共查询到20条相似文献,搜索用时 46 毫秒
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We consider subordinators in the domain of attraction at 0 of a stable subordinator (where ); thus, with the property that , the tail function of the canonical measure of , is regularly varying of index as . We also analyse the boundary case, , when is slowly varying at 0. When , we show that converges in distribution, as , to the random variable . This latter random variable, as a function of , converges in distribution as to the inverse of an exponential random variable. We prove these convergences, also generalised to functional versions (convergence in ), and to trimmed versions, whereby a fixed number of its largest jumps up to a specified time are subtracted from the process. The case produces convergence to an extremal process constructed from ordered jumps of a Cauchy subordinator. Our results generalise random walk and stable process results of Darling, Cressie, Kasahara, Kotani and Watanabe. 相似文献
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Yongsheng Song 《Stochastic Processes and their Applications》2019,129(6):2066-2085
As is known, if is a -Brownian motion, a process of form , , is a non-increasing -martingale. In this paper, we shall show that a non-increasing -martingale cannot be form of or , , which implies that the decomposition for generalized -Itô processes is unique: For arbitrary , and non-increasing -martingales , if then we have , and. As an application, we give a characterization to the -Sobolev spaces introduced in Peng and Song (2015). 相似文献
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Jean Bertoin 《Stochastic Processes and their Applications》2019,129(4):1443-1454
This work concerns the Ornstein–Uhlenbeck type process associated to a positive self-similar Markov process which drifts to , namely . We point out that is always a (topologically) recurrent ergodic Markov process. We identify its invariant measure in terms of the law of the exponential functional , where is the dual of the real-valued Lévy process related to by the Lamperti transformation. This invariant measure is infinite (i.e. is null-recurrent) if and only if . In that case, we determine the family of Lévy processes for which fulfills the conclusions of the Darling–Kac theorem. Our approach relies crucially on a remarkable connection due to Patie (Patie, 2008) with another generalized Ornstein–Uhlenbeck process that can be associated to the Lévy process , and properties of time-substitutions based on additive functionals. 相似文献
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Alexander Iksanov Konrad Kolesko Matthias Meiners 《Stochastic Processes and their Applications》2019,129(11):4480-4499
Let be Biggins’ martingale associated with a supercritical branching random walk, and let be its almost sure limit. Under a natural condition for the offspring point process in the branching random walk, we show that if the law of belongs to the domain of normal attraction of an -stable distribution for some , then, as , there is weak convergence of the tail process , properly normalized, to a random scale multiple of a stationary autoregressive process of order one with -stable marginals. 相似文献
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In ergodic stochastic problems the limit of the value function of the associated discounted cost functional with infinite time horizon is studied, when the discounted factor tends to zero. These problems have been well studied in the literature and the used assumptions guarantee that the value function converges uniformly to a constant as . The objective of this work consists in studying these problems under the assumption, namely, the nonexpansivity assumption, under which the limit function is not necessarily constant. Our discussion goes beyond the case of the stochastic control problem with infinite time horizon and discusses also given by a Hamilton–Jacobi–Bellman equation of second order which is not necessarily associated with a stochastic control problem. On the other hand, the stochastic control case generalizes considerably earlier works by considering cost functionals defined through a backward stochastic differential equation with infinite time horizon and we give an explicit representation formula for the limit of , as . 相似文献
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For a random walk on we study the asymptotic behaviour of the associated centre of mass process . For lattice distributions we give conditions for a local limit theorem to hold. We prove that if the increments of the walk have zero mean and finite second moment, is recurrent if and transient if . In the transient case we show that has a diffusive rate of escape. These results extend work of Grill, who considered simple symmetric random walk. We also give a class of random walks with symmetric heavy-tailed increments for which is transient in . 相似文献
