Bessel processes and infinitely divisible laws
01 Jan 1981-pp 285-370
About: The article was published on 1981-01-01. It has received 265 citations till now. The article focuses on the topics: Bessel function & Infinitesimal generator.
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01 Oct 1991
TL;DR: In this paper, the authors discuss aspects of this incipient general theory which are most closely related to topics of current interest in theoretical stochastic processes, aimed at theoretical probabilists.
Abstract: INTRODUCTION Many different models of random trees have arisen in a variety of applied setting, and there is a large but scattered literature on exact and asymptotic results for particular models. For several years I have been interested in what kinds of “general theory” (as opposed to ad hoc analysis of particular models) might be useful in studying asymptotics of random trees. In this paper, aimed at theoretical probabilists, I discuss aspects of this incipient general theory which are most closely related to topics of current interest in theoretical stochastic processes. No prior knowledge of this subject is assumed: the paper is intended as an introduction and survey. To give the really big picture in a paragraph, consider a tree on n vertices. View the vertices as points in abstract (rather than d -dimensional) space, but let the edges have length (= 1, as a default) so that there is metric structure: the distance between two vertices is the length of the path between them. Consider the average distance between pairs of vertices. As n → ∞ this average distance could stay bounded or could grow as order n , but almost all natural random trees fall into one of two categories. In the first (and larger) category, the average distance grows as order logn. This category includes supercritical branching processes, and most “Markovian growth” models such as those occurring in the analysis of algorithms. This paper is concerned with the second category, in which the average distance grows as order n ½ .
507 citations
469 citations
Cites background or methods from "Bessel processes and infinitely div..."
...This last measurability property may be seen from the Yamada-Watanabe paper [27]; another proof of this may be obtained from the following Radon-Nikodym formula of [ 16 ], which will be used again below: for t>0, x>0, d>2:...
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...More generally, for d.x.y>O lexcept if d=x-O, y>0 - see (5.c) below), Qx~d ~, is the law of the process m (5.a) when X is the square of a d-dimensional Bessel process started at ]fx with drift t y, as defined in [16] following Watanabe [21] - see Theorem (5.87 of [ 16 ] for a proof....
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...BESQ ~ One of the ingredients of our proof of a) and b) above is that, from Watanabe [21], the law of Xt under Qa is that of (t2Xl/~) under Qa, X, where () a'x is the distribution of the square of the d-dimensional Bessel process, with drift l/x, starting at 0 (see either Watanabe [21], or Pitman-Yor [ 16 ] for the definitions and notations concerning these generalized Bessel processes)....
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...More generally, for d.x.y>O lexcept if d=x-O, y>0 - see (5.c) below), Qx~d ~, is the law of the process m (5.a) when X is the square of a d-dimensional Bessel process started at ]fx with drift t y, as defined in [ 16 ] following Watanabe [21] - see Theorem (5.87 of [16] for a proof....
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TL;DR: In this article, the distribution of the integral over a fixed time interval [0, T] of the exponential of Brownian motion with drift is computed explicitly, with the help of computations previously made by the author for Bessel processes.
Abstract: In this paper, distributional questions which arise in certain mathematical finance models are studied: the distribution of the integral over a fixed time interval [0, T]of the exponential of Brownian motion with drift is computed explicitly, with the help of computations previously made by the author for Bessel processes. The moments of this integral are obtained independently and take a particularly simple form. A subordination result involving this integral and previously obtained by Bougerol is recovered and related to an important identity for Bessel functions. When the fixed time T is replaced by an independent exponential time, the distribution of the integral is shown to be related to last-exit time distributions and the fixed time case is recovered by inverting Laplace transforms.
410 citations
01 Jan 2014
TL;DR: In this article, Kloeden, P., Ombach, J., Cyganowski, S., Kostrikin, A. J., Reddy, J.A., Pokrovskii, A., Shafarevich, I.A.
