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Journal articles on the topic 'Positive self-similar Markov processes'

Author: Grafiati

Published: 4 June 2021

Last updated: 18 February 2022

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1

Chaumont, Loïc, and Víctor Rivero. "On some transformations between positive self-similar Markov processes." Stochastic Processes and their Applications 117, no. 12 (2007): 1889–909. http://dx.doi.org/10.1016/j.spa.2007.03.007.

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2

Chaumont, Loic, and Juan Carlos Pardo Millan. "The Lower Envelope of Positive Self-Similar Markov Processes." Electronic Journal of Probability 11 (2006): 1321–41. http://dx.doi.org/10.1214/ejp.v11-382.

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3

Pardo, J. C. "The Upper Envelope of Positive Self-Similar Markov Processes." Journal of Theoretical Probability 22, no. 2 (2008): 514–42. http://dx.doi.org/10.1007/s10959-008-0152-z.

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4

Chaumont, L., A. E. Kyprianou, and J. C. Pardo. "Some explicit identities associated with positive self-similar Markov processes." Stochastic Processes and their Applications 119, no. 3 (2009): 980–1000. http://dx.doi.org/10.1016/j.spa.2008.05.001.

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5

Pardo, J. C. "On the future infimum of positive self-similar Markov processes." Stochastics 78, no. 3 (2006): 123–55. http://dx.doi.org/10.1080/17442500600739055.

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6

Caballero, M. E., and L. Chaumont. "Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes." Annals of Probability 34, no. 3 (2006): 1012–34. http://dx.doi.org/10.1214/009117905000000611.

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7

Kyprianou, A. E., and P. Patie. "A Ciesielski–Taylor type identity for positive self-similar Markov processes." Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 47, no. 3 (2011): 917–28. http://dx.doi.org/10.1214/10-aihp398.

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8

Chaumont, Loïc, Andreas Kyprianou, Juan Carlos Pardo, and Víctor Rivero. "Fluctuation theory and exit systems for positive self-similar Markov processes." Annals of Probability 40, no. 1 (2012): 245–79. http://dx.doi.org/10.1214/10-aop612.

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9

Patie, P. "Law of the absorption time of some positive self-similar Markov processes." Annals of Probability 40, no. 2 (2012): 765–87. http://dx.doi.org/10.1214/10-aop638.

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10

Vidmar, Matija. "A temporal factorization at the maximum for certain positive self-similar Markov processes." Journal of Applied Probability 57, no. 4 (2020): 1045–69. http://dx.doi.org/10.1017/jpr.2020.62.

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AbstractFor a spectrally negative self-similar Markov process on $[0,\infty)$ with an a.s. finite overall supremum, we provide, in tractable detail, a kind of conditional Wiener–Hopf factorization at the maximum of the absorption time at zero, the conditioning being on the overall supremum and the jump at the overall supremum. In a companion result the Laplace transform of this absorption time (on the event that the process does not go above a given level) is identified under no other assumptions (such as the process admitting a recurrent extension and/or hitting zero continuously), generalizing some existing results in the literature.

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11

Kyprianou, Andreas E., Juan Carlos Pardo, and Matija Vidmar. "Double hypergeometric Lévy processes and self-similarity." Journal of Applied Probability 58, no. 1 (2021): 254–73. http://dx.doi.org/10.1017/jpr.2020.86.

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AbstractMotivated by a recent paper (Budd (2018)), where a new family of positive self-similar Markov processes associated to stable processes appears, we introduce a new family of Lévy processes, called the double hypergeometric class, whose Wiener–Hopf factorisation is explicit, and as a result many functionals can be determined in closed form.

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12

Kyprianou, A. E., and J. C. Pardo. "Continuous-State Branching Processes and Self-Similarity." Journal of Applied Probability 45, no. 04 (2008): 1140–60. http://dx.doi.org/10.1017/s0021900200005039.

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In this paper we study the α-stable continuous-state branching processes (for α ∈ (1, 2]) and the α-stable continuous-state branching processes conditioned never to become extinct in the light of positive self-similarity. Understanding the interaction of the Lamperti transformation for continuous-state branching processes and the Lamperti transformation for positive, self-similar Markov processes gives access to a number of explicit results concerning the paths of α-stable continuous-state branching processes and α-stable continuous-state branching processes conditioned never to become extinct.

