arXiv:2102.00002 (Published 2021-01-31)
Input-output behaviour of a model neuron with alternating drift
Journal: BioSystems (2002) 67, 27-34Keywords: input-output behaviour, model neuron, wiener neuronal model subject, alternating drift, drift changes occurTags: journal articleThe input-output behaviour of the Wiener neuronal model subject to alternating input is studied under the assumption that the effect of such an input is to make the drift itself of an alternating type. Firing densities and related statistics are obtained via simulations of the sample-paths of the process in the following three cases: the drift changes occur during random periods characterized by (i) exponential distribution, (ii) Erlang distribution with a preassigned shape parameter, and (iii) deterministic distribution. The obtained results are compared with those holding for the Wiener neuronal model subject to sinusoidal input
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arXiv:2101.11578 (Published 2021-01-27)
Simulation of first-passage times for alternating Brownian motions
Journal: Meth. Comp. In Applied Probab. (2005) 7, 161--181Categories: math.PRKeywords: alternating brownian motions, first-passage times, first-passage-time density, estimate first-passage-time densities, constant boundaryTags: journal articleThe first-passage-time problem for a Brownian motion with alternating infinitesimal moments through a constant boundary is considered under the assumption that the time intervals between consecutive changes of these moments are described by an alternating renewal process. Bounds to the first-passage-time density and distribution function are obtained, and a simulation procedure to estimate first-passage-time densities is constructed. Examples of applications to problems in environmental sciences and mathematical finance are also provided.
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arXiv:2101.10583 (Published 2021-01-26)
On the connection between orthant probabilities and the first passage time problem
Comments: 2 tablesJournal: Journal of Statistical Computation & Simulation (2005), vol. 75, 437--445Keywords: first passage time problem, orthant probabilities, connection, non-singular gaussian discrete time-series, non-singular gaussian discrete-time seriesTags: journal articleThis article describes a new Monte Carlo method for the evaluation of the orthant probabilities by sampling first passage times of a non-singular Gaussian discrete time-series across an absorbing boundary. This procedure makes use of a simulation of several time-series sample paths, aiming to record their first crossing instants. Thus, the computation of the orthant probabilities is traced back to the accurate simulation of a non-singular Gaussian discrete-time series. Moreover, if the simulation is also efficient, this method is shown to be more speedy than the others proposed in the literature. As example, we make use of the Davies-Harte algorithm in the evaluation of the orthant probabilities associated to the ARFIMA$(0,d,0)$ model. Test results are presented that compare this method with currently available software.
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arXiv:2101.04481 (Published 2021-01-12)
A fractional generalization of the Dirichlet distribution and related distributions
Categories: math.PRThis paper is devoted to a fractional generalization of the Dirichlet distribution. The form of the multivariate distribution is derived assuming that the $n$ partitions of the interval $[0,W_n]$ are independent and identically distributed random variables following the generalized Mittag-Leffler distribution. The expected value and variance of the one-dimensional marginal are derived as well as the form of its probability density function. A related generalized Dirichlet distribution is studied that provides a reasonable approximation for some values of the parameters. The relation between this distribution and other generalizations of the Dirichlet distribution is discussed. Monte Carlo simulations of the one-dimensional marginals for both distributions are presented.
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arXiv:2006.12073 (Published 2020-06-22)
A cumulant approach for the first-passage-time problem of the Feller square-root process
Categories: math.PRThe paper focuses on an approximation of the first passage time probability density function of a Feller stochastic process by using cumulants and a Laguerre-Gamma polynomial approximation. The feasibility of the method relies on closed form formulae for cumulants and moments recovered from the Laplace transform of the probability density function and using the algebra of formal power series. To improve the approximation, sufficient conditions on cumulants are stated. The resulting procedure is made easier to implement by the symbolic calculus and a rational choice of the polynomial degree depending on skewness, kurtosis and hyperskewness. Some case-studies coming from neuronal and financial fields show the goodness of the approximation even for a low number of terms. Open problems are addressed at the end of the paper.
