Search

In this paper, we study the speed of extinction of continuous state branching processes in subcritical Lévy environments. More precisely, when the associated Lévy process to the environment drifts to -8 and, under a suitable exponential change of measure (Esscher transform), the environment either drifts to -8 or oscillates. We extend recent results of Palau et al. (2016) and Li and Xu (2018), where the branching term is associated to a spectrally positive stable Lévy process and complement the recent article of Bansaye et al. (2021) where the critical case was studied. Our methodology combines a path analysis of the branching process together with its Lévy environment, fluctuation theory for Lévy processes and the asymptotic behaviour of exponential functionals of Lévy processes. As an application of the aforementioned results, we characterise the process conditioned to survival also known as the Q-process. [ABSTRACT FROM AUTHOR]

For a spectrally negative Lévy process, scale functions appear in the solution of twosided exit problems, and in particular in relation with the Laplace transform of the first time it exits a closed interval. In this paper, we consider the Laplace transform of more general functionals, which can depend simultaneously on the values of the process and its supremum up to the exit time. These quantities will be expressed in terms of generalized scale functions, which can be defined using excursion theory. In the case the functional does not depend on the supremum, these scale functions coincide with the ones found on the literature, see e.g. Li and Palmowski (2018) and therefore the results in this work are an extension of them. [ABSTRACT FROM AUTHOR]

Suppose X is a Markov process on the real line (or some interval). Do the distributions of its first passage times downwards (fptd) determine its law? In this paper we treat some special cases of this question. We prove that if the fptd process has the law of a subordinator, then necessarily X is a Lévy process with no negative jumps; specifying the law of the subordinator determines the law of X uniquely. We further show that, likewise, the classes of continuous-state branching processes and of self-similar processes without negative jumps are also respectively characterised by a certain structure of their fptd distributions; and each member of these classes separately is determined uniquely by the precise family of its fptd laws. The road to these results is paved by (i) the identification of Markov processes without negative jumps in terms of the nature of their fptd laws, and (ii) some general results concerning the identification of the fptd distributions for such processes. [ABSTRACT FROM AUTHOR]

Around the 1960s a celebrated collection of papers emerged offering a number of explicit identities for the class of isotropic Lévy stable processes in one and higher dimensions; these include, for example, the lauded works of Ray (1958); Widom (1961); Rogozin (1972) (in one dimension) and Blumenthal et al. (1961); Getoor (1966); Port (1969) (in higher dimension), see also more recently Byczkowski et al. (2009); Luks (2013). Amongst other things, these results nicely exemplify the use of standard Riesz potential theory on the unit open ball 픹d:= {x ε ℝd : ∣x∣ < 1}, ℝd\픹dand the sphere 핊d-1 := {x ε ℝd : ∣x∣ = 1} with the, then, modern theory of potential analysis for Markov processes. Following initial observations of Lamperti (1972), with the occasional sporadic contributions such as Kiu (1980); Vuolle-Apiala and Graversen (1986); Graversen and Vuolle-Apiala (1986), an alternative understanding of Lévy stable processes through the theory of self-similar Markov processes has prevailed in the last decade or more. This point of view offers deeper probabilistic insights into some of the aforementioned potential analytical relations; see for example Bertoin and Yor (2002); Bertoin and Caballero (2002); Caballero and Chaumont (2006a,b); Chaumont et al. (2009); Patie (2009, 2012); Bogdan and Zak (2006); Patie (2012); Kyprianou et al. (2014); Kuznetsov et al. (2014); Kyprianou and Watson (2014); Kuznetsov and Pardo (2013); Kyprianou (2016); Kyprianou et al. (2018, 2016); Alili et al. (2017). In this review article, we will rediscover many of the aforementioned classical identities in relation to the unit ball by combining elements of these two theories, which have otherwise been largely separated by decades in the literature. We present a dialogue that appeals as much as possible to path decompositions. Most notable in this respect is the Lamperti-Kiu decomposition of self-similar Markov processes given in Kiu (1980); Chaumont et al. (2013); Alili et al. (2017) and the Riesz-Bogdan-Zak transformation given in Bogdan and Zak (2006). Some of the results and proofs we give are known (marked ♡), some are mixed with new results or methods, respectively, (marked ) and some are completely new (marked ♣). We assume that the reader has a degree of familiarity with the bare basics of Lévy processes but, nonetheless, we often include reminders of standard material that can be found in e.g. Bertoin (1996), Sato (2013) or Kyprianou (2014). [ABSTRACT FROM AUTHOR]