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13

Kyprianou, A. E., and J. C. Pardo. "Continuous-State Branching Processes and Self-Similarity." Journal of Applied Probability 45, no. 4 (2008): 1140–60. http://dx.doi.org/10.1239/jap/1231340239.

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In this paper we study the α-stable continuous-state branching processes (for α ∈ (1, 2]) and the α-stable continuous-state branching processes conditioned never to become extinct in the light of positive self-similarity. Understanding the interaction of the Lamperti transformation for continuous-state branching processes and the Lamperti transformation for positive, self-similar Markov processes gives access to a number of explicit results concerning the paths of α-stable continuous-state branching processes and α-stable continuous-state branching processes conditioned never to become extinct.

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14

Bertoin, Jean, and Alexander R. Watson. "Probabilistic aspects of critical growth-fragmentation equations." Advances in Applied Probability 48, A (2016): 37–61. http://dx.doi.org/10.1017/apr.2016.41.

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AbstractThe self-similar growth-fragmentation equation describes the evolution of a medium in which particles grow and divide as time proceeds, with the growth and splitting of each particle depending only upon its size. The critical case of the equation, in which the growth and division rates balance one another, was considered in Doumic and Escobedo (2015) for the homogeneous case where the rates do not depend on the particle size. Here, we study the general self-similar case, using a probabilistic approach based on Lévy processes and positive self-similar Markov processes which also permits us to analyse quite general splitting rates. Whereas existence and uniqueness of the solution are rather easy to establish in the homogeneous case, the equation in the nonhomogeneous case has some surprising features. In particular, using the fact that certain self-similar Markov processes can enter (0,∞) continuously from either 0 or ∞, we exhibit unexpected spontaneous generation of mass in the solutions.

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15

Horton, Emma L., and Andreas E. Kyprianou. "More on hypergeometric Lévy processes." Advances in Applied Probability 48, A (2016): 153–58. http://dx.doi.org/10.1017/apr.2016.47.

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AbstractKuznetsov and co-authors in 2011‒14 introduced the family of hypergeometric Lévy processes. They appear naturally in the study of fluctuations of stable processes when one analyses stable processes through the theory of positive self-similar Markov processes. Hypergeometric Lévy processes are defined through their characteristic exponent, which, as a complex-valued function, has four independent parameters. In 2014 it was shown that the definition of a hypergeometric Lévy process could be taken to include a greater range of the aforesaid parameters than originally specified. In this short article, we push the parameter range even further.

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16

Vidmar, Matija. "First passage upwards for state-dependent-killed spectrally negative Lévy processes." Journal of Applied Probability 56, no. 2 (2019): 472–95. http://dx.doi.org/10.1017/jpr.2019.23.

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AbstractFor a spectrally negative Lévy process X, killed according to a rate that is a function ω of its position, we complement the recent findings of [12] by analysing (in greater generality) the exit probability of the one-sided upwards passage problem. When ω is strictly positive, this problem is related to the determination of the Laplace transform of the first passage time upwards for X that has been time-changed by the inverse of the additive functional $$\int_0^ \cdot \omega ({X_u}){\kern 1pt} {\rm{d}}u$$. In particular, our findings thus shed extra light on related results concerning first passage times downwards (resp. upwards) of continuous-state branching processes (resp. spectrally negative positive self-similar Markov processes).

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17

Caballero, M. E., and L. Chaumont. "Conditioned stable Lévy processes and the Lamperti representation." Journal of Applied Probability 43, no. 04 (2006): 967–83. http://dx.doi.org/10.1017/s0021900200002369.

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By variously killing a stable Lévy process when it leaves the positive half-line, conditioning it to stay positive, and conditioning it to hit 0 continuously, we obtain three different, positive, self-similar Markov processes which illustrate the three classes described by Lamperti (1972). For each of these processes, we explicitly compute the infinitesimal generator and from this deduce the characteristics of the underlying Lévy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Lévy processes before their first passage time across level 0, which we describe here. As an application, for a certain class of Lévy processes we give the law of the minimum before an independent exponential time. This provides the explicit form of the spatial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Lévy processes.

