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Journal articles on the topic 'Lévy subordinators'

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Published: 11 March 2023

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1

Barndorff-Nielsen, Ole E., Jan Pedersen, and Ken-Iti Sato. "Multivariate subordination, self-decomposability and stability." Advances in Applied Probability 33, no. 1 (March 2001): 160–87. http://dx.doi.org/10.1017/s0001867800010685.

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Multivariate subordinators are multivariate Lévy processes that are increasing in each component. Various examples of multivariate subordinators, of interest for applications, are given. Subordination of Lévy processes with independent components by multivariate subordinators is defined. Multiparameter Lévy processes and their subordination are introduced so that the subordinated processes are multivariate Lévy processes. The relations between the characteristic triplets involved are established. It is shown that operator self-decomposability and the operator version of the class Lm property are inherited from the multivariate subordinator to the subordinated process under the condition of operator stability of the subordinand.

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2

Sun, Yunpeng, Rafael Mendoza-Arriaga, and Vadim Linetsky. "Marshall–Olkin distributions, subordinators, efficient simulation, and applications to credit risk." Advances in Applied Probability 49, no. 2 (June 2017): 481–514. http://dx.doi.org/10.1017/apr.2017.10.

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Abstract In the paper we present a novel construction of Marshall–Olkin (MO) multivariate exponential distributions of failure times as distributions of the first-passage times of the coordinates of multidimensional Lévy subordinator processes above independent unit-mean exponential random variables. A time-inhomogeneous version is also given that replaces Lévy subordinators with additive subordinators. An attractive feature of MO distributions for applications, such as to portfolio credit risk, is its singular component that yields positive probabilities of simultaneous defaults of multiple obligors, capturing the default clustering phenomenon. The drawback of the original MO fatal shock construction of MO distributions is that it requires one to simulate 2n-1 independent exponential random variables. In practice, the dimensionality is typically on the order of hundreds or thousands of obligors in a large credit portfolio, rendering the MO fatal shock construction infeasible to simulate. The subordinator construction reduces the problem of simulating a rich subclass of MO distributions to simulating an n-dimensional subordinator. When one works with the class of subordinators constructed from independent one-dimensional subordinators with known transition distributions, such as gamma and inverse Gaussian, or their Sato versions in the additive case, the simulation effort is linear in n. To illustrate, we present a simulation of 100,000 samples of a credit portfolio with 1,000 obligors that takes less than 18 seconds on a PC.

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3

Covo, Shai. "One-dimensional distributions of subordinators with upper truncated Lévy measure, and applications." Advances in Applied Probability 41, no. 2 (June 2009): 367–92. http://dx.doi.org/10.1239/aap/1246886616.

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Given a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with continuous Lévy measure ν, we derive a formula for the distribution function Fs (x; t) at time t of the associated subordinator whose Lévy measure is the restriction of ν to (0,s]. It will be expressed in terms of ν and the marginal distribution function F (⋅; t) of the original process. A generalization concerning an arbitrary truncation of ν will follow. Under certain conditions, an analogous formula will be obtained for the nth derivative, ∂nFs (x; t) ∂ xn. The requirement that ν is continuous is shown to have no intrinsic meaning. A number of interesting results involving the size ordered jumps of subordinators will be derived. An appropriate approximation for the small jumps of a gamma process will be considered, leading to a revisiting of the generalized Dickman distribution.

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4

Covo, Shai. "One-dimensional distributions of subordinators with upper truncated Lévy measure, and applications." Advances in Applied Probability 41, no. 02 (June 2009): 367–92. http://dx.doi.org/10.1017/s0001867800003347.

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Given a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with continuous Lévy measure ν, we derive a formula for the distribution functionFs(x;t) at timetof the associated subordinator whose Lévy measure is the restriction of ν to (0,s]. It will be expressed in terms of ν and the marginal distribution functionF(⋅;t) of the original process. A generalization concerning an arbitrary truncation of ν will follow. Under certain conditions, an analogous formula will be obtained for thenth derivative, ∂nFs(x;t) ∂xn. The requirement that ν is continuous is shown to have no intrinsic meaning. A number of interesting results involving the size ordered jumps of subordinators will be derived. An appropriate approximation for the small jumps of a gamma process will be considered, leading to a revisiting of the generalized Dickman distribution.

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5

Levajković, Tijana, Hermann Mena, and Martin Zarfl. "Lévy processes, subordinators and crime modelling." Novi Sad Journal of Mathematics 46, no. 2 (August 26, 2016): 65–86. http://dx.doi.org/10.30755/nsjom.03903.

