14 research outputs found
Parametric family of SDEs driven by L\'evy noise
In this article we study the existence and uniqueness of strong solutions of
a class of parameterized family of SDEs driven by L\'evy noise. These SDEs
occurs in connection with a class of stochastic PDEs, which take values in the
space of tempered distributions . This correspondence for
diffusion processes was proved in [Rajeev, Translation invariant diffusion in
the space of tempered distributions, Indian J. Pure Appl. Math. 44 (2013),
no.~2, 231--258]
Stationary solutions of stochastic partial differential equations in the space of tempered distributions
In Rajeev (2013), 'Translation invariant diffusion in the space of tempered
distributions', it was shown that there is an one to one correspondence between
solutions of a class of finite dimensional SDEs and solutions of a class of
SPDEs in , the space of tempered distributions, driven by the
same Brownian motion. There the coefficients of the
finite dimensional SDEs were related to the coefficients of the SPDEs in
in a special way, viz. through convolution with the initial
value of the SPDEs.
In this paper, we consider the situation where the solutions of the finite
dimensional SDEs are stationary and ask whether the corresponding solutions of
the equations in are also stationary. We provide an affirmative
answer, when the initial random variable takes value in a certain set
, which ensures that the coefficients of the finite dimensional
SDEs are related to the coefficients of the SPDEs in the above `special'
manner
Co-variance Operator of Banach Valued Random Elements: U-Statistic Approach
This article proposes a co-variance operator for Banach valued random
elements using the concept of -statistic. We then study the asymptotic
distribution of the proposed co-variance operator along with related large
sample properties. Moreover, specifically for Hilbert space valued random
elements, the asymptotic distribution of the proposed estimator is derived even
for dependent data under some mixing conditions.Comment: Preliminary version of an ongoing work. Comments are welcom
Testing Independence of Infinite Dimensional Random Elements: A Sup-norm Approach
In this article, we study the test for independence of two random elements
and lying in an infinite dimensional space (specifically, a
real separable Hilbert space equipped with the inner product ). In the course of this study, a measure of association is
proposed based on the sup-norm difference between the joint probability density
function of the bivariate random vector and the product of marginal probability
density functions of the random variables
and , where and
are two arbitrary elements. It is established that the
proposed measure of association equals zero if and only if the random elements
are independent. In order to carry out the test whether and are
independent or not, the sample version of the proposed measure of association
is considered as the test statistic after appropriate normalization, and the
asymptotic distributions of the test statistic under the null and the local
alternatives are derived. The performance of the new test is investigated for
simulated data sets and the practicability of the test is shown for three real
data sets related to climatology, biological science and chemical science.Comment: Remark 2.4 has been adde
Existence and Uniqueness of Stochastic PDEs associated with the Forward Equations: An Approach using Alternate Norms
We consider stochastic PDEs
and associated PDEs with regular initial
conditions. Here, and are certain partial differential operators
involving multiplication by smooth functions and are of the order two and one
respectively, and in special cases are associated with finite dimensional
diffusion processes. This PDE also includes Kolmogorov's Forward Equation
(Fokker-Planck Equation) as a special case. We first prove a Monotonicity
inequality for the pair and using this inequality, we obtain the
existence and uniqueness of strong solutions to the Stochastic PDE and the PDE.
In addition, a stochastic representation for the solution to the PDE is also
established