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18

Caballero, M. E., and L. Chaumont. "Conditioned stable Lévy processes and the Lamperti representation." Journal of Applied Probability 43, no. 4 (2006): 967–83. http://dx.doi.org/10.1239/jap/1165505201.

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By variously killing a stable Lévy process when it leaves the positive half-line, conditioning it to stay positive, and conditioning it to hit 0 continuously, we obtain three different, positive, self-similar Markov processes which illustrate the three classes described by Lamperti (1972). For each of these processes, we explicitly compute the infinitesimal generator and from this deduce the characteristics of the underlying Lévy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Lévy processes before their first passage time across level 0, which we describe here. As an application, for a certain class of Lévy processes we give the law of the minimum before an independent exponential time. This provides the explicit form of the spatial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Lévy processes.

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19

González, M., R. Martínez, and M. Mota. "Rates of Growth in a Class of Homogeneous Multidimensional Markov Chains." Journal of Applied Probability 43, no. 01 (2006): 159–74. http://dx.doi.org/10.1017/s0021900200001431.

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We investigate the asymptotic behaviour of homogeneous multidimensional Markov chains whose states have nonnegative integer components. We obtain growth rates for these models in a situation similar to the near-critical case for branching processes, provided that they converge to infinity with positive probability. Finally, the general theoretical results are applied to a class of controlled multitype branching process in which random control is allowed.

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20

González, M., R. Martínez, and M. Mota. "Rates of Growth in a Class of Homogeneous Multidimensional Markov Chains." Journal of Applied Probability 43, no. 1 (2006): 159–74. http://dx.doi.org/10.1239/jap/1143936250.

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We investigate the asymptotic behaviour of homogeneous multidimensional Markov chains whose states have nonnegative integer components. We obtain growth rates for these models in a situation similar to the near-critical case for branching processes, provided that they converge to infinity with positive probability. Finally, the general theoretical results are applied to a class of controlled multitype branching process in which random control is allowed.

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21

McNickle, Don. "Correlations in Output and Overflow Traffic Processes in Simple Queues." Journal of Applied Mathematics and Decision Sciences 2007 (September 24, 2007): 1–13. http://dx.doi.org/10.1155/2007/51801.

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We consider some simple Markov and Erlang queues with limited storage space. Although the departure processes from some such systems are known to be Poisson, they actually consist of the superposition of two complex correlated processes, the overflow process and the output process. We measure the cross-correlation between the counting processes for these two processes. It turns out that this can be positive, negative, or even zero (without implying independence). The models suggest some general principles on how big these correlations are, and when they are important. This may suggest when renewal or moment approximations to similar processes will be successful, and when they will not.

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22

Avram, Florin, Danijel Grahovac та Ceren Vardar-Acar. "The W,Z/ν,δ Paradigm for the First Passage of Strong Markov Processes without Positive Jumps". Risks 7, № 1 (2019): 18. http://dx.doi.org/10.3390/risks7010018.

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As is well-known, the benefit of restricting Lévy processes without positive jumps is the “ W , Z scale functions paradigm”, by which the knowledge of the scale functions W , Z extends immediately to other risk control problems. The same is true largely for strong Markov processes X t , with the notable distinctions that (a) it is more convenient to use as “basis” differential exit functions ν , δ , and that (b) it is not yet known how to compute ν , δ or W , Z beyond the Lévy, diffusion, and a few other cases. The unifying framework outlined in this paper suggests, however, via an example that the spectrally negative Markov and Lévy cases are very similar (except for the level of work involved in computing the basic functions ν , δ ). We illustrate the potential of the unified framework by introducing a new objective () for the optimization of dividends, inspired by the de Finetti problem of maximizing expected discounted cumulative dividends until ruin, where we replace ruin with an optimally chosen Azema-Yor/generalized draw-down/regret/trailing stopping time. This is defined as a hitting time of the “draw-down” process Y t = sup 0 ≤ s ≤ t X s - X t obtained by reflecting X t at its maximum. This new variational problem has been solved in a parallel paper.

