19 results on '"Larbi Alili"'
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2. On the semi-group of a scaled skew Bessel process
- Author
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Larbi Alili and Andrew Aylwin
- Subjects
Statistics and Probability ,Time inversion ,Bessel process ,Group (mathematics) ,010102 general mathematics ,Mathematical analysis ,Skew ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Green's function ,symbols ,Point (geometry) ,Differentiable function ,0101 mathematics ,Statistics, Probability and Uncertainty ,Bessel function ,Mathematics - Abstract
We define scaled skew Bessel processes and determine their Green’s functions and semi-group densities. We prove that these processes satisfy the time inversion property although the corresponding densities of the semi-groups are not, in general, twice differentiable in the density and starting point arguments. Some characterizations and semi-martingale properties are given.
- Published
- 2019
3. On Doney’s Striking Factorization of the Arc-Sine Law
- Author
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Larbi Alili, Carine Bartholmé, Loïc Chaumont, Pierre Patie, Mladen Savov, and Stavros Vakeroudis
- Published
- 2021
4. Further studies on square-root boundaries for Bessel processes
- Author
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Hiroyuki Matsumoto and Larbi Alili
- Subjects
Statistics and Probability ,Pure mathematics ,01 natural sciences ,Bessel processes ,010104 statistics & probability ,symbols.namesake ,Square root ,FOS: Mathematics ,60J65 ,Perpetuity ,exponential functionals ,0101 mathematics ,QA ,60G40 ,Mathematics ,60J60 ,010102 general mathematics ,Probability (math.PR) ,random affine equations ,60G18 ,symbols ,square-root boundaries ,Statistics, Probability and Uncertainty ,60G40, 60J65, 60G18, 60J60 ,Bessel function ,Mathematics - Probability - Abstract
We look at decompositions of perpetuities and apply them to the study of the distributions of hitting times of Bessel processes of two types of square root boundaries. These distributions are linked giving a new proof of some Mellin transforms results obtained by DeLong [6] and Yor [17]. Several random factorizations and characterizations of the studied distributions are established.\ud \ud
- Published
- 2018
5. Space and time inversions of stochastic processes and Kelvin transform
- Author
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Larbi Alili, Loïc Chaumont, Tomasz Żak, Piotr Graczyk, Department of Mathematics, University of Warwick, Warwick Mathematics Institute (WMI), University of Warwick [Coventry]-University of Warwick [Coventry], Laboratoire Angevin de Recherche en Mathématiques (LAREMA), Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS), Technical university of Wroclaw, Panorisk, and Graczyk, Piotr
- Subjects
Spacetime ,Stochastic process ,General Mathematics ,Gaussian ,010102 general mathematics ,Mathematical analysis ,Probability (math.PR) ,Markov process ,Inversion (meteorology) ,[MATH] Mathematics [math] ,01 natural sciences ,law.invention ,010101 applied mathematics ,symbols.namesake ,60J45, 31C05 (Primary), 60J65, 60J60 (Secondary) ,law ,symbols ,FOS: Mathematics ,0101 mathematics ,[MATH]Mathematics [math] ,Mathematics - Probability ,Bessel function ,Brownian motion ,Kelvin transform ,Mathematics - Abstract
Let $X$ be a standard Markov process. We prove that a space inversion property of $X$ implies the existence of a Kelvin transform of $X$-harmonic, excessive and operator-harmonic functions and that the inversion property is inherited by Doob $h$-transforms. We determine new classes of processes having space inversion properties amongst transient processes {satisfying the} time inversion property. {For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly.} We treat in details the examples of free scaled power Bessel processes, non-colliding Bessel particles, Wishart processes, Gaussian Ensemble and Dyson Brownian Motion.
- Published
- 2017
- Full Text
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6. Boundary crossing identities for Brownian motion and some nonlinear ode’s
- Author
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Larbi Alili and Pierre Patie
- Subjects
Nonlinear system ,Operator (computer programming) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Family of curves ,Lie group ,Heat equation ,Algebraic number ,First-hitting-time model ,Brownian motion ,Mathematics - Abstract
We start by introducing a nonlinear involution operator which maps the space of solutions of Sturm-Liouville equations into the space of solutions of the associated equations which turn out to be nonlinear ordinary differential equations. We study some algebraic and analytical properties of this involution operator as well as some properties of a two-parameter family of operators describing the set of solutions of Sturm-Liouville equations. Next, we show how a specific composition of these mappings allows us to connect, by means of a simple analytical expression, the law of the first passage time of a Brownian motion over a curve to a two-parameter family of curves. We offer three different proofs of this fact which may be of independent interests. In particular, one is based on the construction of parametric time-space harmonic transforms of the law of some Gauss-Markov processes. Another one, which is of algebraic nature, relies on the Lie group symmetry methods applied to the heat equation and reveals that our two-parameter transformation is the unique nontrivial one.