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6

Al Masry, Zeina, Landy Rabehasaina, and Ghislain Verdier. "Change-level detection for Lévy subordinators." Stochastic Processes and their Applications 147 (May 2022): 423–55. http://dx.doi.org/10.1016/j.spa.2022.01.022.

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7

Beghin, Luisa, and Costantino Ricciuti. "Lévy Processes Linked to the Lower-Incomplete Gamma Function." Fractal and Fractional 5, no. 3 (July 17, 2021): 72. http://dx.doi.org/10.3390/fractalfract5030072.

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We start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study Lévy processes that are time-changed by these subordinators with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fractional Brownian motion produces a model of anomalous diffusion, which exhibits a sub-diffusive behavior.

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8

Hering, Christian, Marius Hofert, Jan-Frederik Mai, and Matthias Scherer. "Constructing hierarchical Archimedean copulas with Lévy subordinators." Journal of Multivariate Analysis 101, no. 6 (July 2010): 1428–33. http://dx.doi.org/10.1016/j.jmva.2009.10.005.

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9

Schneider, Jan, and Roman Urban. "Lévy Subordinators in Cones of Fuzzy Sets." Journal of Theoretical Probability 32, no. 4 (August 9, 2018): 1909–24. http://dx.doi.org/10.1007/s10959-018-0853-x.

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10

Covo, Shai. "On Approximations of Small Jumps of Subordinators with Particular Emphasis on a Dickman-Type Limit." Journal of Applied Probability 46, no. 3 (September 2009): 732–55. http://dx.doi.org/10.1239/jap/1253279849.

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Let X be a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with infinite Lévy measure, let Xε be the sum of jumps not exceeding ε, and let µ(ε)=E[Xε(1)]. We study the question of weak convergence of Xε/µ(ε) as ε ↓0, in terms of the limit behavior of µ(ε)/ε. The most interesting case reduces to the weak convergence of Xε/ε to a subordinator whose marginals are generalized Dickman distributions; we give some necessary and sufficient conditions for this to hold. For a certain significant class of subordinators for which the latter convergence holds, and whose most prominent representative is the gamma process, we give some detailed analysis regarding the convergence quality (in particular, in the context of approximating X itself). This paper completes, in some respects, the study made by Asmussen and Rosiński (2001).

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11

Covo, Shai. "On Approximations of Small Jumps of Subordinators with Particular Emphasis on a Dickman-Type Limit." Journal of Applied Probability 46, no. 03 (September 2009): 732–55. http://dx.doi.org/10.1017/s0021900200005854.

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Let X be a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with infinite Lévy measure, let X ε be the sum of jumps not exceeding ε, and let µ(ε)=E[X ε(1)]. We study the question of weak convergence of X ε/µ(ε) as ε ↓0, in terms of the limit behavior of µ(ε)/ε. The most interesting case reduces to the weak convergence of X ε/ε to a subordinator whose marginals are generalized Dickman distributions; we give some necessary and sufficient conditions for this to hold. For a certain significant class of subordinators for which the latter convergence holds, and whose most prominent representative is the gamma process, we give some detailed analysis regarding the convergence quality (in particular, in the context of approximating X itself). This paper completes, in some respects, the study made by Asmussen and Rosiński (2001).

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12

Belomestny, Denis, Shota Gugushvili, Moritz Schauer, and Peter Spreij. "Nonparametric Bayesian inference for Gamma-type Lévy subordinators." Communications in Mathematical Sciences 17, no. 3 (2019): 781–816. http://dx.doi.org/10.4310/cms.2019.v17.n3.a8.

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13

Huzak, Miljenko, Mihael Perman, Hrvoje Šikić, and Zoran Vondraček. "Ruin probabilities for competing claim processes." Journal of Applied Probability 41, no. 3 (September 2004): 679–90. http://dx.doi.org/10.1239/jap/1091543418.

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LetC1,C2,…,Cmbe independent subordinators with finite expectations and denote their sum byC. Consider the classical risk processX(t) =x+ct-C(t). The ruin probability is given by the well-known Pollaczek–Khinchin formula. If ruin occurs, however, it will be caused by a jump of one of the subordinatorsCi. Formulae for the probability that ruin is caused byCiare derived. These formulae can be extended to perturbed risk processes of the typeX(t) =x+ct-C(t) +Z(t), whereZis a Lévy process with mean 0 and no positive jumps.