Abstract: Algebra and Famous Inpossibilities Differential Systems Dumortier.: Qualitative Theory of Planar Jost, J.: Dynamical Systems. Examples of Complex Behaviour Jost, J.: Postmodern Analysis Jost, J.: Riemannian Geometry and Geometric Analysis Kac, V.; Cheung, P.: Quantum Calculus Kannan, R.; Krueger, C.K.: Advanced Analysis on the Real Line Kelly, P.; Matthews, G.: The NonEuclidean Hyperbolic Plane Kempf, G.: Complex Abelian Varieties and Theta Functions Kitchens, B. P.: Symbolic Dynamics Kloeden, P.; Ombach, J.; Cyganowski, S.: From Elementary Probability to Stochastic Differential Equations with MAPLE Kloeden, P. E.; Platen; E.; Schurz, H.: Numerical Solution of SDE Through Computer Experiments Kostrikin, A. I.: Introduction to Algebra Krasnoselskii, M.A.; Pokrovskii, A.V.: Systems with Hysteresis Kurzweil, H.; Stellmacher, B.: The Theory of Finite Groups. An Introduction Lang, S.: Introduction to Differentiable Manifolds Luecking, D.H., Rubel, L.A.: Complex Analysis. A Functional Analysis Approach Ma, Zhi-Ming; Roeckner, M.: Introduction to the Theory of (non-symmetric) Dirichlet Forms Mac Lane, S.; Moerdijk, I.: Sheaves in Geometry and Logic Marcus, D.A.: Number Fields Martinez, A.: An Introduction to Semiclassical and Microlocal Analysis Matoušek, J.: Using the Borsuk-Ulam Theorem Matsuki, K.: Introduction to the Mori Program Mazzola, G.; Milmeister G.; Weissman J.: Comprehensive Mathematics for Computer Scientists 1 Mazzola, G.; Milmeister G.; Weissman J.: Comprehensive Mathematics for Computer Scientists 2 Mc Carthy, P. J.: Introduction to Arithmetical Functions McCrimmon, K.: A Taste of Jordan Algebras Meyer, R.M.: Essential Mathematics for Applied Field Meyer-Nieberg, P.: Banach Lattices Mikosch, T.: Non-Life Insurance Mathematics Mines, R.; Richman, F.; Ruitenburg, W.: A Course in Constructive Algebra Moise, E. E.: Introductory Problem Courses in Analysis and Topology Montesinos-Amilibia, J.M.: Classical Tessellations and Three Manifolds Morris, P.: Introduction to Game Theory Nikulin, V.V.; Shafarevich, I. R.: Geometries and Groups Oden, J. J.; Reddy, J. N.: Variational Methods in Theoretical Mechanics Øksendal, B.: Stochastic Differential Equations Øksendal, B.; Sulem, A.: Applied Stochastic Control of Jump Diffusions Poizat, B.: A Course in Model Theory Polster, B.: A Geometrical Picture Book Porter, J. R.; Woods, R.G.: Extensions and Absolutes of Hausdorff Spaces Radjavi, H.; Rosenthal, P.: Simultaneous Triangularization Ramsay, A.; Richtmeyer, R.D.: Introduction to Hyperbolic Geometry Rees, E.G.: Notes on Geometry Reisel, R. B.: Elementary Theory of Metric Spaces Rey, W. J. J.: Introduction to Robust and Quasi-Robust Statistical Methods Ribenboim, P.: Classical Theory of Algebraic Numbers Rickart, C. E.: Natural Function Algebras Roger G.: Analysis II Rotman, J. J.: Galois Theory Jost, J.: Compact Riemann Surfaces Applications ́ Introductory Lectures on Fluctuations of Levy Processes with Kyprianou, A. : Rautenberg, W.; A Concise Introduction to Mathematical Logic Samelson, H.: Notes on Lie Algebras Schiff, J. L.: Normal Families Sengupta, J.K.: Optimal Decisions under Uncertainty Séroul, R.: Programming for Mathematicians Seydel, R.: Tools for Computational Finance Shafarevich, I. R.: Discourses on Algebra Shapiro, J. H.: Composition Operators and Classical Function Theory Simonnet, M.: Measures and Probabilities Smith, K. E.; Kahanpää, L.; Kekäläinen, P.; Traves, W.: An Invitation to Algebraic Geometry Smith, K.T.: Power Series from a Computational Point of View Smoryński, C.: Logical Number Theory I. An Introduction Stichtenoth, H.: Algebraic