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

Maximal inequality of Stochastic convolution driven by compensated Poisson random measures in Banach spaces

Let $(E, \| \cdot\|)$ be a Banach space such that, for some $q\geq 2$, the function $x\mapsto \|x\|^q$ is of $C^2$ class and its first and second Fr\'{e}chet derivatives are bounded by some constant multiples of $(q-1)$-th power of the norm and $(q-2)$-th power of the norm and let $S$ be a $C_0$-semigroup of contraction type on $(E, \| \cdot\|)$. 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 $\tilde{N}$ is a compensated Poisson random measure on a measurable space $(Z,\mathcal{Z})$ and $\xi:[0,\infty)\times\Omega\times Z\rightarrow E$ is an $\mathbb{F}\otimes \mathcal{Z}$-predictable function. We prove that there exists a c\`{a}dl\`{a}g modification a $\tilde{u}$ of the process $u$ 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 $q^\prime \geq q$ and $1<p\leq 2$ with $C=C(q,p)$.Comment: This version is only very slightly updated as compared to the one from September 201