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14

Fink, Holger. "Conditional Characteristic Functions of Molchan-Golosov Fractional Lévy Processes with Application to Credit Risk." Journal of Applied Probability 50, no. 4 (December 2013): 983–1005. http://dx.doi.org/10.1239/jap/1389370095.

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Molchan-Golosov fractional Lévy processes (MG-FLPs) are introduced by way of a multivariate componentwise Molchan-Golosov transformation based on an n-dimensional driving Lévy process. Using results of fractional calculus and infinitely divisible distributions, we are able to calculate the conditional characteristic function of integrals driven by MG-FLPs. This leads to important predictions concerning multivariate fractional Brownian motion, fractional subordinators, and general fractional stochastic differential equations. Examples are the fractional Lévy Ornstein-Uhlenbeck and Cox-Ingersoll-Ross models. As an application we present a fractional credit model with a long range dependent hazard rate and calculate bond prices.

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15

Fink, Holger. "Conditional Characteristic Functions of Molchan-Golosov Fractional Lévy Processes with Application to Credit Risk." Journal of Applied Probability 50, no. 04 (December 2013): 983–1005. http://dx.doi.org/10.1017/s0021900200013759.

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Molchan-Golosov fractional Lévy processes (MG-FLPs) are introduced by way of a multivariate componentwise Molchan-Golosov transformation based on ann-dimensional driving Lévy process. Using results of fractional calculus and infinitely divisible distributions, we are able to calculate the conditional characteristic function of integrals driven by MG-FLPs. This leads to important predictions concerning multivariate fractional Brownian motion, fractional subordinators, and general fractional stochastic differential equations. Examples are the fractional Lévy Ornstein-Uhlenbeck and Cox-Ingersoll-Ross models. As an application we present a fractional credit model with a long range dependent hazard rate and calculate bond prices.

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16

MARFÈ, ROBERTO. "A MULTIVARIATE PURE-JUMP MODEL WITH MULTI-FACTORIAL DEPENDENCE STRUCTURE." International Journal of Theoretical and Applied Finance 15, no. 04 (June 2012): 1250028. http://dx.doi.org/10.1142/s0219024912500288.

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In this work we propose a new approach to build multivariate pure jump processes. We introduce linear and nonlinear dependence, without restrictions on marginal properties, by imposing a multi-factorial structure separately on both positive and negative jumps. Such a new approach provides higher flexibility in calibrating nonlinear dependence than in other comparable Lévy models in the literature. Using the notion of multivariate subordinator, this modeling approach can be applied to the class of univariate Lévy processes which can be written as the difference of two subordinators. A common example in the financial literature is the variance gamma process, which we extend to the multivariate (multi-factorial) case. The model is tractable and a straightforward multivariate simulation procedure is available. An empirical analysis documents an accurate multivariate fit of stock index returns in terms of both linear and nonlinear dependence. An example of multi-asset option pricing emphasizes the importance of the proposed multivariate approach.

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17

MAI, JAN-FREDERIK, and MATTHIAS SCHERER. "A TRACTABLE MULTIVARIATE DEFAULT MODEL BASED ON A STOCHASTIC TIME-CHANGE." International Journal of Theoretical and Applied Finance 12, no. 02 (March 2009): 227–49. http://dx.doi.org/10.1142/s0219024909005208.

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A stochastic time-change is applied to introduce dependence to a portfolio of credit-risky assets whose default times are modeled as random variables with arbitrary distribution. The dependence structure of the vector of default times is completely separated from its marginal default probabilities, making the model analytically tractable. This separation is achieved by restricting the time-change to suitable Lévy subordinators which preserve the marginal distributions. Jump times of the Lévy subordinator are interpreted as times of excess default clustering. Relevant for practical implementations is that the parameters of the time-change allow for an intuitive economical explanation and can be calibrated independently of the marginal default probabilities. On a theoretical level, a so-called time normalization allows to compute the resulting copula of the default times. Moreover, the exact portfolio-loss distribution and an approximation for large portfolios under a homogeneous portfolio assumption are derived. Given these results, the pricing of complex portfolio derivatives is possible in closed-form. Three different implementations of the model are proposed, including a compound Poisson subordinator, a Gamma subordinator, and an Inverse Gaussian subordinator. Using two parameters to adjust the dependence structure in each case, the model is capable of capturing the full range of dependence patterns from independence to complete comonotonicity. A simultaneous calibration to portfolio-CDS spreads and CDO tranche spreads is carried out to demonstrate the model's applicability.

