Search ResultsShowing 1-13 of 13
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arXiv:1604.06028 (Published 2016-04-20)
Efficient computation of first passage times in Kuo's jump-diffusion model
Categories: math.PRS. G. Kuo 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 Kuo model. They also propose methods to invert these transforms. In the present paper, we give a much simpler expressions of the Laplace transform of the joint distribution, we also show that these Laplace transform can be extended to the complex plane and give efficient methods to invert them. The improvement in the computing times and accuracy is remarkable.
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arXiv:1308.5541 (Published 2013-08-26)
On the norming constants for normal maxima
Categories: math.PRIn 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.
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arXiv:1307.3433 (Published 2013-07-12)
Approximating Mills ratio
Categories: math.PRConsider the Mills ratio $f(x)=\big(1-\Phi(x)\big)/\phi(x), \, x\ge 0$, where $\phi$ is the density function of the standard Gaussian law and $\Phi$ 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.
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Gaussian Mills ratio is completely monotone
Comments: This paper has been withdrawn by the authors due that the results appear in Arpad Baricz. Mills'ratio: Monotonicity patterns and functional inequalities. Journal of Mathematical Analysis and Applications, 340 (2008) 1362-1370 M. R. Sampford. Inequalities on Mill's ratio and related functions, The Annals of Mathematical Statistics, Vol. 24, No. 1 (1953) 130-132Categories: math.PRConsider the Mills ratio corresponding to the standard Gaussian law, $f(x)=\big(1-\Phi(x)\big)/\phi(x), \, x\ge 0$, where $\phi$ is the density function of this law and $\Phi$ 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.
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arXiv:1210.1156 (Published 2012-10-03)
Local Malliavin Calculus for Lévy Processes and Applications
Categories: math.PRIn this paper a Malliavin calculus for L\'evy processes based on a family of true derivative operators is developed. The starting point is an extension to L\'evy 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 L\'evy process. As an application, we analyze the absolute continuity of the law of the solution of some stochastic differential equations.
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arXiv:1009.1543 (Published 2010-09-08)
Inversion of analytic characteristic functions and infinite convolutions of exponential and Laplace densities
Categories: math.PRWe 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.
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arXiv:1007.2516 (Published 2010-07-15)
Lévy area for Gaussian processes: A double Wiener-Itô integral approach
Categories: math.PRLet $\{X_{1}(t)\}_{0\leq t\leq1}$ 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{\'e}vy area can be defined as a double Wiener--It\`o 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{\'e}vy area are studied.
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arXiv:0902.2154 (Published 2009-02-12)
A new look at the Heston characteristic function
Categories: math.PRA 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.
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arXiv:0807.5035 (Published 2008-07-31)
Stein's method and normal approximation of Poisson functionals
Comments: 32 pagesCategories: math.PRWe combine Stein's method with a version of Malliavin calculus on the Poisson space. As a result, we obtain explicit Berry-Ess\'een bounds in Central Limit Theorems (CLTs) involving multiple Wiener-It\^o integrals with respect to a general Poisson measure. We provide several applications to CLTs related to Ornstein-Uhlenbeck L\'evy processes.
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arXiv:0804.2585 (Published 2008-04-16)
On the orthogonal polynomials associated with a Lévy process
Comments: Published in at http://dx.doi.org/10.1214/07-AOP343 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)Journal: Annals of Probability 2008, Vol. 36, No. 2, 765-795DOI: 10.1214/07-AOP343Categories: math.PRTags: journal articleLet $X=\{X_t, t\ge0\}$ be a c\`{a}dl\`{a}g L\'{e}vy 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 $p^{\sigma}_n(x)$ with respect to the measure $\sigma^2\delta_0(dx)+x^2 \nu(dx)$, where $\sigma^2$ is the variance of the Gaussian part of $X$ and $\nu$ its L\'{e}vy measure. These polynomials are the building blocks of a kind of chaotic representation of the square functionals of the L\'{e}vy 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 L\'{e}vy 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 L\'{e}vy 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 $X$.
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arXiv:0803.0829 (Published 2008-03-06)
Time--space harmonic polynomials relative to a Lévy process
Comments: Published in at http://dx.doi.org/10.3150/07-BEJ6173 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)Journal: Bernoulli 2008, Vol. 14, No. 1, 1-13DOI: 10.3150/07-BEJ6173Keywords: time-space harmonic polynomials relative, lévy process, recurrence relation, closed form, relationshipTags: journal articleIn this work, we give a closed form and a recurrence relation for a family of time--space harmonic polynomials relative to a L\'{e}vy process. We also state the relationship with the Kailath--Segall (orthogonal) polynomials associated to the process.
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Multiple Stratonovich integral and Hu--Meyer formula for Lévy processes
Comments: Published in at http://dx.doi.org/10.1214/10-AOP528 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)Journal: Annals of Probability 2010, Vol. 38, No. 6, 2136-2169DOI: 10.1214/10-AOP528Categories: math.PRKeywords: multiple stratonovich integral, lévy processes, stratonovich multiple integral, stochastic multiple integral, product random measureTags: journal articleIn 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 It\^{o} multiple integral and a Stratonovich multiple integral with respect to a L\'{e}vy 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 It\^{o} 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.
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arXiv:0711.2879 (Published 2007-11-19)
A family of martingales generated by a process with independent increments
Categories: math.PRAn 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