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  • 1
    Electronic Resource
    Electronic Resource
    Integration par parties dans l'espace de Wiener et approximation du temps local (1991)
    Springer
    Probability theory and related fields 90 (1991), S. 83-109 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Let be a real-valued stochastic process having a continuous local timeL(u,t),u∈ —, 0≦t≦T andX ε(t) = (Ψ ε *X)(t),t ⪴ 0, the regularization ofX by means of the convolution with the approximation of unityΨ ε. The main theorem in this paper (Theorem 3.5) is a generalization of various results about the approximation (for fixedu) of the local timeL(u, •) by means of a convenient normalization of the numberN X ε (u;•) of crossings of the processX ε with the levelu. Especially, this Theorem extends to a class of not necessarily Markovian continuous martingales, a result of this type for one-dimensional diffusions due to Azais [A2]). The methods of proof combine some estimations of the moments of the number of crossings with a level of a regular stochastic processes with stochastic analysis techniques based upon integration by parts in the Wiener space.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01321135
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  • 2
    Electronic Resource
    Electronic Resource
    Approximation du temps local des surfaces gaussiennes (1993)
    Springer
    Probability theory and related fields 96 (1993), S. 1-32 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary LetX be a centered stationary Gaussian stochastic process with ad-dimensional parameter (d≧2),F its spectral measure, $$\int\limits_{R^d } {||x||^2 F(dx) = + \infty } $$ (‖x‖ denotes the Euclidean norm ofx). We consider regularizations of the trajectories ofX by means of convolutions of the formX ε(t)=(Ψ ε*X)(t) where Ψε stands for an approximation of unity (as ε tends to zero) satisfying certain regularity conditions. The aim of this paper is to recover the local time ofX at a given levelu, as a limit of appropriate normalizations of the geometric measure of theu-level set of the regular approximating processesX ε. A part of the difficulties comes from the fact that the geometric behavior of the covariance of the Gaussian processX ε can be a complex one as ε approaches O. The results are onL 2-convergence and include bounds for the speed of convergence.L presults may be obtained in similar ways, but almost sure convergence or simultaneous convergence for the various values ofu do not seem to follow from our methods. In Sect. 3 we have included examples showing a diversity of geometric behaviors, especially in what concerns the dependence on the thickness of the set in which the covariance of the original processX is irregular. Some technical results of analytic nature are included as appendices in Sect. 4.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01195880
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Sur un théorème de Léonard Shepp (1973)
    Springer
    Probability theory and related fields 27 (1973), S. 179-184 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00535847
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Formule de Rice en dimension d (1982)
    Springer
    Probability theory and related fields 60 (1982), S. 393-401 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Let {x(t): t∈R d} a stochastic process with parameter in R d, and u a fixed real number. Denote by C u, Au, Bu respectively the random sets {t: x(t)= u}, {t: x(t)〈u}, {t: x(t)〉u}. The paper contains two main results for processes with continuously differentiable paths plus some additional requirements: First, a formula for the expectation of Q T(Au) and Q T(Bu), where for a given bounded open set T in R d, QT(B) denotes the “perimeter of B relative to T” and second, sufficient conditions on the process, so that it does not have local extrema on the barrier u. The second result can also be used to interpret the first in terms of C u.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00535722
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  • 5
    Electronic Resource
    Electronic Resource
    On the increments of the Wiener process (1984)
    Springer
    Probability theory and related fields 65 (1984), S. 329-339 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Let {W(t), t≧0} be a standard Wiener process and 0〈a t ≦t a nondecreasing function of t. The asymptotic behaviour of several increment processes, obtained from W and a t , is investigated in terms of their upper classes. In some cases we characterize these classes by means of an integral test. Two such processes are (W(t+a t) − W(t))a t −1/2 and $$\mathop {\sup }\limits_{o \leqq t \leqq T} \mathop {\sup }\limits_{o 〈 s \leqq a_t \wedge (T - t)} (W(t + s) - W(t))a_t^{ - 1/2}$$
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00533740
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  • 6
    Book
    Book
    Level sets and extrema of random processes and fields / (2009)
    Hoboken, NJ :Wiley,
    Details
    Title: Level sets and extrema of random processes and fields /
    Author: Azaïs, Jean-Marc
    Contributer: Wschebor, Mario
    Publisher: Hoboken, NJ :Wiley,
    Year of publication: 2009
    Pages: XI, 393 S.
    ISBN: 978-0-470-40933-6
    Type of Medium: Book
    Language: English
    Parallel Title: Level sets and extrema of random processes and fields
    URL: 01  (lizenzfrei)
    URL: Table of contents only
    URL: 04
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    Zuse Institute Berlin get availability status
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