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We show that the result from Da Prato and Lunardi is valid for stochastic convolutions driven by L\'evy processes.
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In this paper we are interested in a reaction diffusion equation driven by Poissonian noise respective L\'evy noise. For this aim we first show existence of a martingale solution for an SPDE of parabolic type driven by a Poisson random... more
In this paper we are interested in a reaction diffusion equation driven by Poissonian noise respective L\'evy noise. For this aim we first show existence of a martingale solution for an SPDE of parabolic type driven by a Poisson random measure with only continuous and bounded coefficients. This result is transferred to an parabolic SPDE driven by L\'evy noise. In a second step, we show existence of a martingale solution of reaction diffusion type, also driven by Poissonian noise respective L\'evy noise. Our results answer positively a long standing open question about existence of martingale solutions driven by genuine L\'evy processes.
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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... more
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 $C=C(q,p)$.
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Research Interests: Statistics and Space Time
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The solution of stochastic evolution equations generally relies on numerical computation. Here, usually the main idea is to discretize the SPDE spatially obtaining a system of SDEs that can be solved by e.g., the Euler scheme. In this... more
The solution of stochastic evolution equations generally relies on numerical computation. Here, usually the main idea is to discretize the SPDE spatially obtaining a system of SDEs that can be solved by e.g., the Euler scheme. In this paper, we investigate the discretization error of semilinear stochastic evolution equations in Lp-spaces, resp. Banach spaces. The space discretization may be done by Galerkin approximation, for the time discretization we consider the implicit Euler, the explicit Euler scheme and the Crank-Nicholson scheme. In the last section, we give some examples, i.e., we consider an SPDEs driven by nuclear Wiener noise approximated by wavelets and delay equation approximated by finite differences.
Research Interests: Applied Mathematics, Numerical Analysis, Applied Mathematics and Computational Science, Recurrence Relation, Finite Difference, and 7 moreNumerical Analysis and Computational Mathematics, Numerical computation, Galerkin Method, Difference equation, Electrical And Electronic Engineering, Stochastic Partial Differential Equations, and Euler scheme
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The Radon transform arise in, if X-rays traverses a body, like in a computer tomograph. Thus, finding out the internal structure of a body in computer tomography corresponds to reconstruct a function from their Radon transform. This... more
The Radon transform arise in, if X-rays traverses a body, like in a computer tomograph. Thus, finding out the internal structure of a body in computer tomography corresponds to reconstruct a function from their Radon transform. This report deals with a parallelized numerical reconstruction algorithm, in particular the filtered backprojection. The filtered backprojection uses high dimensional integration. This integration is
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Our goal is achieved by proving general results about the existence of maximal and global solution to an abstract stochastic partial differential equations with locally Lipschitz continuous coefficients. The method of the proofs are based... more
Our goal is achieved by proving general results about the existence of maximal and global solution to an abstract stochastic partial differential equations with locally Lipschitz continuous coefficients. The method of the proofs are based on some truncation and fixed point methods.
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The aim of this note is to give some Burkholder-Davis-Gundy type inequalities which are valid for the Ito stochastic integral with respect to Banach valued Levy noise.
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In this paper we introduce probabilistic methods for solving partial differential equations by approximating stochastic systems. The underlying idea is to search for stochastic systems, for example particle systems or diffusion processes,... more
In this paper we introduce probabilistic methods for solving partial differential equations by approximating stochastic systems. The underlying idea is to search for stochastic systems, for example particle systems or diffusion processes, where the corresponding infinitesimal generator is equal to the differential operator of the differential equation. These system can be approximated by certain methods. Furthermore, for example problems arising in financial mathematics can also be solved by these methods. Why are we interested in these methods despite of having finite difference methods, finite element methods and other deterministic algorithms for solving partial differential equations? Because of their inherent parallelism probabilistic methods are much more efficient when using parallel systems than deterministic methods. Now, because of the new facilities of parallel computing, simulating stochastic systems with a large sample size can be done in a short time. Therefore these m...
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... Its originally assumed Gaussian nature was questioned in the climate physics literature byDitlevsen [2, 1] who analyzed global temper-ature proxies in a Greenland ice-core time series from the last glacial period and suggested jump... more
... Its originally assumed Gaussian nature was questioned in the climate physics literature byDitlevsen [2, 1] who analyzed global temper-ature proxies in a Greenland ice-core time series from the last glacial period and suggested jump diffusion models with α-stable noise instead ...
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... for parallel architectures, probabilistic methods are much more efficient. ... For solving parabolicPDE's with mixed boundary conditions we need a quantity which measures the time the process ... But there exists a few algorithms... more
... for parallel architectures, probabilistic methods are much more efficient. ... For solving parabolicPDE's with mixed boundary conditions we need a quantity which measures the time the process ... But there exists a few algorithms which are difficult to implement (See eg [6, 17, 14]). ...
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The Radon transform arise in, if X-rays traverses a body, like in a computer tomograph. Thus, finding out the internal structure of a body in computer tomography corresponds to reconstruct a function from their Radon transform. This... more
The Radon transform arise in, if X-rays traverses a body, like in a computer tomograph. Thus, finding out the internal structure of a body in computer tomography corresponds to reconstruct a function from their Radon transform. This report deals with a parallelized numerical reconstruction algorithm, in particular the filtered backprojection. The filtered backprojection uses high dimensional integration. This integration is
Research Interests:
The aim of this note is to give some Burkholder-Davis-Gundy type inequalities which are valid for the Ito stochastic integral with respect to Banach valued Levy noise.