This paper studies stability of the paths of stochastic differential equations (SDE) driven by time-changed Lévy noise. The conditions for the solution of time-changed SDE to be path stable and exponentially path stable are given. We consider the time-changed Lévy noises with small and large jumps. Moreover, we reveal the important role of the time drift in determining the path stability of the solution. Related examples are provided. This extends the moment stability of related SDEs studied in Nane and Ni (2017) and Wu (2016). [ABSTRACT FROM AUTHOR]

This paper is about lower and upper bounds for the Hausdorff dimension of the level and collision sets for a class of Feller processes. Our approach is motivated by analogous results for Levy processes by Hawkes (1974) (for level sets) and Taylor (1966) and Jain and Pruitt (1969) (for collision sets). Since Feller processes lack independent or stationary increments, the methods developed for Levy processes cannot be used in a straightforward manner. Under the assumption that the Feller process possesses a transition probability density, which admits lower and upper bounds of a certain type, we derive sufficient conditions for regularity and non-polarity of points; together with suitable time changes this allows us to get upper and lower bounds for the Hausdorff dimension. [ABSTRACT FROM AUTHOR]

We consider a branching stable process with positive jumps, i.e. a continuous-time branching process in which the particles evolve independently as stable Lévy processes with positive jumps. Assuming the branching mechanism is critical or subcritical, we compute the asymptotics of the maximum location ever reached by a particle of the process. [ABSTRACT FROM AUTHOR]

We provide general conditions ensuring that the value functions of some nonlinear stopping problems with finite horizon converge to the value functions of the corresponding problems with infinite horizon. Our result can be formulated as result on stability, with respect to time horizon, of nonlinear f-expectations. We also study the rate of convergence. Many examples are given to illustrate our results. They include the analysis of time asymptotics of the fair prices of American options in a multidimensional exponential Lévy model. [ABSTRACT FROM AUTHOR]

We study the exponential functional ∫0∞ e--ξs-dηs of two one-dimensional independent Levy processes ξ and η, where η is a subordinator. In particular, we derive an integro-differential equation for the density of the exponential functional whenever it exists. Further, we consider the mapping Φξ for a fixed Levy process ξ, which maps the law of η1 to the law of the corresponding exponential functional ∫0∞ e--ξs-dηs and study the behaviour of the range of Φξ for varying characteristics of ξ. Moreover, we derive conditions for selfdecomposable distributions and generalized Gamma convolutions to be in the range. On the way we also obtain new characterizations of these classes of distributions. [ABSTRACT FROM AUTHOR]

Long memory processes driven by Lévy noise with finite second-order moments have been well studied in the literature. They form a very rich class of processes presenting an autocovariance function that decays like a power function. Here, we study a class of Lévy processes whose second-order moments are infinite, the so-called α-stable processes. Based on Samorodnitsky and Taqqu (1994), we construct an isometry that allows us to define stochastic integrals concerning the linear fractional stable motion using Riemann-Liouville fractional integrals. With this construction, an integration by parts formula follows naturally. We then present a family of stationary SαS processes with the property of long-range dependence, using a generalized measure to investigate its dependence structure. At the end, the law of large number's result for a time's sample of the process is shown as an application of the isometry and integration by parts formula. [ABSTRACT FROM AUTHOR]

In this work we introduce a theory of stochastic integration for operator-valued integrands with respect to some classes of cylindrical martingale-valued measures in Hilbert spaces. The integral is constructed via the radonification of cylindrical martingales by a Hilbert-Schmidt operator theorem and unifies several other theories of stochastic integration in Hilbert spaces. In particular, our theory covers the theory of stochastic integration with respect to a Hilbert space valued Lévy process with second moments, with respect to a cylindrical Lévy processes with (weak) second moments and with respect to a Lévy-valued random martingale measures with finite second moment. As an application of our theory of integration we prove existence and uniqueness of solutions for stochastic stochastic partial differential equations driven by multiplicative cylindrical martingale-valued measure noise with rather general coefficients. Existence and uniqueness of solutions in the presence of multiplicative Lévy noise (with no moments assumptions) is also proved. [ABSTRACT FROM AUTHOR]

