arXiv subjects

Frederic Utzet

Publications and source records attributed to Frederic Utzet.

14 recordsLinked to original sources

arXiv2016

Efficient computation of first passage times in Kou's jump-diffusion model

S. G. Kou and H. Wang [First Passage times of a Jump Diffusion Process \textit{Ann. Appl. Probab.} {\bf 35} (2003) 504--531] give expressions of both the (real) Laplace transform of the distribution of first passage time and the (real) Laplace transform of the joint distribution of the first passage time and the running maxima of a jump-diffusion model called Kou model. %However, to invert the last Laplace transform it is needed Kuo and Wang invert the first Laplace transform by using Gaver-Stehfest algorithm, and the inversion of second one involves a large computing time with an algebra computer system. In the present paper, we give a much simpler expression of the Laplace transform of the joint distribution, and we also show, using Complex Analysis techniques, that both Laplace transform can be extended to the complex plane. Hence, we can use variants of the Fourier-series methods to invert that Laplace transfoms, which are very efficent. The improvement in the computing times and accuracy is remarkable.

math.PR↗

arXiv2013

Maxima of Weibull-like distributions and the Lambert W function

The Weibull--like distributions form a large class of probability distributions that belong to the domain of attraction for the maxima of the Gumbel law. Besides the Weibull distribution, it includes important distributions as the Gamma laws and, in particular, the $χ^2$ distributions. In order to have explicit expressions of the norming constants for the maxima it is necessary to solve asymptotically a nonlinear equation; however, for some members of that family, numerical and simulation studies show that the constants that are usual suggested are inaccurate for moderate or even large sample sizes. In this paper we propose other norming constants computed with the asymptotics of the Lambert W function that significantly improve the accuracy of the approximation to the Gumbel law. These results are applied to the computation of the constants for the maxima of Gamma random variables that appear in some applied problems.

math.ST↗

arXiv2013

On the norming constants for normal maxima

In a remarkable paper, Peter Hall [{\it On the rate of convergence of normal extremes}, J. App. Prob, {\bf 16} (1979) 433--439] proved that the supremum norm distance between the distribution function of the normalized maximum of $n$ independent standard normal random variables and the distribution function of the Gumbel law is bounded by $3/\log n$. In the present paper we prove that choosing a different set of norming constants that bound can be reduced to $1/\log n$. As a consequence, using the asymptotic expansion of a Lambert $W$ type function, we propose new explicit constants for the maxima of normal random variables.

math.PR↗

arXiv2013

Approximating Mills ratio

Consider the Mills ratio $f(x)=\big(1-Φ(x)\big)/ϕ(x), \, x\ge 0$, where $ϕ$ is the density function of the standard Gaussian law and $Φ$ its cumulative distribution.We introduce a general procedure to approximate $f$ on the whole $[0,\infty)$ which allows to prove interesting properties where $f$ is involved. As applications we present a new proof that $1/f$ is strictly convex, and we give new sharp bounds of $f$ involving rational functions, functions with square roots or exponential terms. Also Chernoff type bounds for the Gaussian $Q$--function are studied.

math.PR↗

arXiv2013

Gaussian Mills ratio is completely monotone

Consider the Mills ratio corresponding to the standard Gaussian law, $f(x)=\big(1-Φ(x)\big)/ϕ(x), \, x\ge 0$, where $ϕ$ is the density function of this law and $Φ$ its cumulative distribution function. We prove that this function is completely monotone. In the proof we obtain a sequence of rational functions that are sharp bounds for $f$; it turns out that these rational functions are the convergents of the continued fraction defined by $f$, and provide an approximation procedure that allows to prove interesting properties where $f$ or its derivatives are involved. As an application we show that $1/f$ is strictly convex.

math.PR↗

arXiv2012

Local Malliavin Calculus for Lévy Processes and Applications

In this paper a Malliavin calculus for Lévy processes based on a family of true derivative operators is developed. The starting point is an extension to Lévy processes of the pioneering paper by Carlen and Pardoux [8] for the Poisson process, and our approach includes also the classical Malliavin derivative for Gaussian processes. We obtain a sufficient condition for the absolute continuity of functionals of the Lévy process. As an application, we analyze the absolute continuity of the law of the solution of some stochastic differential equations.

