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1993 | Buch

Barcelona Seminar on Stochastic Analysis

St.Feliu de Guíxols, 1991

herausgegeben von: David Nualart, Marta Sanz Solé

Verlag: Birkhäuser Basel

Buchreihe : Progress in Probability

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SUCHEN

Über dieses Buch

During the of Fall 1991, The Centre de Recerca Matematica, a research institute sponsored by the Institut d'Estudis Catalans, devoted a quarter to the study of stochastic analysis. Prominent workers in this field visited the Center from all over the world for periods ranging from a few days to several weeks. To take advantage of the presence in Barcelona of so many special­ ists in stochastic analysis, we organized a workshop on the subject in Sant Feliu de Guixols (Girona) that provided an opportunity for them to ex­ change information and ideas about their current work. Topics discussed included: Analysis on the Wiener space, Anticipating Stochastic Calculus and its Applications, Correlation Inequalities, Stochastic Flows, Reflected Semimartingales, and others. This volume contains a refereed selection of contributions from some of the participants in this workshop. We are deeply indebted to the authors of the articles for these exposi­ tions of their valuable research contributions. We also would like to thank all the referees for their helpful advice in making the volume a reflection of the dynamic interchange that characterized the workshop. The success of the Seminar was due essentially to the enthusiasm and stimulating discus­ sions of all the participants in an informal and pleasant atmosphere. To all of them our warm gratitude.

Inhaltsverzeichnis

Frontmatter
Modulus of Continuity for Stochastic Flows
Abstract
In this article we determine the modulus of continuity for a class of stochastic flows. We also give an application to anticipating stochastic differential equations of the Stratonovich type.
Paolo Baldi, Marta Sanz-Solé
Nonlinear Skorohod Stochastic Differential Equations
Abstract
Let σ ∈ C 4 b(R1). We provide assumptions on the random variable G and the process b = (b t (x)) possibly anticipating the driving Wiener process (W t ) under which the anticipative stochastic differential equation with Skorohod integral 21-1
Rainer Buckdahn
Ornstein-Uhlenbeck Processes as Bernstein Processes
Abstract
Bernstein diffusions belong to a new class of time symmetric (but not time homogeneous) stochastic processes associated with the quantum dynamics of nonrelativistic particles in potentials. It is shown in which sense Feynman’s formal path integral method can be interpreted in terms of those processes, specially for the subset of Gaussian Bernstein diffusions. The familiar Ornstein-Uhlenbeck process becomes, in this framework, a particular Gaussian process in a large class of Bernstein diffusions, all associated with the same dynamics. The method is also illustrated for some Hilbert space valued Gaussian Diffusions
A. B. Cruzeiro, J. C. Zambrini
A Convergence Criterion for Measure-Valued Processes, and Application to Continuous Superprocesses
Abstract
A criterion for weak convergence of measure-valued processes is proved, and it is exemplified by showing convergence of branching particle systems to continuous superprocesses.
Luis G. Gorostiza, J. Alfredo López-Mimbela
A simple proof for a large deviation theorem
Abstract
The Davies method [2] allows to get estimates outside the diagonal of a heat kernel. In [3] one gives a probabilistic analogue of that method, which is local instead of being global. We apply this method when we want to estimate probabilities instead of densities. We get a simple proof of the continuity lemma of Azencott which plays an important role in the theory of large deviations. We can also integrate some exponential functionals associated to elliptic diffusions; this allows to find again the upper-bound of the large deviations without using the continuity lemma of Azencott; we are close to some ideas of [4], by using stopping times instead of cutting the time interval into a lot of pieces.
Rémi Léandre
Universal Wiener Space
Abstract
In the 1950’s Irving Segal developed for the needs of Quantum Field Theory, an abstract theory of integration on an abstract Hilbert space. In the 1960’s Leonard Gross has built the theory of gaussian borelian measures on an arbitrary Banach space. Looking for the greatest generality combined with the easiest approach, we shall follow an approach close to Segal’s one but in a version more concrete that his original papers. We have been greatly encouraged to take this point of view by several unpublished manuscripts of Kiyosi Itô. We thank him heartly for his kind communications from which we have greatly beneficied.
Paul Malliavin
On the Support of a Skorohod Anticipating Stochastic Differential Equation
Abstract
This paper deals with an anticipating one-dimensional quasilinear stochastic differential equation of the Skorohod type. Some regularity properties of the solution are proved and we characterize the support of the law of the solution
Annie Millet, Marta Sanz-Solé
Positive and Strongly Positive Wiener Functionals
Abstract
Let F(W) = ∑∞ n=0 I (f) be the representation of the Wiener functional F. The positivity index of F is defined to be supremum of all λ > 0 such that ∑∞n=0λn I(f) is a positive functional. It is shown that, in a suitable setup, if the index of positivity of two functionals is non zero, so is the index of positivity of their Wick product and characterizations of the case where the index of positivity is infinite (i.e., F is strongly positive) are presented
David Nualart, Moshe Zakai
A Symmetry Characterization of Conditionally Independent Increment Martingales
Abstract
We show that a càdlàg, local martingale has conditionally independent increments and symmetric jumps if and only if its law is invariant under integral transformations which preserve quadratic variation.
Daniel L. Ocone
The Stochastic Volterra Equation
Abstract
We study the stochastic (Skorohod) integral equation of the Volterra type
$$ {X_t}(\omega ) = {Y_t}(\omega ) + \int\limits_0^t {b(t,s){X_s}(\omega )ds} + \int\limits_0^t {\sigma (t,s){X_s}(\omega )\delta {B_s}(\omega )} $$
where Y, b and a are given functions; b and a are bounded, deterministic and Yt is stochastic, not necessarily adapted. The stochastic integral (δB) is taken in the Skorohod sense. In general there need not exist a classical stochastic process Xt(w) satisfying this equation. However, we show that a unique solution exists in the following extended senses:
(I)
As a functional process
 
(II)
As a generalized white noise functional (Hida distribution).
 
Moreover, in both cases we find explicit solution formulas. The formulas are similar to the formulas in the deterministic case (σ≡0), but with Wick products in stead of ordinary (pointwise) products.
Bernt Øksendal, Tu-Sheng Zhang
Exponential estimates for convex norms and some applications
Abstract
The role of correlation inequalities and martingale arguments in establishing conditional exponential bounds is reviewed. Applications to the computation of the Onsager Machlup functional for diffusions under non supremum norms follow.
L. A. Shepp, O. Zeitouni
Reflected Brownian Motion: Hunt Processes and Semimartingale Representation
Abstract
The Hunt process associated with a regular Dirichlet form for reflected Brownian motion on a bounded domain is considered. It is shown that a necessary condition for this process to be a semimartingale whose bounded variation part has an associated smooth measure with finite energy integral is that the domain be a Caccioppolis set.
R. J. Williams
The Fractional Calculus and Stochastic Evolution Equations
Abstract
The paper is devoted to the factorization method of infinite dimensional stochastic calculus. The applications of the method to the existence of weak solutions to stochastic PDEs and to the derivation of the maximal inequalities for stochastic convolutions are presented
J. Zabczyk
Backmatter
Metadaten
Titel
Barcelona Seminar on Stochastic Analysis
herausgegeben von
David Nualart
Marta Sanz Solé
Copyright-Jahr
1993
Verlag
Birkhäuser Basel
Electronic ISBN
978-3-0348-8555-3
Print ISBN
978-3-0348-9677-1
DOI
https://doi.org/10.1007/978-3-0348-8555-3