We prove sharp two-sided estimates on the tail probability of the first hitting time of bounded interval as well as its asymptotic behaviour for general non-symmetric processes which satisfy an integral condition ... To this end, we first prove and then apply the global scale invariant Harnack inequality. Results are obtained under certain conditions on the characteristic exponent. We provide a wide class of Lévy processes which satisfy these assumptions. [ABSTRACT FROM AUTHOR]

We investigate the simulation methods for a large family of stable random fields that appeared in the recent literature, known as the Karlin stable setindexed processes. We exploit a new representation and implement the procedure introduced by Asmussen and Rosinski (2001) by first decomposing the random fields into large-jump and small-jump parts, and simulating each part separately. As special cases, simulations for several manifold-indexed processes are considered, and adjustments are introduced accordingly in order to improve the computational efficiency. [ABSTRACT FROM AUTHOR]

The concept of a Lévy subordinator is generalized to a family of nondecreasing stochastic processes, which are parameterized in terms of two Bernstein functions. Whereas the independent increments property is only maintained in the Lévy subordinator special case, the considered family is always strongly infinitely divisible with respect to time, meaning that a path can be represented in distribution as a finite sum with arbitrarily many summands of independent and identically distributed paths of another process. Besides distributional properties of the process, we present two applications to the design of accurate and efficient simulation algorithms. First, each member of the considered family corresponds uniquely to an exchangeable max-stable sequence of random variables, and we demonstrate how the associated extreme-value copula can be simulated exactly and efficiently from its Pickands dependence measure. Second, we show how one obtains different series and integral representations for infinitely divisible probability laws by varying the parameterizing pair of Bernstein functions, without changing the law of one-dimensional margins of the process. As a particular example, we present an exact simulation algorithm for compound Poisson distributions from the Bondesson class, for which the generalized inverse of the distribution function of the associated Stieltjes measure can be evaluated accurately. [ABSTRACT FROM AUTHOR]

Lévy-type perpetuities being the a.s. limits of particular generalized Ornstein-Uhlenbeck processes are a natural continuous-time generalization of discrete-time perpetuities. These are random variables of the form ..., where (X, Z) is a two-dimensional Lévy process, and Z is a drift-free Lévy process of bounded variation. We prove an ultimate criterion for the finiteness of power moments of S. This result and the previously known assertion due to Erickson and Maller (2005) concerning the a.s. finiteness of S are then used to derive ultimate necessary and sufficient conditions for the Lp-convergence for p > 1 and p = 1, respectively, of Biggins' martingales associated to branching Lévy processes. In particular, we provide final versions of results obtained recently in Bertoin and Mallein (2018). [ABSTRACT FROM AUTHOR]

We modify the global Skorokhod topology, on the space of cadlag paths, by localising with respect to space variable, in order to include the eventual explo- sions. The tightness of families of probability measures on the paths space endowed with this local Skorokhod topology is studied and a characterization of Aldous type is obtained. The local and global Skorokhod topologies are compared by using a time change transformation. [ABSTRACT FROM AUTHOR]

Markov Additive Processes are bi-variate Markov processes of the form (Ϊ, J) = ((Ϊt, Jt), t ≥ 0 ) which should be thought of as a multi-type Lévy process: the second component J is a Markov chain on a finite space {1,…,K}, and the first component Ϊ behaves locally as a Lévy process with dynamics depending on J. In the subordinator-like case where Ϊ is nondecreasing, we establish several results concerning the moments of Ϊ and of its exponential functional IΪ = ∫ ∞ 0 e-Ϊtdt, extending the work of Carmona et al. (1997), and Bertoin and Yor (2001). We then apply these results to the study of multi-type self-similar fragmentation processes: these are self-similar transformations of Bertoin's homogeneous multitype fragmentation processes, introduced in Bertoin (2008). Notably, we encode the genealogy of the process in an R-tree as in Haas and Miermont (2004), and under some Malthusian hypotheses, compute its Hausdorff dimension in a generalisation of our previous results in Stephenson (2013). [ABSTRACT FROM AUTHOR]