math.PR↗

arXiv2010

Multiple Stratonovich integral and Hu--Meyer formula for Lévy processes

In the framework of vector measures and the combinatorial approach to stochastic multiple integral introduced by Rota and Wallstrom [Ann. Probab. 25 (1997) 1257--1283], we present an Itô multiple integral and a Stratonovich multiple integral with respect to a Lévy process with finite moments up to a convenient order. In such a framework, the Stratonovich multiple integral is an integral with respect to a product random measure whereas the Itô multiple integral corresponds to integrate with respect to a random measure that gives zero mass to the diagonal sets. A general Hu--Meyer formula that gives the relationship between both integrals is proved. As particular cases, the classical Hu--Meyer formulas for the Brownian motion and for the Poisson process are deduced. Furthermore, a pathwise interpretation for the multiple integrals with respect to a subordinator is given.

math.PR↗

arXiv2010

Inversion of analytic characteristic functions and infinite convolutions of exponential and Laplace densities

We prove that certain quotients of entire functions are characteristic functions. Under some conditions, the probability measure corresponding to a characteristic function of that type has a density which can be expressed as a generalized Dirichlet series, which in turn is an infinite linear combination of exponential or Laplace densities. These results are applied to several examples.

math.PR↗

arXiv2010

Lévy area for Gaussian processes: A double Wiener-Itô integral approach

Let $\{X_{1}(t)\}_{0\leq t\leq1}$ and $\{X_{2}(t)\}_{0\leq t\leq1}$ be two independent continuous centered Gaussian processes with covariance functions$R_{1}$ and $R_{2}$. This paper shows that if the covariance functions are of finite $p$-variation and $q$-variation respectively and such that $p^{-1}+q^{-1}>1$,then the L{é}vy area can be defined as a double Wiener--Itò integral with respect to an isonormal Gaussian process induced by $X_{1}$ and $X_{2}$. Moreover, some properties of the characteristic function of that generalised L{é}vy area are studied.

math.PR↗

arXiv2009

A new look at the Heston characteristic function

A new expression for the characteristic function of log-spot in Heston model is presented. This expression more clearly exhibits its properties as an analytic characteristic function and allows us to compute the exact domain of the moment generating function. This result is then applied to the volatility smile at extreme strikes and to the control of the moments of spot. We also give a factorization of the moment generating function as product of Bessel type factors, and an approximating sequence to the law of log-spot is deduced.

math.PR↗

arXiv2008

Stein's method and normal approximation of Poisson functionals

We combine Stein's method with a version of Malliavin calculus on the Poisson space. As a result, we obtain explicit Berry-Esséen bounds in Central Limit Theorems (CLTs) involving multiple Wiener-Itô integrals with respect to a general Poisson measure. We provide several applications to CLTs related to Ornstein-Uhlenbeck Lévy processes.

math.PR↗

arXiv2008

On the orthogonal polynomials associated with a Lévy process

Let $X=\{X_t, t\ge0\}$ be a càdlàg Lévy process, centered, with moments of all orders. There are two families of orthogonal polynomials associated with $X$. On one hand, the Kailath--Segall formula gives the relationship between the iterated integrals and the variations of order $n$ of $X$, and defines a family of polynomials $P_1(x_1), P_2(x_1,x_2),...$ that are orthogonal with respect to the joint law of the variations of $X$. On the other hand, we can construct a sequence of orthogonal polynomials $p^σ_n(x)$ with respect to the measure $σ^2δ_0(dx)+x^2 ν(dx)$, where $σ^2$ is the variance of the Gaussian part of $X$ and $ν$ its Lévy measure. These polynomials are the building blocks of a kind of chaotic representation of the square functionals of the Lévy process proved by Nualart and Schoutens. The main objective of this work is to study the probabilistic properties and the relationship of the two families of polynomials. In particular, the Lévy processes such that the associated polynomials $P_n(x_1,...,x_n)$ depend on a fixed number of variables are characterized. Also, we give a sequence of Lévy processes that converge in the Skorohod topology to $X$, such that all variations and iterated integrals of the sequence converge to the variations and iterated integrals of $X$.

math.PR↗

arXiv2008

Time--space harmonic polynomials relative to a Lévy process

In this work, we give a closed form and a recurrence relation for a family of time--space harmonic polynomials relative to a Lévy process. We also state the relationship with the Kailath--Segall (orthogonal) polynomials associated to the process.

math.PR↗

arXiv2007

A family of martingales generated by a process with independent increments

An explicit procedure to construct a family of martingales generated by a process with independent increments is presented. The main tools are the polynomials that give the relationship between the moments and cumulants, and a set of martingales related to the jumps of the process called Teugels martingales

math.PR↗