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18

Yakubovich, Yuri. "A simple proof of the Lévy–Khintchine formula for subordinators." Statistics & Probability Letters 176 (September 2021): 109136. http://dx.doi.org/10.1016/j.spl.2021.109136.

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19

Riva-Palacio, Alan, and Fabrizio Leisen. "Compound vectors of subordinators and their associated positive Lévy copulas." Journal of Multivariate Analysis 183 (May 2021): 104728. http://dx.doi.org/10.1016/j.jmva.2021.104728.

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20

Liu, Ruixuan. "A competing risks model with time‐varying heterogeneity and simultaneous failure." Quantitative Economics 11, no. 2 (2020): 535–77. http://dx.doi.org/10.3982/qe1159.

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This paper proposes a new bivariate competing risks model in which both durations are the first passage times of dependent Lévy subordinators with exponential thresholds and multiplicative covariates effects. Our specification extends the mixed proportional hazards model, as it allows for the time‐varying heterogeneity represented by the unobservable Lévy processes and it generates the simultaneous termination of both durations with positive probability. We obtain nonparametric identification of all model primitives given competing risks data. A flexible semiparametric estimation procedure is provided and illustrated through the analysis of a real dataset.

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21

Huzak, Miljenko, Mihael Perman, Hrvoje Šikić, and Zoran Vondraček. "Ruin probabilities for competing claim processes." Journal of Applied Probability 41, no. 03 (September 2004): 679–90. http://dx.doi.org/10.1017/s0021900200020477.

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Abstract:

Let C 1, C 2,…,C m be independent subordinators with finite expectations and denote their sum by C. Consider the classical risk process X(t) = x + ct - C(t). The ruin probability is given by the well-known Pollaczek–Khinchin formula. If ruin occurs, however, it will be caused by a jump of one of the subordinators C i . Formulae for the probability that ruin is caused by C i are derived. These formulae can be extended to perturbed risk processes of the type X(t) = x + ct - C(t) + Z(t), where Z is a Lévy process with mean 0 and no positive jumps.

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22

Toaldo, Bruno. "Lévy mixing related to distributed order calculus, subordinators and slow diffusions." Journal of Mathematical Analysis and Applications 430, no. 2 (October 2015): 1009–36. http://dx.doi.org/10.1016/j.jmaa.2015.05.024.

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23

Letemplier, Julien, and Thomas Simon. "On the law of homogeneous stable functionals." ESAIM: Probability and Statistics 23 (2019): 82–111. http://dx.doi.org/10.1051/ps/2018028.

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LetAbe theLq-functional of a stable Lévy process starting from one and killed when crossing zero. We observe thatAcan be represented as the independent quotient of two infinite products of renormalized Beta random variables. The proof relies on Markovian time change, the Lamperti transformation, and an explicit computation performed in [38] on perpetuities of hypergeometric Lévy processes. This representation allows us to retrieve several factorizations previously shown by various authors, and also to derive new ones. We emphasize the connections betweenAand more standard positive random variables. We also investigate the law of Riemannian integrals of stable subordinators. Finally, we derive several distributional properties ofArelated to infinite divisibility, self-decomposability, and the generalized Gamma convolution.

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24

Shu, Yin, Qianmei Feng, and David W. Coit. "Life distribution analysis based on Lévy subordinators for degradation with random jumps." Naval Research Logistics (NRL) 62, no. 6 (August 31, 2015): 483–92. http://dx.doi.org/10.1002/nav.21642.

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25

Debbi, Latifa. "Explicit solutions of some fractional partial differential equations via stable subordinators." Journal of Applied Mathematics and Stochastic Analysis 2006 (February 26, 2006): 1–18. http://dx.doi.org/10.1155/jamsa/2006/93502.

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The aim of this work is to represent the solutions of one-dimensional fractional partial differential equations (FPDEs) of order (α∈ℝ+\ℕ) in both quasi-probabilistic and probabilistic ways. The canonical processes used are generalizations of stable Lévy processes. The fundamental solutions of the fractional equations are given as functionals of stable subordinators. The functions used generalize the functions given by the Airy integral of Sirovich (1971). As a consequence of this representation, an explicit form is given to the density of the 3/2-stable law and to the density of escaping island vicinity in vortex medium. Other connected FPDEs are also considered.

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26

Urban, Roman. "A note on Lévy subordinators in cones of fuzzy sets in Banach spaces." Mathematica Slovaca 72, no. 3 (June 1, 2022): 787–96. http://dx.doi.org/10.1515/ms-2022-0053.

