2,541 research outputs found
Poisson stochastic integration in Banach spaces
We prove new upper and lower bounds for Banach space-valued stochastic
integrals with respect to a compensated Poisson random measure. Our estimates
apply to Banach spaces with non-trivial martingale (co)type and extend various
results in the literature. We also develop a Malliavin framework to interpret
Poisson stochastic integrals as vector-valued Skorohod integrals, and prove a
Clark-Ocone representation formula.Comment: 26 page
Levy processes and stochastic integrals in Banach spaces
We review in¯nite divisibility and Levy processes in Banach spaces and discuss the relationship with notions of type and cotype. The Levy-It^o decomposition is described. Strong, weak and Pettis-style notions of stochastic integral are introduced and applied to construct
generalised Ornstein-Uhlenbeck processes
It\^{o} isomorphisms for -valued Poisson stochastic integrals
Motivated by the study of existence, uniqueness and regularity of solutions
to stochastic partial differential equations driven by jump noise, we prove
It\^{o} isomorphisms for -valued stochastic integrals with respect to a
compensated Poisson random measure. The principal ingredients for the proof are
novel Rosenthal type inequalities for independent random variables taking
values in a (noncommutative) -space, which may be of independent interest.
As a by-product of our proof, we observe some moment estimates for the operator
norm of a sum of independent random matrices.Comment: Published in at http://dx.doi.org/10.1214/13-AOP906 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Maximal inequality of Stochastic convolution driven by compensated Poisson random measures in Banach spaces
Let be a Banach space such that, for some , the
function is of class and its first and second
Fr\'{e}chet derivatives are bounded by some constant multiples of -th
power of the norm and -th power of the norm and let be a
-semigroup of contraction type on . We consider the
following stochastic convolution process \begin{align*}
u(t)=\int_0^t\int_ZS(t-s)\xi(s,z)\,\tilde{N}(\mathrm{d} s,\mathrm{d} z), \;\;\;
t\geq 0, \end{align*} where is a compensated Poisson random measure
on a measurable space and is an -predictable function. We
prove that there exists a c\`{a}dl\`{a}g modification a of the
process which satisfies the following maximal inequality \begin{align*}
\mathbb{E} \sup_{0\leq s\leq t} \|\tilde{u}(s)\|^{q^\prime}\leq C\ \mathbb{E}
\left(\int_0^t\int_Z \|\xi(s,z) \|^{p}\,N(\mathrm{d} s,\mathrm{d}
z)\right)^{\frac{q^\prime}{p}}, \end{align*} for all and
with .Comment: This version is only very slightly updated as compared to the one
from September 201
On maximal inequalities for purely discontinuous martingales in infinite dimensions
The purpose of this paper is to give a survey of a class of maximal
inequalities for purely discontinuous martingales, as well as for stochastic
integral and convolutions with respect to Poisson measures, in infinite
dimensional spaces. Such maximal inequalities are important in the study of
stochastic partial differential equations with noise of jump type.Comment: 19 pages, no figure
Radonifying operators and infinitely divisible Wiener integrals
In this article we illustrate the relation between the existence of Wiener
integrals with respect to a Levy process in a separable Banach space and
radonifying operators. For this purpose, we introduce the class of
theta-radonifying operators, i.e. operators which map a cylindrical measure
theta to a genuine Radon measure. We study this class of operators for various
examples of infinitely divisible cylindrical measures theta and highlight the
differences from the Gaussian case.Comment: 16 page
Cylindrical Levy processes in Banach spaces
Cylindrical probability measures are finitely additive measures on Banach
spaces that have sigma-additive projections to Euclidean spaces of all
dimensions. They are naturally associated to notions of weak (cylindrical)
random variable and hence weak (cylindrical) stochastic processes. In this
paper we focus on cylindrical Levy processes. These have (weak) Levy-Ito
decompositions and an associated Levy-Khintchine formula. If the process is
weakly square integrable, its covariance operator can be used to construct a
reproducing kernel Hilbert space in which the process has a decomposition as an
infinite series built from a sequence of uncorrelated bona fide one-dimensional
Levy processes. This series is used to define cylindrical stochastic integrals
from which cylindrical Ornstein-Uhlenbeck processes may be constructed as
unique solutions of the associated Cauchy problem. We demonstrate that such
processes are cylindrical Markov processes and study their (cylindrical)
invariant measures.Comment: 31 page
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