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In this paper we study the domain of the generator of stable processes, stable-like processes and more general pseudo- and integro-differential operators which naturally arise both in analysis and as infinitesimal generators of Lévy- and Lévy-type (Feller) processes. In particular we obtain conditions on the symbol of the operator ensuring that certain (variable order) Hölder and Hölder–Zygmund spaces are in the domain. We use tools from probability theory to investigate the small-time asymptotics of the generalized moments of a Lévy or Lévy-type process , for functions f which are not necessarily bounded or differentiable. The pointwise limit exists for fixed if f satisfies a Hölder condition at x. Moreover, we give sufficient conditions which ensure that the limit exists uniformly in the space of continuous functions vanishing at infinity. As an application we prove that the domain of the generator of contains certain Hölder spaces of variable order. Our results apply, in particular, to stable-like processes, relativistic stable-like processes, solutions of Lévy-driven SDEs and Lévy processes. 相似文献
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Le Chen Yaozhong Hu David Nualart 《Stochastic Processes and their Applications》2019,129(12):5073-5112
This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: where is the space–time white noise, , , and . Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang’s condition: . In some cases, the initial data can be measures. When , we prove the sample path regularity of the solution. 相似文献
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Irmina Czarna José-Luis Pérez Tomasz Rolski Kazutoshi Yamazaki 《Stochastic Processes and their Applications》2019,129(12):5406-5449
A level-dependent Lévy process solves the stochastic differential equation , where is a spectrally negative Lévy process. A special case is a multi-refracted Lévy process with . A general rate function that is non-decreasing and locally Lipschitz continuous is also considered. We discuss solutions of the above stochastic differential equation and investigate the so-called scale functions, which are counterparts of the scale functions from the theory of Lévy processes. We show how fluctuation identities for can be expressed via these scale functions. We demonstrate that the derivatives of the scale functions are solutions of Volterra integral equations. 相似文献
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This paper derives a diffusion approximation for a sequence of discrete-time one-sided limit order book models with non-linear state dependent order arrival and cancellation dynamics. The discrete time sequences are specified in terms of an -valued best bid price process and an -valued volume process. It is shown that under suitable assumptions the sequence of interpolated discrete time models is relatively compact in a localized sense and that any limit point satisfies a certain infinite dimensional SDE. Under additional assumptions on the dependence structure we construct two classes of models, which fit in the general framework, such that the limiting SDE admits a unique solution and thus the discrete dynamics converge to a diffusion limit in a localized sense. 相似文献
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I. Berkes 《Stochastic Processes and their Applications》2019,129(11):4500-4509
The St. Petersburg paradox (Bernoulli, 1738) concerns the fair entry fee in a game where the winnings are distributed as . The tails of are not regularly varying and the sequence of accumulated gains has, suitably centered and normalized, a class of semistable laws as subsequential limit distributions (Martin-Löf, 1985; Csörg? and Dodunekova, 1991). This has led to a clarification of the paradox and an interesting and unusual asymptotic theory in past decades. In this paper we prove that can be approximated by a semistable Lévy process with a.s. error and, surprisingly, the error term is asymptotically normal, exhibiting an unexpected central limit theorem in St. Petersburg theory. 相似文献
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Leonardo P.C. da Cruz Douglas D. Novaes Joan Torregrosa 《Journal of Differential Equations》2019,266(7):4170-4203
We study the number of limit cycles bifurcating from a piecewise quadratic system. All the differential systems considered are piecewise in two zones separated by a straight line. We prove the existence of 16 crossing limit cycles in this class of systems. If we denote by the extension of the Hilbert number to degree n piecewise polynomial differential systems, then . As fas as we are concerned, this is the best lower bound for the quadratic class. Moreover, in the studied cases, all limit cycles appear nested bifurcating from a period annulus of a isochronous quadratic center. 相似文献
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Ion Grama Ronan Lauvergnat Émile Le Page 《Stochastic Processes and their Applications》2019,129(7):2485-2527
Let be a branching process in a random environment defined by a Markov chain with values in a finite state space . Let be the probability law generated by the trajectories of starting at We study the asymptotic behaviour of the joint survival probability , as in the critical and strongly, intermediate and weakly subcritical cases. 相似文献
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