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Abstract Let K be a fixed proper convex cone contained in a separable Banach space E. Our main result in this note is a construction of a fuzzy process taking values in K-positive fuzzy sets, i.e. in a cone F c c o n v K ( E ) $ \mathcal F_{\mathrm{cconv}}^{K}(E) $ of fuzzy sets contained in K. We prove that this process can be considered as a fuzzy Lévy subordinator (with values in F c c o n v K ( E ) $ \mathcal F_{\mathrm{cconv}}^{K}(E) $ ) or simply fuzzy F c c o n v K ( E ) $ \mathcal F_{\mathrm{cconv}}^{K}(E) $ -subordinator.

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27

Yin, Chuancun, Kam Chuen Yuen, and Ying Shen. "Convexity of Ruin Probability and Optimal Dividend Strategies for a General Lévy Process." Scientific World Journal 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/354129.

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We consider the optimal dividends problem for a company whose cash reserves follow a general Lévy process with certain positive jumps and arbitrary negative jumps. The objective is to find a policy which maximizes the expected discounted dividends until the time of ruin. Under appropriate conditions, we use some recent results in the theory of potential analysis of subordinators to obtain the convexity properties of probability of ruin. We present conditions under which the optimal dividend strategy, among all admissible ones, takes the form of a barrier strategy.

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28

Butko, Yana A. "Chernoff approximation of subordinate semigroups." Stochastics and Dynamics 18, no. 03 (May 18, 2018): 1850021. http://dx.doi.org/10.1142/s0219493718500211.

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This note is devoted to the approximation of evolution semigroups generated by some Markov processes and hence to the approximation of transition probabilities of these processes. The considered semigroups correspond to processes obtained by subordination (i.e. by a time-change) of some original (parent) Markov processes with respect to some subordinators, i.e. Lévy processes with a.s. increasing paths (they play the role of the new time). If the semigroup, corresponding to a parent Markov process, is not known explicitly then neither the subordinate semigroup, nor even the generator of the subordinate semigroup are known explicitly too. In this note, some (Chernoff) approximations are constructed for subordinate semigroups (in the case when subordinators have either known transitional probabilities, or known and bounded Lévy measure) under the condition that the parent semigroups are not known but are already Chernoff-approximated. As it has been shown in the recent literature, this condition is fulfilled for several important classes of Markov processes. This fact allows, in particular, to use the constructed Chernoff approximations of subordinate semigroups, in order to approximate semigroups corresponding to subordination of Feller processes and (Feller type) diffusions in Euclidean spaces, star graphs and Riemannian manifolds. Such approximations can be used for direct calculations and simulation of stochastic processes. The method of Chernoff approximation is based on the Chernoff theorem and can be interpreted also as a construction of Markov chains approximating a given Markov process and as the numerical path integration method of solving the corresponding PDE/SDE.

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29

Barral, Julien, and Stéphane Seuret. "A class of multifractal semi-stable processes including Lévy subordinators and Mandelbrot multiplicative cascades." Comptes Rendus Mathematique 341, no. 9 (November 2005): 579–82. http://dx.doi.org/10.1016/j.crma.2005.09.020.

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30

Mai, Jan-Frederik. "The de Finetti structure behind some norm-symmetric multivariate densities with exponential decay." Dependence Modeling 8, no. 1 (October 1, 2020): 210–20. http://dx.doi.org/10.1515/demo-2020-0012.

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AbstractWe derive a sufficient condition on the symmetric norm ||·|| such that the probability distribution associated with the density function f (x) ∝exp(−λ ||x||) is conditionally independent and identically distributed in the sense of de Finetti’s seminal theorem. The criterion is mild enough to comprise the ℓp-norms as special cases, in which f is shown to correspond to a polynomially tilted stable mixture of products of transformed Gamma densities. In another special case of interest f equals the density of a time-homogeneous load sharing model, popular in reliability theory, whose motivation is a priori unrelated to the concept of conditional independence. The de Finetti structure reveals a surprising link between time-homogeneous load sharing models and the concept of Lévy subordinators.

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31

HIROSHIMA, FUMIO, TAKASHI ICHINOSE, and JÓZSEF LŐRINCZI. "PATH INTEGRAL REPRESENTATION FOR SCHRÖDINGER OPERATORS WITH BERNSTEIN FUNCTIONS OF THE LAPLACIAN." Reviews in Mathematical Physics 24, no. 06 (June 17, 2012): 1250013. http://dx.doi.org/10.1142/s0129055x12500134.