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arXiv:2001.08912 (Published 2020-01-24)
Flexible models for overdispersed and underdispersed count data
Categories: math.PRWithin the framework of probability models for overdispersed count data, we propose the generalized fractional Poisson distribution (gfPd), which is a natural generalization of the fractional Poisson distribution (fPd), and the standard Poisson distribution. We derive some properties of gfPd and more specifically we study moments, limiting behavior and other features of fPd. The skewness suggests that fPd can be left-skewed, right-skewed or symmetric; this makes the model flexible and appealing in practice. We apply the model to a real big count data and estimate the model parameters using maximum likelihood. Then, we turn to the very general class of weighted Poisson distributions (WPD's) to allow both overdispersion and underdispersion. Similar to Kemp's generalized hypergeometric probability distribution, based on hypergeometric functions, we introduce a novel WPD case where the weight function is chosen as a suitable ratio of three-parameter Mittag--Leffler functions. The proposed family includes the well-known COM-Poisson and the hyper-Poisson models. We characterize conditions on the parameters allowing for overdispersion and underdispersion, and analyze two special cases of interest which have not yet appeared in the literature.
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arXiv:1310.4254 (Published 2013-10-16)
Multivariate time-space harmonic polynomials: a symbolic approach
Comments: In pressJournal: Math. Methods Econ. Finance (2013)Categories: math.PRKeywords: multivariate time-space harmonic polynomials, symbolic approach, multivariate levy-sheffer systems, simple closed-form expressions, multivariate levy processesTags: journal articleBy means of a symbolic method, in this paper we introduce a new family of multivariate polynomials such that multivariate L\'evy processes can be dealt with as they were martingales. In the univariate case, this family of polynomials is known as time-space harmonic polynomials. Then, simple closed-form expressions of some multivariate classical families of polynomials are given. The main advantage of this symbolic representation is the plainness of the setting which reduces to few fundamental statements but also of its implementation in any symbolic software. The role played by cumulants is emphasized within the generalized Hermite polynomials. The new class of multivariate L\'evy-Sheffer systems is introduced.
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arXiv:1310.4005 (Published 2013-10-15)
On some applications of a symbolic representation of non-centered Lévy processes
Journal: Communications in Statistics - Theory and Methods (2013),42:21, 3974-3988,Categories: math.PRKeywords: non-centered lévy processes, symbolic representation, first kind meixner polynomials, applications, time-space harmonic polynomialsTags: journal articleBy using a symbolic technique known in the literature as the classical umbral calculus, we characterize two classes of polynomials related to L\'evy processes: the Kailath-Segall and the time-space harmonic polynomials. We provide the Kailath-Segall formula in terms of cumulants and we recover simple closed-forms for several families of polynomials with respect to not centered L\'evy processes, such as the Hermite polynomials with the Brownian motion, the Poisson-Charlier polynomials with the Poisson processes, the actuarial polynomials with the Gamma processes, the first kind Meixner polynomials with the Pascal processes, the Bernoulli, Euler and Krawtchuk polynomials with suitable random walks.
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arXiv:1304.0294 (Published 2013-04-01)
On a representation of time space-harmonic polynomials via symbolic Lévy processes
Journal: Scientiae Mathematicae Japonicae, Vol. 76, No. 1, 2013Categories: math.PRKeywords: symbolic lévy processes, time space-harmonic property, symbolic moment representation, hermite polynomials, symbolic deviceTags: journal articleIn this paper, we review the theory of time space-harmonic polynomials developed by using a symbolic device known in the literature as the classical umbral calculus. The advantage of this symbolic tool is twofold. First a moment representation is allowed for a wide class of polynomial stochastic involving the L\'evy processes in respect to which they are martingales. This representation includes some well-known examples such as Hermite polynomials in connection with Brownian motion. As a consequence, characterizations of many other families of polynomials having the time space-harmonic property can be recovered via the symbolic moment representation. New relations with Kailath-Segall polynomials are stated. Secondly the generalization to the multivariable framework is straightforward. Connections with cumulants and Bell polynomials are highlighted both in the univariate case and in the multivariate one. Open problems are addressed at the end of the paper.