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23

Piau, Didier. "Harmonic moments of inhomogeneous branching processes." Advances in Applied Probability 38, no. 02 (2006): 465–86. http://dx.doi.org/10.1017/s0001867800001051.

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We study the harmonic moments of Galton-Watson processes that are possibly inhomogeneous and have positive values. Good estimates of these are needed to compute unbiased estimators for noncanonical branching Markov processes, which occur, for instance, in the modelling of the polymerase chain reaction. By convexity, the ratio of the harmonic mean to the mean is at most 1. We prove that, for every square-integrable branching mechanism, this ratio lies between 1-A/k and 1-A/k for every initial population of size k&gt;A. The positive constants A and Aͤ are such that A≥Aͤ, are explicit, and depend only on the generation-by-generation branching mechanisms. In particular, we do not use the distribution of the limit of the classical martingale associated with the Galton-Watson process. Thus, emphasis is put on nonasymptotic bounds and on the dependence of the harmonic mean upon the size of the initial population. In the Bernoulli case, which is relevant for the modelling of the polymerase chain reaction, we prove essentially optimal bounds that are valid for every initial population size k≥1. Finally, in the general case and for sufficiently large initial populations, similar techniques yield sharp estimates of the harmonic moments of higher degree.

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24

Piau, Didier. "Harmonic moments of inhomogeneous branching processes." Advances in Applied Probability 38, no. 2 (2006): 465–86. http://dx.doi.org/10.1239/aap/1151337080.

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We study the harmonic moments of Galton-Watson processes that are possibly inhomogeneous and have positive values. Good estimates of these are needed to compute unbiased estimators for noncanonical branching Markov processes, which occur, for instance, in the modelling of the polymerase chain reaction. By convexity, the ratio of the harmonic mean to the mean is at most 1. We prove that, for every square-integrable branching mechanism, this ratio lies between 1-A/k and 1-A/k for every initial population of size k>A. The positive constants A and Aͤ are such that A≥Aͤ, are explicit, and depend only on the generation-by-generation branching mechanisms. In particular, we do not use the distribution of the limit of the classical martingale associated with the Galton-Watson process. Thus, emphasis is put on nonasymptotic bounds and on the dependence of the harmonic mean upon the size of the initial population. In the Bernoulli case, which is relevant for the modelling of the polymerase chain reaction, we prove essentially optimal bounds that are valid for every initial population size k≥1. Finally, in the general case and for sufficiently large initial populations, similar techniques yield sharp estimates of the harmonic moments of higher degree.

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25

FAGNOLA, FRANCO, and VERONICA UMANITÀ. "GENERIC QUANTUM MARKOV SEMIGROUPS, CYCLE DECOMPOSITION AND DEVIATION FROM EQUILIBRIUM." Infinite Dimensional Analysis, Quantum Probability and Related Topics 15, no. 03 (2012): 1250016. http://dx.doi.org/10.1142/s0219025712500166.

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A generic quantum Markov semigroup [Formula: see text] of a d-level quantum open system with a faithful normal invariant state ρ admits a dual semigroup [Formula: see text] with respect to the scalar product induced by ρ. We show that the difference of the generators [Formula: see text] can be written as the sum of a derivation 2i[H, ⋅] and a weighted difference of automorphisms [Formula: see text] where [Formula: see text] is a family of cycles on the d levels of the system, wc are positive weights and [Formula: see text] are unitaries. This formula allows us to represent the deviation from equilibrium (in a "small" time interval) as the superposition of cycles of the system where the difference between the forward and backward evolution is written as the difference of a reversible evolution and its time reversal. Moreover, it generalises cycle decomposition of Markov jump processes. We also find a similar formula with partial isometries instead of unitaries.

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26

Lee, Chihoon. "A Geometric Drift Inequality for a Reflected Fractional Brownian Motion Process on the Positive Orthant." Journal of Applied Probability 48, no. 3 (2011): 820–31. http://dx.doi.org/10.1239/jap/1316796917.

