Continuity of symmetric stable processes (original) (raw)
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On stochastic integral representation of stable processes with sample paths in Banach spaces
Journal of Multivariate Analysis, 1986
Certain path properties of a symmetric a-stable process X(r) = Is h(t, s) d&f(s), to T, are studied in terms of the kernel h. The existence of an appropriate modification of the kernel h enables one to use results from stable measures on Banach spaces in studying X. Bounds for the moments of the norm of sample paths of X are obtained. This yields definite bounds for the moments of a double a-stable integral. Also, necessary and suflicient conditions for the absolute continuity of sample paths of X are given. Along with the above stochastic integral representation of stable processes, the representation of stable random vectors due to R. LePage, M. Woodroofe, and J. Zinn (1981, Ann. Probab. 9, 624632) is extensively used and the relationship between these two representations is discussed.
A functional LIL for symmetric stable processes
The Annals of Probability, 2000
A functional law of the iterated logarithm is obtained for symmetric stable processes with stationary independent increments. This extends the classical liminf results of Chung for Brownian motion, and of Taylor for such remaining processes. It also extends an earlier result of Wichura on Brownian motion. Proofs depend on small ball probability estimates and yield the small ball probabilities of the weighted sup-norm for these processes.
On the spectral representation of symmetric stable processes
Journal of Multivariate Analysis, 1982
The so-called spectral representation theorem for stable processes linearly imbeds each symmetric stable process of index p into Lp (0 < p < 2). We use the theory of Lp isometries for 0 < p < 2 to study the uniqueness of this representation for the non-Gaussian stable processes. We also determine the form of this representation for stationary processes and for substable processes. Complex stable processes are defined, and a complex version of the spectral representation theorem is proved. As a corollary to the complex theory we exhibit an imbedding of complex L' into real or complex Lp for 0 < p < q < 2.
Transactions of the American Mathematical Society
In this paper we study mutual absolute continuity and singularity of probability measures on the path space which are induced by an isotropic stable Lévy process and the purely discontinuous Girsanov transform of this process. We also look at the problem of finiteness of the relative entropy of these measures. An important tool in the paper is the question under which circumstances the a.s. finiteness of an additive functional at infinity implies the finiteness of its expected value.
Absolute continuity of symmetric Markov processes
The Annals of Probability, 2004
We study Girsanov's theorem in the context of symmetric Markov processes, extending earlier work of Fukushima-Takeda and Fitzsimmons on Girsanov transformations of "gradient type." We investigate the most general Girsanov transformation leading to another symmetric Markov process. This investigation requires an extension of the forward-backward martingale method of Lyons-Zheng, to cover the case of processes with jumps.
Small Deviations of Stable Processes via Metric Entropy
Journal of Theoretical Probability, 2000
Let X=(X(t)) t ¥ T be a symmetric a-stable, 0 < a < 2, process with paths in the dual E g of a certain Banach space E. Then there exists a (bounded, linear) operator u from E into some L a (S, s) generating X in a canonical way. The aim of this paper is to compare the degree of compactness of u with the small deviation (ball) behavior of f(e)=−log P(||X|| E* < e) as e Q 0. In particular, we prove that a lower bound for the metric entropy of u implies a lower bound for f(e) under an additional assumption on E. As applications we obtain upper small deviation estimates for weighted a-stable Levy motions, linear fractional a-stable motions and d-dimensional a-stable sheets. Our results rest upon an integral representation of L a -valued operators as well as on small deviation results for Gaussian processes due to Kuelbs and Li and to the authors.
The variation of a stable path is stable
Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1969
Let X(t) be a separable symmetric stable process of index c~. Let P be a finite partition of [0, 1], and .r a collection of partitions. The variation of a path X(t) is defined in three ways in terms of the sum ~ [X(t~)-X(q_l) f and the collection N. Under certain conditions on r and on the para-t~P meters ~ and/~, the distribution of the variation is shown to be a stable law. Under other conditions the distribution of the variational sum converges to a stable distribution.
The Stability of some stochastic processes
2010
We formulate and prove a new criterion for stability of e-processes. It says that any e-process which is averagely bounded and concentrating is asymptotically stable. In the second part, we show how this general result applies to some shell models (the Goy and the Sabra model). Indeed, we manage to prove that the processes corresponding to these models satisfy the
A Multiplier Related to Symmetric Stable Processes
Hacettepe Journal of Mathematics and Statistics, 2016
In two recent papers [5] and [6], we generalized some classical results of Harmonic Analysis using probabilistic approach by means of a ddimensional rotationally symmetric stable process. These results allow one to discuss some boundedness conditions with weaker hypotheses. In this paper, we study a multiplier theorem using these more general results. We consider a product process consisting of a d-dimensional symmetric stable process and a 1-dimensional Brownian motion, and use properties of jump processes to obtain bounds on jump terms and the L p (R d)-norm of a new operator.