Function Fields and Codes Stillwell, J.: Geometry of Surfaces Stroock, D.W.: An Introduction to the Theory of Large Deviations Sunder, V. S.: An Invitation to von Neumann Algebras Tamme, G.: Introduction to Étale Cohomology Tondeur, P.: Foliations on Riemannian Manifolds Toth, G.: Finite Möbius Groups, Minimal Immersions of Spheres, and Moduli Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems Wong, M.W.: Weyl Transforms Xambó-Descamps, S.: Block Error-Correcting Codes Zaanen, A.C.: Continuity, Integration and Fourier Theory Zhang, F.: Matrix Theory Zong, C.: Sphere Packings Zong, C.: Strange Phenomena in Convex and Discrete Geometry Zorich, V.A.: Mathematical Analysis I Zorich, V.A.: Mathematical Analysis II Rybakowski, K. P.: The Homotopy Index and Partial Differential Equations Sagan, H.: Space-Filling Curves Ruiz-Tolosa, J. R.; Castillo E.: From Vectors to Tensors Runde, V.: A Taste of Topology Rubel, L.A.: Entire and Meromorphic Functions Weintraub, S.H.: Galois Theory
401 citations
TL;DR: In this article, the first hitting times of squared Bessel processes and radial Ornstein-Uhlenbeck processes with negative dimensions or negative starting points are studied. But the authors focus on the first time a Bessel process hits a given barrier.
Abstract: Bessel processes play an important role in financial mathematics because of their strong relation to financial models such as geometric Brownian motion or Cox-Ingersoll-Ross processes. We are interested in the first time Bessel processes and, more generally, radial Ornstein-Uhlenbeck processes hit a given barrier. We give explicit expressions of the Laplace transforms of first hitting times by (squared) radial Ornstein-Uhlenbeck processes, that is, Cox-Ingersoll-Ross processes. As a natural extension we study squared Bessel processes and squared Ornstein-Uhlenbeck processes with negative dimensions or negative starting points and derive their properties.
336 citations
Cites background or methods from "Bessel processes and infinitely div..."
...c2) in Pitman–Yor [45], and inserting in (18) we have...
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...Remark 4 In disguised versions the formulae above also appear in Pitman–Yor [45], Chapter 12, and Eisenbaum [19]....
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...We know (see Getoor [25], Pitman–Yor [45]): L̂0→x (law) = x(2) 2Zν̂ , (11)...
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...7) and Pitman–Yor [45]) for y < x, y 6= 0: E x [ exp ( −α2 2 Tx→y )] = E δ̂ 0 [ exp ( −α2 2 L̂y↗x )]...
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...For a study of Bessel processes we refer to Revuz–Yor [49] and Pitman–Yor [45, 46], see also Appendix A....
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987 citations
543 citations
01 Jul 1967
TL;DR: In this paper, the entropy of very long flexible molecules in the presence of topological constraints is studied, and a formula deduced which needs the probability that a random walk will have a particular topological specification.
Abstract: The entropy of very long flexible molecules in the presence of topological constraints is studied, and a formula deduced which needs the probability that a random walk will have a particular topological specification. Examples are solved, including a plane random walk sweeping out a given angle around a point in the plane which is generalized to three dimensions including the passage of a random walk past many lines in space, and the probability that a random walk will penetrate through or become multiply entangled with a closed ring.
375 citations
375 citations