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We study a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = ℝ+d, with drift r0 ∈ ℝd and Hurst parameter H ∈ (½, 1). Under a natural stability condition on the drift vector r0 and reflection directions, we establish a geometric drift towards a compact set for the 1-skeleton chain Ž̆ of the RFBM process Z; that is, there exist β, b ∈ (0, ∞) and a compact set C ⊂ S such that ΔV(x):= Ex[V(Ž̆(1))] − V(x) ≤ −βV(x) + b1C(x), x ∈ S, for an exponentially growing Lyapunov function V : S → [1, ∞). For a wide class of Markov processes, such a drift inequality is known as a necessary and sufficient condition for exponential ergodicity. Indeed, similar drift inequalities have been established for reflected processes driven by standard Brownian motions, and our result can be viewed as their fractional Brownian motion counterpart. We also establish that the return times to the set C itself are geometrically bounded. Motivation for this study is that RFBM appears as a limiting workload process for fluid queueing network models fed by a large number of heavy-tailed ON/OFF sources in heavy traffic.

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27

Lee, Chihoon. "A Geometric Drift Inequality for a Reflected Fractional Brownian Motion Process on the Positive Orthant." Journal of Applied Probability 48, no. 03 (2011): 820–31. http://dx.doi.org/10.1017/s0021900200008342.

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We study a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = ℝ+ d , with drift r 0 ∈ ℝ d and Hurst parameter H ∈ (½, 1). Under a natural stability condition on the drift vector r 0 and reflection directions, we establish a geometric drift towards a compact set for the 1-skeleton chain Ž̆ of the RFBM process Z; that is, there exist β, b ∈ (0, ∞) and a compact set C ⊂ S such that ΔV(x):= E x [V(Ž̆(1))] − V(x) ≤ −βV(x) + b 1 C (x), x ∈ S, for an exponentially growing Lyapunov function V : S → [1, ∞). For a wide class of Markov processes, such a drift inequality is known as a necessary and sufficient condition for exponential ergodicity. Indeed, similar drift inequalities have been established for reflected processes driven by standard Brownian motions, and our result can be viewed as their fractional Brownian motion counterpart. We also establish that the return times to the set C itself are geometrically bounded. Motivation for this study is that RFBM appears as a limiting workload process for fluid queueing network models fed by a large number of heavy-tailed ON/OFF sources in heavy traffic.

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28

Graversen, S. E., and J. Vuolle-Apiala. "?-self-similar Markov processes." Probability Theory and Related Fields 71, no. 1 (1986): 149–58. http://dx.doi.org/10.1007/bf00366277.

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29

Liao, Ming, and Longmin Wang. "Isotropic self-similar Markov processes." Stochastic Processes and their Applications 121, no. 9 (2011): 2064–71. http://dx.doi.org/10.1016/j.spa.2011.05.008.

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30

Kyprianou, Andreas E., Víctor M. Rivero, and Weerapat Satitkanitkul. "Conditioned real self-similar Markov processes." Stochastic Processes and their Applications 129, no. 3 (2019): 954–77. http://dx.doi.org/10.1016/j.spa.2018.04.001.

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31

Chaumont, Loïc, and Salem Lamine. "On Rd-valued multi-self-similar Markov processes." Stochastic Processes and their Applications 130, no. 5 (2020): 3174–92. http://dx.doi.org/10.1016/j.spa.2019.09.009.

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32

Barczy, Mátyás, and Leif Döring. "On Entire Moments of Self-Similar Markov Processes." Stochastic Analysis and Applications 31, no. 2 (2013): 191–98. http://dx.doi.org/10.1080/07362994.2013.741397.

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33

Vuolle-Apiala, J. "Ito Excursion Theory for Self-Similar Markov Processes." Annals of Probability 22, no. 2 (1994): 546–65. http://dx.doi.org/10.1214/aop/1176988721.

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34

Modarresi, N., and S. Rezakhah. "Spectral analysis of multi-dimensional self-similar Markov processes." Journal of Physics A: Mathematical and Theoretical 43, no. 12 (2010): 125004. http://dx.doi.org/10.1088/1751-8113/43/12/125004.

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35

Pantí, H., J. C. Pardo, and V. M. Rivero. "Recurrent Extensions of Real-Valued Self-Similar Markov Processes." Potential Analysis 53, no. 3 (2019): 899–920. http://dx.doi.org/10.1007/s11118-019-09791-x.