- Published
- 2014
7. Boundary-Crossing Identities for Diffusions Having the Time-Inversion Property
- Author
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Larbi Alili and Pierre Patie
- Subjects
Statistics and Probability ,Bessel process ,General Mathematics ,Mathematical analysis ,Function (mathematics) ,symbols.namesake ,Diffusion process ,Simple (abstract algebra) ,Local time ,symbols ,Statistics, Probability and Uncertainty ,QA ,Realization (systems) ,Bessel function ,Brownian motion ,Mathematics - Abstract
We review and study a one-parameter family of functional transformations, denoted by (S (β)) β∈ℝ, which, in the case β
- Published
- 2009
8. Further results on some singular linear stochastic differential equations
- Author
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Larbi Alili and Ching-Tang Wu
- Subjects
Statistics and Probability ,Type (model theory) ,Measure (mathematics) ,Stochastic differential equation ,Linear differential equation ,Canonical decomposition ,Modelling and Simulation ,Goursat kernels ,FOS: Mathematics ,Quantitative Biology::Populations and Evolution ,Applied mathematics ,Gramian matrices ,Brownian motion ,Mathematics ,Gramian matrix ,Stochastic process ,Self-reproducing kernels ,Applied Mathematics ,Enlargement of filtrations ,Probability (math.PR) ,Mathematical analysis ,26C05: 60G65 ,Connection (mathematics) ,Volterra transform ,Modeling and Simulation ,Stochastic differential equations ,Mathematics - Probability - Abstract
A class of Volterra transforms, preserving the Wiener measure, with kernels of Goursat type is considered. Such kernels satisfy a self-reproduction property. We provide some results on the inverses of the associated Gramian matrices which lead to a new self-reproduction property. A connection to the classical reproduction property is given. Results are then applied to the study of a class of singular linear stochastic differential equations together with the corresponding decompositions of filtrations. The studied equations are viewed as non-canonical decompositions of some generalized bridges.
- Published
- 2009
9. Representations of the First Hitting Time Density of an Ornstein-Uhlenbeck Process1
- Author
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Larbi Alili, Jesper Pedersen, and Pierre Patie
- Subjects
Statistics and Probability ,Pure mathematics ,Laplace transform ,Group (mathematics) ,Applied Mathematics ,Mathematical analysis ,Hitting time ,Ornstein–Uhlenbeck process ,Density estimation ,Parabolic cylinder function ,symbols.namesake ,Special functions ,Modeling and Simulation ,symbols ,Bessel function ,Mathematics - Abstract
Three expressions are provided for the first hitting time density of an Ornstein-Uhlenbeck process to reach a fixed level. The first hinges on an eigenvalue expansion involving zeros of the parabolic cylinder functions. The second is an integral representation involving some special functions whereas the third is given in terms of a functional of a 3-dimensional Bessel bridge. The expressions are used for approximating the density. 1Research supported by RiskLab, Switzerland, funded by Credit Suisse Group, Swiss Re and UBS AG. The third author was supported by a Steno grant from the Danish Natural Science Research Council.
- Published
- 2005
10. On the first crossing times of a Brownian motion and a family of continuous curves
- Author
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Pierre Patie and Larbi Alili
- Subjects
symbols.namesake ,Geometric Brownian motion ,Fractional Brownian motion ,Wiener process ,Diffusion process ,Reflected Brownian motion ,Mathematical analysis ,Family of curves ,symbols ,General Medicine ,Brownian excursion ,Heavy traffic approximation ,Mathematics - Abstract
We review the analytic transformations allowing to construct standard Brownian bridges from a Brownian motion. These are generalized and some of their properties are studied. The new family maps the space of continuous positive functions into a family of curves which is the topic of our study. We establish a simple and explicit formula relating the distributions of the first hitting times of each of these curves by a standard Brownian motion. To cite this article: L. Alili, P. Patie, C. R. Acad. Sci. Paris, Ser. I 340 (2005).