In this work, we study asymptotics of multitype Galton-Watson forests with finitely many types. We consider critical and irreducible offspring distributions such that they belong to the domain of attraction of a stable law, where the stability indices may differ. We show that after a proper rescaling, their corresponding height process converges to the continuous-time height process associated with a strictly stable spectrally positive Lévy process. This gives an analog of a result obtained by Miermont (2008) in the case of multitype Galton-Watson forests with finite covariance matrices of the offspring distribution. Our approach relies on a remarkable decomposition for multitype trees into monotype trees introduced in Miermont (2008). [ABSTRACT FROM AUTHOR]

In this article we consider continuous time moving averages observed on a lattice, driven by an infinite variance Lévy process with regularly varying tails with index α ∈ (0, 2). We show that the asymptotic distribution of the sample mean and sample autocovariance function is a stable law. The parameters of the stable limit distribution are explicitly given in terms of the kernel and characteristics of the Lévy process. [ABSTRACT FROM AUTHOR]

We introduce a continuous time autoregressive moving average process with random Lévy coefficients, termed RC-CARMA(p,q) process, of order p and q < p via a subclass of multivariate generalized Ornstein-Uhlenbeck processes. Sufficient conditions for the existence of a strictly stationary solution and the existence of moments are obtained. We further investigate second order stationarity properties, calculate the autocovariance function and spectral density, and give sufficient conditions for their existence. [ABSTRACT FROM AUTHOR]

Consider a reflected jump-diffusion on the positive half-line. Assume it is stochastically ordered. We apply the theory of Lyapunov functions and find explicit estimates for the rate of exponential convergence to the stationary distribution, as time goes to infinity. This continues the work of Lund et al. (1996). We apply these results to systems of two competing L'evy particles with rank-dependent dynamics. [ABSTRACT FROM AUTHOR]

Suppose we have a high-frequency sample from the Lévy process of the form Xϑt = βt + γZt + Ut, where Z is a possibly asymmetric locally α-stable Levy process, and U is a nuisance Lévy process less active than Z. We prove the LAN property about 3the explicit parameter ϑ = (β,γ) under very mild conditions without specific form of the Lévy measure of Z, thereby generalizing the LAN result of Ai-Sahalia and Jacob (2007). In particular, it is clarified that a non-diagonal norming may be necessary in the truly asymmetric case. Due to the special nature of the local α-stable property, the asymptotic Fisher information matrix takes a clean-cut form. [ABSTRACT FROM AUTHOR]

We study jump-diffusion processes with parameters switching at random times. Being motivated by possible applications, we characterise equivalent martingale measures for these processes by means of the relative entropy. The minimal entropy approach is also developed. It is shown that in contrast to the case of Levy processes, for this model an Esscher transformation does not produce the minimal relative entropy. [ABSTRACT FROM AUTHOR]

We consider a random forest F*, defined as a sequence of i.i.d. birth-death (BD) trees, each started at time 0 from a single ancestor, stopped at the first tree having survived up to a fixed time T. We denote by (ξ*t, 0 ≤ t ≤ T) the population size process associated to this forest, and we prove that if the BD trees are supercritical, then the time-reversed process (ξ*t, 0 ≤ t ≤ T), has the same distribution as (ξ*t, 0 ≤ t ≤ T), the corresponding population size process of an equally defined forest F*, but where the underlying BD trees are subcritical, obtained by swapping birth and death rates. We generalize this result to splitting trees (i.e. life durations of individuals are not necessarily exponential), provided that the i.i.d. lifetimes of the ancestors have a specific explicit distribution, different from that of their descendants. The results are based on an identity between the contour of these random forests truncated up to T and the duality property of Levy processes, which allow to derive other useful properties of the forests with potential applications in epidemiology. [ABSTRACT FROM AUTHOR]

Using a method developed by Durrett and Resnick (1978) we establish general criteria for the convergence of properly rescaled clock processes of random dynamics in random environments on infinite graphs. This complements the results of Gayrard (2012), Bovier and Gayrard (2013), and Bovier et al. (2013): put together these results provide a unified framework for proving convergence of clock processes. As a first application we prove that Bouchaud's asymmetric trap model on ℤd exhibits a normal aging behavior for all d ≥ 2. Namely, we show that certain two-time correlation functions, among which the classical probability to find the process at the same site at two time points, converge, as the age of the process diverges, to the distribution function of the arcsine law. As a byproduct we prove that the fractional kinetics process ages. [ABSTRACT FROM AUTHOR]