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Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard Feynman–Kac formula is taken here by subordinate Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which an Lp-Lq bound of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.

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32

Rusakov, O., and Yu Yakubovich. "Poisson processes directed by subordinators, stuttering poisson and pseudo-poisson processes, with applications to actuarial mathematics." Journal of Physics: Conference Series 2131, no. 2 (December 1, 2021): 022107. http://dx.doi.org/10.1088/1742-6596/2131/2/022107.

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Abstract Weconsider a PSI-process, that is a sequence of random variables (&), i = 0.1,…, which is subordinated by a continuous-time non-decreasing integer-valued process N(t): <K0 = ÇN(ty Our main example is when /V(t) itself is obtained as a subordination of the standard Poisson process 77(s) by a non-decreasing Lévy process S(t): N(t) = 77(S(t)).The values (&)one interprets as random claims, while their accumulated intensity S(t) is itself random. We show that in this case the process 7V(t) is a compound Poisson process of the stuttering type and its rate depends just on the value of theLaplace exponent of S(t) at 1. Under the assumption that the driven sequence (&) consists of i.i.d. random variables with finite variance we calculate a correlation function of the constructed PSI-process. Finally, we show that properly rescaled sums of such processes converge to the Ornstein-Uhlenbeck process in the Skorokhod space. We suppose that the results stated in the paper mightbe interesting for theorists and practitioners in insurance, in particular, for solution of reinsurance tasks.

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33

Kyprianou, A. E., and Z. Palmowski. "Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process." Journal of Applied Probability 44, no. 2 (June 2007): 428–43. http://dx.doi.org/10.1239/jap/1183667412.

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We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muler (2005), and Avram et al. (2007), which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically, we build on recent work in the actuarial literature concerning calculations of the nth moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than the existing literature, in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér–Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and, for the case of the nth moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.

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34

Kyprianou, A. E., and Z. Palmowski. "Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process." Journal of Applied Probability 44, no. 02 (June 2007): 428–43. http://dx.doi.org/10.1017/s0021900200117930.

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We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muler (2005), and Avram et al. (2007), which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically, we build on recent work in the actuarial literature concerning calculations of the nth moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than the existing literature, in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér–Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and, for the case of the nth moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.

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35

Kyprianou, A. E., and Z. Palmowski. "Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process." Journal of Applied Probability 44, no. 02 (June 2007): 428–43. http://dx.doi.org/10.1017/s0021900200003077.

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We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muler (2005), and Avram et al. (2007), which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically, we build on recent work in the actuarial literature concerning calculations of the nth moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than the existing literature, in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér–Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and, for the case of the nth moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.

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Guadagnini, A., M. Riva, and S. P. Neuman. "Extended power-law scaling of heavy-tailed random air-permeability fields in fractured and sedimentary rocks." Hydrology and Earth System Sciences 16, no. 9 (September 10, 2012): 3249–60. http://dx.doi.org/10.5194/hess-16-3249-2012.

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Abstract. We analyze the scaling behaviors of two field-scale log permeability data sets showing heavy-tailed frequency distributions in three and two spatial dimensions, respectively. One set consists of 1-m scale pneumatic packer test data from six vertical and inclined boreholes spanning a decameters scale block of unsaturated fractured tuffs near Superior, Arizona, the other of pneumatic minipermeameter data measured at a spacing of 15 cm along three horizontal transects on a 21 m long and 6 m high outcrop of the Upper Cretaceous Straight Cliffs Formation, including lower-shoreface bioturbated and cross-bedded sandstone near Escalante, Utah. Order q sample structure functions of each data set scale as a power ξ(q) of separation scale or lag, s, over limited ranges of s. A procedure known as extended self-similarity (ESS) extends this range to all lags and yields a nonlinear (concave) functional relationship between ξ(q) and q. Whereas the literature tends to associate extended and nonlinear power-law scaling with multifractals or fractional Laplace motions, we have shown elsewhere that (a) ESS of data having a normal frequency distribution is theoretically consistent with (Gaussian) truncated (additive, self-affine, monofractal) fractional Brownian motion (tfBm), the latter being unique in predicting a breakdown in power-law scaling at small and large lags, and (b) nonlinear power-law scaling of data having either normal or heavy-tailed frequency distributions is consistent with samples from sub-Gaussian random fields or processes subordinated to tfBm or truncated fractional Gaussian noise (tfGn), stemming from lack of ergodicity which causes sample moments to scale differently than do their ensemble counterparts. Here we (i) demonstrate that the above two data sets are consistent with sub-Gaussian random fields subordinated to tfBm or tfGn and (ii) provide maximum likelihood estimates of parameters characterizing the corresponding Lévy stable subordinators and tfBm or tfGn functions.

