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The identification function for the convergence in probability with an application in the estimation theory
Kříž, Pavel ; Hlubinka, Daniel (referee) ; Štěpán, Josef (advisor)
In the present work we introduce the concept of probability limit identification function (PLIF) as it is done in [6]. This function identifies almost surely the value of the probability limit of a sequence of random variables on the basis of one realization of the sequence. According to the same article we show the construction of PLIF for real valued random variables from the special PLIF for 0-1 valued random variables. Following the method described in [8] we prove the existence of the universal PLIF for real valued random variables under the continuum hypothesis. Next we show that there are no borel measurable special PLIFs for 0-1 valued random variables (as well as PLIFs for real valued random variables). We use the proof that is published in [2]. Then we extend the construction of PLIF from R to any separable metrizable topological space. This PLIF may be used e.g. for creating functional representations of stochastic integrals and weak solutions of stochastic differential equations.
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Deterministic and Stochastic Epidemic Models
Staněk, Jakub ; Štěpán, Josef (advisor) ; Hlubinka, Daniel (referee) ; Dohnal, Gejza (referee)
Kermack-McKendrick model and its version with vaccination are presented. First, we introduce a model with vaccination and then a numerical study that includes comparison of di erent vaccination strategies and searching for optimal vaccination strategy is presented. We proceed to introduce a stochastic model with migration and consequently we suggest its generalization and prove the existence and uniqueness of a solution to the stochastic di erential equation (henceforth SDE) describing this model. Three stochastic versions of Kermack-McKendrick model with vaccination are suggested and compared. A procedure of nding the optimal vaccination strategy is presented. We also prove the theorem on the existence and uniqueness of a solution to the SDE that drives a model with multiple pathogens. Finally, the stochastic di erential equation describing the general model is presented. We study properties of a solution to this SDE and present sufficient conditions for the existence of a solution that is absorbed by the natural barrier of the model.
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