- Published
- 2005
11. Müntz linear transforms of Brownian motion
- Author
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Ching-Tang Wu and Larbi Alili
- Subjects
Statistics and Probability ,Enlargement of filtration ,Gaussian ,Volterra representation ,Muntz metal ,symbols.namesake ,Mathematics::Probability ,noncanonical representation ,45D05 ,QA ,Gaussian process ,Martingale representation theorem ,Brownian motion ,Mathematics ,Sequence ,Fractional Brownian motion ,Mathematical analysis ,M\"untz polynomials ,self-reproducing kernel ,45D05, 60G15 (Primary) 26C05, 46E22 (Secondary) ,Semimartingale ,60G15 ,symbols ,Statistics, Probability and Uncertainty ,Mathematics - Probability - Abstract
We consider a class of Volterra linear transforms of Brownian motion associated to a sequence of Müntz Gaussian spaces and determine explicitly their kernels; the kernels take a simple form when expressed in terms of Müntz-Legendre polynomials. These are new explicit examples of progressive Gaussian enlargement of a Brownian filtration. We give a necessary and sufficient condition for the existence of kernels of infinite order associated to an infinite dimensional Müntz Gaussian space; we also examine when the transformed Brownian motion remains a semimartingale in the filtration of the original process. This completes some already obtained partial answers to the aforementioned problems in the infinite dimensional case.
- Published
- 2014
12. Martin boundaries associated with a killed random walk
- Author
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Ron Doney and Larbi Alili
- Subjects
Statistics and Probability ,Rest (physics) ,Combinatorics ,Heterogeneous random walk in one dimension ,Distribution (mathematics) ,Space time ,Mathematical analysis ,Boundary (topology) ,Renewal theory ,Statistics, Probability and Uncertainty ,Moment-generating function ,Random walk ,Mathematics - Abstract
We start by studying the connection between the full Martin boundary associated with a space time version of a random walk which is killed on entering the negative half-line, and that associated with the bivariate renewal process of weak increasing ladder heights and times in the random walk. We show that although the corresponding spatial boundaries are isomorphic, the space time boundaries are not. The rest of the paper is devoted to determining these boundaries explicitly in the special case that the moment generating function of the step distribution exists in a non-empty interval.
- Published
- 2001
13. Wiener – hopf factorization revisited and some applications
- Author
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Ron Doney and Larbi Alili
- Subjects
Distribution (number theory) ,Stochastic process ,Mathematical analysis ,Function (mathematics) ,Bivariate analysis ,Random walk ,Wiener–Hopf method ,symbols.namesake ,Mathematics::Probability ,Factorization ,symbols ,Applied mathematics ,Condensed Matter::Strongly Correlated Electrons ,Renewal theory ,Mathematics - Abstract
A reformulation of the classical Wiener-Hopf factorization for random walks is given; this is applied to the study of the asymptotic behaviour of the ladder variables, the distribution of the maximum and the renewal mass function in the bivariate renewal process of ladder times and heights
- Published
- 1999
14. On inversions and Doob $h$-transforms of linear diffusions
- Author
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Larbi Alili, Piotr Graczyk, Tomasz Zak, Department of Statistics [Warwick], University of Warwick [Coventry], Laboratoire Angevin de Recherche en Mathématiques (LAREMA), Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS), Institute of Mathematics and Computer Science [Wroclaw] (IMCS), Wroclaw University of Science and Technology, and Graczyk, Piotr
- Subjects
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,Doob conditional process ,Involution ,Bessel process ,01 natural sciences ,60G60, 60J45 ,010104 statistics & probability ,symbols.namesake ,Positive harmonic function ,Euclidean geometry ,FOS: Mathematics ,Diffusion process ,Dual process ,Infinitesimal generator ,0101 mathematics ,Brownian motion ,Mathematics ,60G60 ,60J45 ,Probability (math.PR) ,010102 general mathematics ,Mathematical analysis ,Inversion ,Inversion (meteorology) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,symbols ,Bessel function ,Mathematics - Probability - Abstract
Let $X$ be a regular linear diffusion whose state space is an open interval $E\subseteq\mathbb{R}$. We consider a diffusion $X^*$ which probability law is obtained as a Doob $h$-transform of the law of $X$, where $h$ is a positive harmonic function for the infinitesimal generator of $X$ on $E$. This is the dual of $X$ with respect to $h(x)m(dx)$ where $m(dx)$ is the speed measure of $X$. Examples include the case where $X^*$ is $X$ conditioned to stay above some fixed level. We provide a construction of $X^*$ as a deterministic inversion of $X$, time changed with some random clock. The study involves the construction of some inversions which generalize the Euclidean inversions. Brownian motion with drift and Bessel processes are considered in details., 19 pages
- Published
- 2012
15. On the Joint Law of the L1 and L2 Norms of a 3-Dimensional Bessel Bridge
- Author
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Larbi Alili and Pierre Patie
- Subjects
symbols.namesake ,Quadratic equation ,Laplace transform ,Square root ,Law ,Computation ,Mathematical analysis ,symbols ,First-hitting-time model ,Expression (computer science) ,Bessel function ,Brownian motion ,Mathematics - Abstract
We give an analytical expression for the joint Laplace transform of the L-1 and L-2 norms of a 3-dimensional Bessel bridge. We derive the results by using merely probabilistic arguments. More precisely we show that the law of this functional is closely connected with the one of the first passage time of an Ornstein-Uhlenbeck process. The motivation for studying this problem are multiple; as an instance, they include the computation of the density of the first passage time of Brownian motion over some moving boundaries such as the square root and the quadratic ones.