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Guadagnini, A., M. Riva, and S. P. Neuman. "Extended power-law scaling of heavy-tailed random fields or processes." Hydrology and Earth System Sciences Discussions 9, no. 6 (June 12, 2012): 7379–413. http://dx.doi.org/10.5194/hessd-9-7379-2012.

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Abstract. We analyze the scaling behaviors of two log permeability data sets showing heavy-tailed frequency distributions in three and two spatial dimensions, respectively. One set consists of 1-m scale pneumatic packer test data from six vertical and inclined boreholes spanning a decameters scale block of unsaturated fractured tuffs near Superior, Arizona, the other of pneumatic minipermeameter data measured at a spacing of 15 cm along two horizontal transects on a 21 m long outcrop of lower-shoreface bioturbated sandstone near Escalante, Utah. Order q sample structure functions of each data set scale as a power ξ (q) of separation scale or lag, s, over limited ranges of s. A procedure known as Extended Self-Similarity (ESS) extends this range to all lags and yields a nonlinear (concave) functional relationship between ξ (q) and q. Whereas the literature tends to associate extended and nonlinear power-law scaling with multifractals or fractional Laplace motions, we have shown elsewhere that (a) ESS of data having a normal frequency distribution is theoretically consistent with (Gaussian) truncated (additive, self-affine, monofractal) fractional Brownian motion (tfBm), the latter being unique in predicting a breakdown in power-law scaling at small and large lags, and (b) nonlinear power-law scaling of data having either normal or heavy-tailed frequency distributions is consistent with samples from sub-Gaussian random fields or processes subordinated to tfBm, stemming from lack of ergodicity which causes sample moments to scale differently than do their ensemble counterparts. Here we (i) demonstrate that the above two data sets are consistent with sub-Gaussian random fields subordinated to tfBm and (ii) provide maximum likelihood estimates of parameters characterizing the corresponding Lévy stable subordinators and tfBm functions.

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Buchmann, Boris, and Kevin W. Lu. "Necessity of weak subordination for some strongly subordinated Lévy processes." Journal of Applied Probability 58, no. 4 (November 22, 2021): 868–79. http://dx.doi.org/10.1017/jpr.2021.17.

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AbstractConsider the strong subordination of a multivariate Lévy process with a multivariate subordinator. If the subordinate is a stack of independent Lévy processes and the components of the subordinator are indistinguishable within each stack, then strong subordination produces a Lévy process; otherwise it may not. Weak subordination was introduced to extend strong subordination, always producing a Lévy process even when strong subordination does not. Here we prove that strong and weak subordination are equal in law under the aforementioned condition. In addition, we prove that if strong subordination is a Lévy process then it is necessarily equal in law to weak subordination in two cases: firstly when the subordinator is deterministic, and secondly when it is pure-jump with finite activity.

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Kumar, A., A. Wyłomańska, and J. Gajda. "Stable Lévy motion with inverse Gaussian subordinator." Physica A: Statistical Mechanics and its Applications 482 (September 2017): 486–500. http://dx.doi.org/10.1016/j.physa.2017.04.097.

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Dassios, Angelos, Jia Wei Lim, and Yan Qu. "Exact Simulation of a Truncated Lévy Subordinator." ACM Transactions on Modeling and Computer Simulation 30, no. 3 (July 9, 2020): 1–17. http://dx.doi.org/10.1145/3368088.

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Huang, Jianhua, Yuhong Li, and Jinqiao Duan. "Random Dynamics of the Stochastic Boussinesq Equations Driven by Lévy Noises." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/653160.

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This paper is devoted to the investigation of random dynamics of the stochastic Boussinesq equations driven by Lévy noise. Some fundamental properties of a subordinator Lévy process and the stochastic integral with respect to a Lévy process are discussed, and then the existence, uniqueness, regularity, and the random dynamical system generated by the stochastic Boussinesq equations are established. Finally, some discussions on the global weak solution of the stochastic Boussinesq equations driven by general Lévy noise are also presented.

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SEMERARO, PATRIZIA. "A MULTIVARIATE VARIANCE GAMMA MODEL FOR FINANCIAL APPLICATIONS." International Journal of Theoretical and Applied Finance 11, no. 01 (February 2008): 1–18. http://dx.doi.org/10.1142/s0219024908004701.

