共查询到20条相似文献,搜索用时 0 毫秒
1.
Frederik S. Herzberg 《Bulletin des Sciences Mathématiques》2010,134(4):423-445
This article links the hyperfinite theory of stochastic integration with respect to certain hyperfinite Lévy processes with the elementary theory of pathwise stochastic integration with respect to pure-jump Lévy processes with finite-variation jump part. Since the hyperfinite Itô integral is also defined pathwise, these results show that hyperfinite stochastic integration provides a pathwise definition of the stochastic integral with respect to Lévy jump-diffusions with finite-variation jump part.As an application, we provide a short and direct nonstandard proof of the generalized Itô formula for stochastic differentials of smooth functions of Lévy jump-diffusions whose jumps are bounded from below in norm. 相似文献
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In this paper, we establish sharp two-sided estimates for the Green functions of relativistic stable processes (i.e. Green functions for non-local operators m−(m2/α−Δ)α/2) in half-space-like C1,1 open sets. The estimates are uniform in m∈(0,M] for each fixed M∈(0,∞). When m↓0, our estimates reduce to the sharp Green function estimates for −(−Δ)α/2 in such kind of open sets that were obtained recently in Chen and Tokle [12]. As a tool for proving our Green function estimates, we show that a boundary Harnack principle for Xm, which is uniform for all m∈(0,∞), holds for a large class of non-smooth open sets. 相似文献
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A stochastic integral of Banach space valued deterministic functions with respect to Banach space valued Lévy processes is defined. There are no conditions on the Banach spaces or on the Lévy processes. The integral is defined analogously to the Pettis integral. The integrability of a function is characterized by means of a radonifying property of an integral operator associated with the integrand. The integral is used to prove a Lévy–Itô decomposition for Banach space valued Lévy processes and to study existence and uniqueness of solutions of stochastic Cauchy problems driven by Lévy processes. 相似文献
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We construct a white noise theory for Lévy processes. The starting point of this theory is a chaos expansion for square integrable random variables. We use this approach to Malliavin calculus to prove the following white noise generalization of the Clark-Haussmann-Ocone formula for Lévy processes
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Sergio Albeverio 《Bulletin des Sciences Mathématiques》2006,130(8):697-706
An internal lifting for an arbitrary measurable Lévy process is constructed. This lifting reflects our intuitive notion of a process which is the infinitesimal sum of its infinitesimal increments, those in turn being independent from and closely related to each other - for short, the process can be regarded as some kind of random walk (where the step size generically will vary). The proof uses the existence of càdlàg modifications of Lévy processes and certain features of hyperfinite adapted probability spaces, commonly known as the “model theory of stochastic processes”. 相似文献
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We study tail probabilities of the suprema of Lévy processes with subexponential or exponential marginal distributions over compact intervals. Several of the processes for which the asymptotics are studied here for the first time have recently become important to model financial time series. Hence our results should be important, for example, in the assessment of financial risk. 相似文献
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We estimate a median of f(Xt) where f is a Lipschitz function, X is a Lévy process and t is an arbitrary time. This leads to concentration inequalities for f(Xt). In turn, corresponding fluctuation estimates are obtained under assumptions typically satisfied if the process has a regular behavior in small time and a, possibly different, regular behavior in large time. 相似文献
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Björn Böttcher 《Stochastic Processes and their Applications》2011,121(6):1201-1216
We construct optimal Markov couplings of Lévy processes, whose Lévy (jump) measure has an absolutely continuous component. The construction is based on properties of subordinate Brownian motions and the coupling of Brownian motions by reflection. 相似文献
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We consider a linear heat equation on a half line with an additive noise chosen properly in such a manner that its invariant measures are a class of distributions of Lévy processes. Our assumption on the corresponding Lévy measure is, in general, mild except that we need its integrability to show that the distributions of Lévy processes are the only invariant measures of the stochastic heat equation. 相似文献
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Serge CohenMakoto Maejima 《Statistics & probability letters》2011,81(11):1664-1669
We study the relationships between the selfdecomposability of marginal distributions or finite dimensional distributions of moving average fractional Lévy processes and distributions of their driving Lévy processes. 相似文献
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We prove exponential convergence to the invariant measure, in the total variation norm, for solutions of SDEs driven by α-stable noises in finite and in infinite dimensions. Two approaches are used. The first one is based on Liapunov’s function approach by Harris, and the second on Doeblin’s coupling argument in [8]. Irreducibility and uniform strong Feller property play an essential role in both approaches. We concentrate on two classes of Markov processes: solutions of finite dimensional equations, introduced in [27], with Hölder continuous drift and a general, non-degenerate, symmetric α-stable noise, and infinite dimensional parabolic systems, introduced in [29], with Lipschitz drift and cylindrical α-stable noise. We show that if the nonlinearity is bounded, then the processes are exponential mixing. This improves, in particular, an earlier result established in [28], with a different method. 相似文献
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We consider natural exponential families of Lévy processes with randomized parameter. Such processes are Markov, and under suitable assumptions, pairs of such processes with shared randomization can be “stitched together” into a single harness. The stitching consists of deterministic reparametrization of the time for both processes, so that they run on adjacent time intervals, and of the choice of the appropriate law at the boundary. 相似文献
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Jan Kallsen 《Journal of multivariate analysis》2006,97(7):1551-1572
This paper suggests Lévy copulas in order to characterize the dependence among components of multidimensional Lévy processes. This concept parallels the notion of a copula on the level of Lévy measures. As for random vectors, a version of Sklar's theorem states that the law of a general multivariate Lévy process is obtained by combining arbitrary univariate Lévy processes with an arbitrary Lévy copula. We construct parametric families of Lévy copulas and prove a limit theorem, which indicates how to obtain the Lévy copula of a multivariate Lévy process X from the ordinary copula of the random vector Xt for small t. 相似文献
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We prove regularity estimates for functions which are harmonic with respect to certain jump processes. The aim of this article is to extend the method of Bass–Levin (2002) [3] and Bogdan–Sztonyk (2005) [6] to more general processes. Furthermore, we establish a new version of the Harnack inequality that implies regularity estimates for corresponding harmonic functions. 相似文献
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Pierre Patie 《Bulletin des Sciences Mathématiques》2009,133(4):355-382
We first introduce and derive some basic properties of a two-parameters (α,γ) family of one-sided Lévy processes, with 1<α<2 and γ>−α. Their Laplace exponents are given in terms of the Pochhammer symbol as follows
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Lévy processes in matrix Lie groups are studied. Subordination (random time change) is used to show that quasi-invariance of the Brownian motion in a Lie group induces absolute continuity of the laws of the corresponding pure jump processes. These results are applied to several examples which are discussed in detail. 相似文献
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In this paper, we consider Girsanov transforms of pure jump type for discontinuous Markov processes. We show that, under some quite natural conditions, the Green functions of the Girsanov transformed process are comparable to those of the original process. As an application of the general results, the drift transform of symmetric stable processes is studied in detail. In particular, we show that the relativistic α-stable process in a bounded C1,1-smooth open set D can be obtained from symmetric α-stable process in D through a combination of a pure jump Girsanov transform and a Feynman-Kac transform. From this, we deduce that the Green functions for these two processes in D are comparable. 相似文献