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A new family of time-space harmonic polynomials with respect to Lévy processes
Comments: Ann. Mat. Pura Appl. In pressBy means of a symbolic method, a new family of time-space harmonic polynomials with respect to L\'evy processes is given. The coefficients of these polynomials involve a formal expression of L\'evy processes by which many identities are stated. We show that this family includes classical families of polynomials such as Hermite polynomials. Poisson-Charlier polynomials result to be a linear combinations of these new polynomials, when they have the property to be time-space harmonic with respect to the compensated Poisson process. The more general class of L\'evy-Sheffer polynomials is recovered as a linear combination of these new polynomials, when they are time-space harmonic with respect to L\'evy processes of very general form. We show the role played by cumulants of L\'evy processes so that connections with boolean and free cumulants are also stated.
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arXiv:1002.4803 (Published 2010-02-25)
Cumulants and convolutions via Abel polynomials
We provide an unifying polynomial expression giving moments in terms of cumulants, and viceversa, holding in the classical, boolean and free setting. This is done by using a symbolic treatment of Abel polynomials. As a by-product, we show that in the free cumulant theory the volume polynomial of Pitman and Stanley plays the role of the complete Bell exponential polynomial in the classical theory. Moreover via generalized Abel polynomials we construct a new class of cumulants, including the classical, boolean and free ones, and the convolutions linearized by them. Finally, via an umbral Fourier transform, we state a explicit connection between boolean and free convolution.
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arXiv:0706.2755 (Published 2007-06-19)
On certain bounds for first-crossing-time probabilities of a jump-diffusion process
Comments: 12 pages, 4 figuresJournal: Sci. Math. Jpn. 64 (2006), no. 2, 449-460Categories: math.PRKeywords: jump-diffusion process, first-crossing-time probabilities, constant jumps driven, first-crossing-time distribution function, usual stochastic orderTags: journal articleWe consider the first-crossing-time problem through a constant boundary for a Wiener process perturbed by random jumps driven by a counting process. On the base of a sample-path analysis of the jump-diffusion process we obtain explicit lower bounds for the first-crossing-time density and for the first-crossing-time distribution function. In the case of the distribution function, the bound is improved by use of processes comparison based on the usual stochastic order. The special case of constant jumps driven by a Poisson process is thoroughly discussed.
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arXiv:math/0412054 (Published 2004-12-02)
Umbral nature of the Poisson random variables
Comments: 23 pagesExtending the rigorous presentation of the classical umbral calculus given by Rota and Taylor in 1994, the so-called partition polynomials are interpreted with the aim to point out the umbral nature of the Poisson random variables. Among the new umbrae introduced, the main tool is the partition umbra that leads also to a simple expression of the functional composition of the exponential power series. Moreover a new short proof of the Lagrange inversion formula is given.
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arXiv:math/0412052 (Published 2004-12-02)
An umbral setting for cumulants and factorial moments
Comments: 20 pagesWe provide an algebraic setting for cumulants and factorial moments through the classical umbral calculus. Main tools are the compositional inverse of the unity umbra, connected with the logarithmic power series, and a new umbra here introduced, the singleton umbra. Various formulae are given expressing cumulants, factorial moments and central moments by umbral functions.
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arXiv:math/0305438 (Published 2003-05-30)
Towards the Modeling of Neuronal Firing by Gaussian Processes
Comments: 10 pages, 3 figures, to be published in Scientiae Mathematicae JaponicaeThis paper focuses on the outline of some computational methods for the approximate solution of the integral equations for the neuronal firing probability density and an algorithm for the generation of sample-paths in order to construct histograms estimating the firing densities. Our results originate from the study of non-Markov stationary Gaussian neuronal models with the aim to determine the neuron's firing probability density function. A parallel algorithm has been implemented in order to simulate large numbers of sample paths of Gaussian processes characterized by damped oscillatory covariances in the presence of time dependent boundaries. The analysis based on the simulation procedure provides an alternative research tool when closed-form results or analytic evaluation of the neuronal firing densities are not available.
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On the asymptotic behavior of first passage time densities for stationary Gaussian processes
Comments: 21 pages, 7 figures, to be published in Methodology and Computing in Applied ProbabilityCategories: math.PRMaking use of a Rice-like series expansion, for a class of stationary Gaussian processes the asymptotic behavior of the first passage time probability density function through certain time-varying boundaries, including periodic boundaries, is determined. Sufficient conditions are then given such that the density asymptotically exhibits an exponential behavior when the boundary is either asymptotically constant or asymptotically periodic.