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In this paper we subordinate a multivariate Brownian motion with independent components by a multivariate gamma subordinator. The resulting process is a generalization of the bivariate variance gamma process proposed by Madan and Seneta [7], mentioned in Cont and Tankov [4] and calibrated in Luciano and Schoutens [5] as a price process. Our main contribution here is to introduce a multivariate subordinator with gamma margins. We investigate the process, determine its Lévy triplet and analyze its dependence structure. At the end we propose an exponential Lévy price model.

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Gajda, J., A. Kumar, and A. Wyłomańska. "Stable Lévy process delayed by tempered stable subordinator." Statistics & Probability Letters 145 (February 2019): 284–92. http://dx.doi.org/10.1016/j.spl.2018.09.008.

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Gajda, Janusz, and Agnieszka Wyłomańska. "Tempered stable Lévy motion driven by stable subordinator." Physica A: Statistical Mechanics and its Applications 392, no. 15 (August 2013): 3168–76. http://dx.doi.org/10.1016/j.physa.2013.03.018.

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Kella, Offer, and Onno Boxma. "Synchronized Lévy queues." Journal of Applied Probability 57, no. 4 (November 23, 2020): 1222–33. http://dx.doi.org/10.1017/jpr.2020.75.

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AbstractWe consider a multivariate Lévy process where the first coordinate is a Lévy process with no negative jumps which is not a subordinator and the others are non-decreasing. We determine the Laplace–Stieltjes transform of the steady-state buffer content vector of an associated system of parallel queues. The special structure of this transform allows us to rewrite it as a product of joint Laplace–Stieltjes transforms. We are thus able to interpret the buffer content vector as a sum of independent random vectors.

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Bekker, R., O. J. Boxma, and O. Kella. "Queues with Delays in Two-State Strategies and Lévy Input." Journal of Applied Probability 45, no. 2 (June 2008): 314–32. http://dx.doi.org/10.1239/jap/1214950350.

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We consider a reflected Lévy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. After D time units, the timer expires and the Lévy exponent of the Lévy process is changed. As soon as the process hits zero again, the Lévy exponent reverses to the original function. If the process has reached the origin before the timer expires then the Lévy exponent does not change. Using martingale techniques, we analyze the steady-state distribution of the resulting process, reflected at the origin. We pay special attention to the cases of deterministic and exponential timers, and to the following three special Lévy processes: (i) a compound Poisson process plus negative drift (corresponding to an M/G/1 queue), (ii) Brownian motion, and (iii) a Lévy process that is a subordinator until the timer expires.

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Bekker, R., O. J. Boxma, and O. Kella. "Queues with Delays in Two-State Strategies and Lévy Input." Journal of Applied Probability 45, no. 02 (June 2008): 314–32. http://dx.doi.org/10.1017/s0021900200004253.

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We consider a reflected Lévy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. After D time units, the timer expires and the Lévy exponent of the Lévy process is changed. As soon as the process hits zero again, the Lévy exponent reverses to the original function. If the process has reached the origin before the timer expires then the Lévy exponent does not change. Using martingale techniques, we analyze the steady-state distribution of the resulting process, reflected at the origin. We pay special attention to the cases of deterministic and exponential timers, and to the following three special Lévy processes: (i) a compound Poisson process plus negative drift (corresponding to an M/G/1 queue), (ii) Brownian motion, and (iii) a Lévy process that is a subordinator until the timer expires.

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Gajda, Janusz, Agnieszka Wylomanska, and Arun Kumar. "Fractional Lévy stable motion time-changed by gamma subordinator." Communications in Statistics - Theory and Methods 48, no. 24 (December 17, 2018): 5953–68. http://dx.doi.org/10.1080/03610926.2018.1523430.

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Kella, Offer. "The Class of Distributions Associated with the Generalized Pollaczek-Khinchine Formula." Journal of Applied Probability 49, no. 3 (September 2012): 883–87. http://dx.doi.org/10.1239/jap/1346955341.

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The goal is to identify the class of distributions to which the distribution of the maximum of a Lévy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the Lévy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing Lévy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczek-Khinchine formula for the stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.

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Kella, Offer. "The Class of Distributions Associated with the Generalized Pollaczek-Khinchine Formula." Journal of Applied Probability 49, no. 03 (September 2012): 883–87. http://dx.doi.org/10.1017/s002190020000961x.

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The goal is to identify the class of distributions to which the distribution of the maximum of a Lévy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the Lévy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing Lévy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczek-Khinchine formula for the stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.

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