71 research outputs found
Copulas in Hilbert spaces
In this article, the concept of copulas is generalised to infinite
dimensional Hilbert spaces. We show one direction of Sklar's theorem and
explain that the other direction fails in infinite dimensional Hilbert spaces.
We derive a necessary and sufficient condition which allows to state this
direction of Sklar's theorem in Hilbert spaces. We consider copulas with
densities and specifically construct copulas in a Hilbert space by a family of
pairwise copulas with densities
Weak approximation of the stochastic wave equation
AbstractWe investigate the accuracy of approximation of E[φ(u(t))], where {u(t):t∈[0,∞)} is the solution of the stochastic wave equation driven by the space–time white noise and φ is an R-valued function defined on the Hilbert space L2(R). The approximation is done by the leap-frog scheme. We show that, under certain conditions on φ, the approximation by the leap-frog scheme is of order two
Numerical Approximation of Parabolic Stochastic Partial Differential Equations
The topic of the talk were the time approximation
of quasi linear stochastic partial differential
equations of parabolic type. The framework were
in the setting of stochastic evolution equations.
An error bounds for the implicit Euler scheme was
given and the stability of the scheme were considered
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
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