736 research outputs found
On a Multiple Stochastic Integral with Respect to a Strictly Semistable Random Measure
The concept of multiple stochastic integration with respect to Brownian motion was introduced by Wiener (1938). Ito (1951) gave a more general construction of multiple stochastic integrals with regard to Brownian motion. Later the study of multiple stochastic integrals with respect to non-Gaussian processes were considered by some authors (e.g., Lin (1981), Surgailis (1981), Engel (1982)). Multiple stochastic integrals have found their applications in areas such as statistics and quantum mechanics. Recently, several authors (e.g., Szulga and Woyczynski (1983), Krakowiak and Szulga (1985), Rosinski and Woyczynski (1986), and Surgailis (1985)), using different approaches, have constructed multiple stochastic integrals with respect to symmetric stable random measures. This dissertation is concerned with the development of the multiple stochastic integrals with respect to semistable random measures.
One of the above mentioned approaches used to construct the multiple stochastic integrals with respect to stable random measures is the Lebesgue-Dunford type construction. This approach reduces the problem of stochastic integration to the problem of integration with respect to a vector measure. Using this approach Krakowiak and Szulga (1985) developed multiple stochastic integrals of Banach valued functions with respect to symmetric and also nonsymmetric stable random measures. In this dissertation, using an approach similar to that of Krakowiak and Szulga (1985), we develop multiple stochastic integrals with respect to symmetric as well as with respect to (nonsymmetric) strictly semistable random measures with index of stability α ∈ (1, 2). Our methods, in the nonsymmetric case, yield results on multiple stochastic integrals relative to strictly stable random measure with index α ∈ (1, 2) considered in [10, 13].
The most crucial role in the development of the integrals here is played by the inequalities (2.29). In these inequalities we establish a comparison theorem between the moments of the integrals of certain simple functions relative to the strictly semistable random measure and the corresponding moments of integrals of these functions relative to symmetric stable random measure. Once these inequalities are established, the methods of construction of the integrals here are similar to those used by Krakowiak and Szulga in [10, 13] to develop the integrals relative to symmetric stable random measure.
In Chapter I, we collect the notation, definitions, and known results that are basic to this dissertation. In Chapter II, we develop necessary tools and prove the crucial inequalities mentioned above. In the first part of Chapter II, we prove a comparison theorem for tail probabilities of nonsymmetric semistable random measures. This uses a distributional property of a strictly semistable random variable. In Chapter III, we define the multiple stochastic integrals of certain Banach valued Borel measurable functions with respect to a strictly semistable random measure of index a. Then, we show that the class of Banach valued integrable functions relative to a semistable random measure of index α coincides with the class of Banach valued integrable functions relative to a symmetric stable random measure of index α
Asymptotic behavior of the Whittle estimator for the increments of a Rosenblatt process
The purpose of this paper is to estimate the self-similarity index of the
Rosenblatt process by using the Whittle estimator. Via chaos expansion into
multiple stochastic integrals, we establish a non-central limit theorem
satisfied by this estimator. We illustrate our results by numerical
simulations
Central limit theorems for sequences of multiple stochastic integrals
We characterize the convergence in distribution to a standard normal law for
a sequence of multiple stochastic integrals of a fixed order with variance
converging to 1. Some applications are given, in particular to study the
limiting behavior of quadratic functionals of Gaussian processes.Comment: Published at http://dx.doi.org/10.1214/009117904000000621 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Krylov-Veretennikov formula for functionals from the stopped Wiener process
We consider a class of measures absolutely continuous with respect to the
distribution of the stopped Wiener process . Multiple
stochastic integrals, that lead to the analogue of the It\^o-Wiener expansions
for such measures, are described. An analogue of the Krylov-Veretennikov
formula for functionals is obtained
Central limit theorems for multiple stochastic integrals and Malliavin calculus
We give a new characterization for the convergence in distribution to a
standard normal law of a sequence of multiple stochastic integrals of a fixed
order with variance one, in terms of the Malliavin derivatives of the sequence.
We extend our result to the multidimensional case and prove a weak convergence
result for a sequence of square integrable random variables.Comment: 16 page
Estimates for multiple stochastic integrals and stochastic Hamilton-Jacobi equations
We study stochastic Hamilton-Jacobi-Bellman equations and the
corresponding Hamiltonian systems driven by jump-type Lévy processes.
The main objective of the present paper is to show existence,
uniqueness and a (locally in time) diffeomorphism property of the solution:
the solution trajectory of the system is a diffeomorphism as a
function of the initial momentum. This result enables us to implement
a stochastic version of the classical method of characteristics for the
Hamilton-Jacobi equations. An –in itself interesting– auxiliary result
are pointwise a.s. estimates for iterated stochastic integrals driven by
a vector of not necessarily independent jump-type semimartingales
Central limit theorems for multiple Skorohod integrals
In this paper, we prove a central limit theorem for a sequence of iterated
Shorohod integrals using the techniques of Malliavin calculus. The convergence
is stable, and the limit is a conditionally Gaussian random variable. Some
applications to sequences of multiple stochastic integrals, and renormalized
weighted Hermite variations of the fractional Brownian motion are discussed.Comment: 32 pages; major changes in Sections 4 and
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