- Published
- 2007
16. Some remarks on first passage of Lévy processes, the American put and pasting principles
- Author
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Andreas E. Kyprianou and Larbi Alili
- Subjects
Statistics and Probability ,Lévy process ,HG ,FOS: Economics and business ,Identity (mathematics) ,FOS: Mathematics ,Optimal stopping ,QA ,American options ,60G40 ,Mathematics ,Discrete mathematics ,principle of smooth pasting ,Series (mathematics) ,Stochastic process ,60G40 (Primary) 60J75, 91B70, 60G51 (Secondary) ,Probability (math.PR) ,principle of continuous pasting ,91B70 ,Lévy processes ,Pricing of Securities (q-fin.PR) ,Statistics, Probability and Uncertainty ,First-hitting-time model ,60J75 ,Quantitative Finance - Pricing of Securities ,Mathematics - Probability ,60G51 - Abstract
The purpose of this article is to provide, with the help of a fluctuation identity, a generic link between a number of known identities for the first passage time and overshoot above/below a fixed level of a Levy process and the solution of Gerber and Shiu [Astin Bull. 24 (1994) 195-220], Boyarchenko and Levendorskii [Working paper series EERS 98/02 (1998), Unpublished manuscript (1999), SIAM J. Control Optim. 40 (2002) 1663-1696], Chan [Original unpublished manuscript (2000)], Avram, Chan and Usabel [Stochastic Process. Appl. 100 (2002) 75-107], Mordecki [Finance Stoch. 6 (2002) 473-493], Asmussen, Avram and Pistorius [Stochastic Process. Appl. 109 (2004) 79-111] and Chesney and Jeanblanc [Appl. Math. Fin. 11 (2004) 207-225] to the American perpetual put optimal stopping problem. Furthermore, we make folklore precise and give necessary and sufficient conditions for smooth pasting to occur in the considered problem., Published at http://dx.doi.org/10.1214/105051605000000377 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
- Published
- 2005
17. On a Fluctuation Identity for Random Walks and Lévy Processes
- Author
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Larbi Alili, Ron Doney, and Loïc Chaumont
- Subjects
Discrete mathematics ,Pure mathematics ,Mathematics::Probability ,Lévy flight ,General Mathematics ,Identity (philosophy) ,media_common.quotation_subject ,Condensed Matter::Strongly Correlated Electrons ,Random walk ,Lévy process ,media_common ,Mathematics - Abstract
Bulletin of the London Mathematical Society, 2005 (37), ISSN:0024-6093, ISSN:1469-2120
- Published
- 2005
- Full Text
- View/download PDF
18. Canonical Decompositions of Certain Generalized Brownian Bridges
- Author
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Larbi Alili
- Subjects
Statistics and Probability ,Pure mathematics ,Fractional Brownian motion ,Mathematical analysis ,Brownian excursion ,Brownian bridge ,Volterra transform ,Diffusion process ,Mathematics::Probability ,26C05 ,Mathematics::K-Theory and Homology ,Canonical decomposition ,Filtration (mathematics) ,Decomposition (computer science) ,60J65 ,Statistics, Probability and Uncertainty ,Brownian motion ,QA ,Martingale representation theorem ,Mathematics - Abstract
We define a generalized Brownian bridge and we provide some information about its filtration. Two decompositions of this process as a semi-martingale are given. The first one is a Volterra decomposition and the second one is its canonical decomposition in its own filtration.
- Published
- 2002
19. On a Triplet of Exponential Brownian Functionals
- Author
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Hiroyuki Matsumoto, Larbi Alili, and Tomoyuki Shiraishi
- Subjects
Distribution (mathematics) ,Laplace transform ,Semigroup ,Joint probability distribution ,Mathematical analysis ,Geometry ,Laplace operator ,Heat kernel ,Brownian motion ,Mathematics ,Exponential function - Abstract
We study the three-dimensional joint distribution of a Brownian motion and the integrals of its exponential and its exponential squared from some points of view. We show an explicit expression of the Laplace transform of the distribution, which gives an extension of Yor’s resutlt on the two-dimensional one. We apply the result to some problems, in particular, to the calculation of an explicit form of the heat kernel of the semigroup generated by the Maass Laplacian on the Poincare upper half plane.
- Published
- 2001
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