Seminar za teoriju vjerojatnosti
Probability Seminar

2009./2010.:

Sažeci / Abstracts:

• Vedran Šohinger: Rast visokih Soboljevovih normi za jednodimenzionalnu nelinearnu Schrodingerovu jednadzbu

  U ovom predavanju cemo promatrati rast Soboljevovih normi rjesenja jednodimenzionalnih nelinearnih Schrodingerovih jednadzbi (NLS) cija ogranicenost ne slijedi iz ocuvanja energije. Prezentirat cemo metodu, zasnovanu na analizi frekvencija, iz koje mozemo izvesti gornje ograde koje su polinomijalne u vremenu. Metodu cemo primjeniti na nelinearnu Hartree jednadzbu sa dovoljno regularnim konvolucijskim potencijalom te na ogradu rasta necjelobrojnih Soboljevovih normi za rjesenja kubicne NLS jednadzbe na pravcu.

• Edward Furman: General Stein-type covariance decompositions with applications to insurance and finance

  A general decomposition of covariances is formulated and proved, and its various applications to numerous phenomena of central importance in insurance and finance are discussed. More specifically, the aforementioned phenomena of interest include, e.g., actuarial and economic approaches to deriving insurance prices, risk measurement and risk capital allocation procedures, the capital asset pricing model. The talk is based on Furman and Zitikis, 2010 [General Steyn-type decomposition of covariances with applications to insurance and finance. ASTIN Bulletin, to appear].

• Philip Protter: Absolutely Continuous Compensators

  Often in applications (for example in Survival Analysis and in Credit Risk) one begins with a totally inaccessible stopping time, and then one assumes the compensator of the indicator of the stopping time has absolutely continuous paths. This gives an interpretation in terms of a "hazard function'' process. Ethier and Kurtz have given sufficient conditions for a given stopping time to have an absolutely continuous compensator, and this condition was extended by Yan Zeng to a necessary and sufficient condition. We take a different approach and make a simple hypothesis on the filtration under which all totally inaccessible stopping times have absolutely continuous compensators. We show such a property is stable under changes of measure, and under the expansion of filtrations; and we detail its limited stability under filtration shrinkage. The talk is based on research performed with Sokhna M'Baye and Svante Janson.

• Zbigniew Palmowski: Ruin Probability with Parisian Delay for a Spectrally Negative Lévy Risk Process

  We analyze so-called Parisian ruin probability that happens when the surplus process stays below zero longer than fixed amount of time $\zeta>0$. We focus on general spectrally negative Lévy insurance risk process. For this class of processes we identify the expression for the ruin probability in terms of some other quantities that could be possibly calculated explicitly in many models. We find its Cram\'{e}r-type and convolution-equivalent asymptotics when reserves tends to infinity. Finally, we analyze few explicit examples. The talk is based on the paper written jointly with I. Czarna.

• Zoran Vondraček: Potential Theory of the Operator $\Delta +\Delta^{\alpha/2}$

  The operator $\Delta +\Delta^{\alpha/2}$ is a sum of the Laplacian, which is a local operator, and the fractional Laplacian, which is (pure) non-local operator. From the probabilistic point of view, the corresponding object is a Markov process $X$ which is the sum of a Brownian motion (process with continuous paths) and an independent symmetric $\alpha$-stable process (a pure jump process). More precisely, $\Delta +\Delta^{\alpha/2}$ is the infinitesimal generator of $X$. The fact that $X$ has both continuous and jump components is the source of many difficulties in investigating the corresponding potential theory. This is particularly the case when the Dirichlet boundary conditions are imposed on $\Delta +\Delta^{\alpha/2}$ (i.e., when the process $X$ is killed upon exiting an open set). In this talk I will report on the progress made on this subject in the last couple of years. The topics discussed will include the Green function and the heat kernel estimates, Harnack inequalities and boundary Harnack principles. The emphasis will be on methods and their scope.

• Björn Böttcher: Approximation of Feller processes

  We show that Feller processes can be approximated by Markov Chains with Levy increments. The approximation can be used for simulations and it is motivated by the attempt to construct Feller processes from a given family of Lévy processes.

• Felix Lindner: Stochastic partial differential equations and Hilbert space valued Lévy processes

  An introduction to stochastic partial differential equations is given and their connection to stochastic differential equations in infinite dimensional spaces is explained. The focus is on Hilbert space valued Lévy processes as driving processes.

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Prošli seminari / Past seminars:

• 2008./2009.
• 2007./2008.