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36

Rivero, Víctor. "Recurrent extensions of self-similar Markov processes and Cramér's condition." Bernoulli 11, no. 3 (2005): 471–509. http://dx.doi.org/10.3150/bj/1120591185.

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37

Chaumont, Loïc, Henry Pantí, and Víctor Rivero. "The Lamperti representation of real-valued self-similar Markov processes." Bernoulli 19, no. 5B (2013): 2494–523. http://dx.doi.org/10.3150/12-bej460.

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38

Caballero, María, and Víctor Rivero. "On the asymptotic behaviour of increasing self-similar Markov processes." Electronic Journal of Probability 14 (2009): 865–94. http://dx.doi.org/10.1214/ejp.v14-637.

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39

Wu, Chuanju, Feng Zhang, and Luqin Liu. "CRITERIA OF STRONG TRANSIENCE FOR OPERATOR-SELF-SIMILAR MARKOV PROCESSES." Acta Mathematica Scientia 26, no. 1 (2006): 41–48. http://dx.doi.org/10.1016/s0252-9602(06)60025-8.

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40

Rivero, Víctor. "Recurrent extensions of self-similar Markov processes and Cramér’s condition II." Bernoulli 13, no. 4 (2007): 1053–70. http://dx.doi.org/10.3150/07-bej6082.

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41

Haas, Bénédicte, and Víctor Rivero. "Quasi-stationary distributions and Yaglom limits of self-similar Markov processes." Stochastic Processes and their Applications 122, no. 12 (2012): 4054–95. http://dx.doi.org/10.1016/j.spa.2012.08.006.

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42

Rivero*, Víctor. "A law of iterated logarithm for increasing self-similar Markov processes." Stochastics and Stochastic Reports 75, no. 6 (2003): 443–72. http://dx.doi.org/10.1080/10451120310001646014.

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43

Fitzsimmons, Patrick. "On the Existence of Recurrent Extensions of Self-similar Markov Processes." Electronic Communications in Probability 11 (2006): 230–41. http://dx.doi.org/10.1214/ecp.v11-1222.

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44

Jacobsen, Martin, and Marc Yor. "Multi-self-similar Markov processes on ? + n and their Lamperti representations." Probability Theory and Related Fields 126, no. 1 (2003): 1–28. http://dx.doi.org/10.1007/s00440-003-0263-5.

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45

Dadoun, Benjamin. "Self-similar Growth Fragmentations as Scaling Limits of Markov Branching Processes." Journal of Theoretical Probability 33, no. 2 (2019): 590–610. http://dx.doi.org/10.1007/s10959-019-00975-0.

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46

Lihu, Huang, Li Bingzhang, and Liu Luqin. "Some fractal properties of a class of self-similar Markov processes." Wuhan University Journal of Natural Sciences 2, no. 3 (1997): 257–62. http://dx.doi.org/10.1007/bf02829899.

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47

LI, TINGTING, QIANQIAN YE, and LIFENG XI. "MARKOV SPECTRA OF SELF-SIMILAR NETWORKS BY SUBSTITUTION RULE." Fractals 26, no. 05 (2018): 1850064. http://dx.doi.org/10.1142/s0218348x18500640.

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For a class of self-similar networks generated by substitution rules, we investigate them in terms of normalized Laplacian spectra. Accordingly, we obtain the recurrent structure of Markov spectra for these self-similar networks, and also estimate the smallest positive eigenvalue for Laplace operator.

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48

Bertoin, Jean, and Igor Kortchemski. "Self-similar scaling limits of Markov chains on the positive integers." Annals of Applied Probability 26, no. 4 (2016): 2556–95. http://dx.doi.org/10.1214/15-aap1157.

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49

Döring, Leif. "A jump-type SDE approach to real-valued self-similar Markov processes." Transactions of the American Mathematical Society 367, no. 11 (2015): 7797–836. http://dx.doi.org/10.1090/s0002-9947-2015-06270-9.

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50

Bertoin, Jean, and Marc Yor. "On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes." Annales de la faculté des sciences de Toulouse Mathématiques 11, no. 1 (2002): 33–45. http://dx.doi.org/10.5802/afst.1016.

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