
Design and verification of an intervention exercise program aimed at improving the fitness of recreational athletes
KŘÍŽ, Pavel
The aim of this work is to compile an intervention program aimed at improving fitness. The goal was achieved. According to the proposal, the intervention exercise program can be practiced by everyone and anywhere, even if the gyms are closed, for example for pandemic reasons. Eleven people took part in the exercise exercise program, and the control group also included eleven people. The EUROFIT test battery, consisting of nine tests, was chosen within the methodology. From the results, it should be emphasized that the experimental group improved on average by 0,1 try for endurance training standing on one leg, 1.89 seconds for tapping, also improved in the overhang in the forward bend by 3.23 centimeters, 13, 96 in the jump from the place, 3.17 repetitions in the situps exercise, 4.79 seconds in the endurance in the squat, the only deterioration of the experimental group occurred in the shuttle run by 0.2 seconds, in the last dynamometry measurement test there was an improvement of 7.14 Newtons. In the control group, there was a slight deterioration in the first test by 0.1 attempts, we could also observe a deterioration in the tapping test by 1.33 seconds, a slight improvement of 0.98 centimeters occurred in the forward bend, the group only further deteriorated in all tests. Specifically, by 0.54 centimeters in the jump, by 0.84 repetitions in the situp, 2.56 seconds in the endurance of the pushup, 0.29 in the shuttle run and by 2.23 Newtons in the dynamometry. All research assumptions have been met.


Optimal FInancial Payoffs Maximizing Utility Function
Kožnar, František ; Večeř, Jan (advisor) ; Kříž, Pavel (referee)
The goal of this thesis is to characterize payoffs that maximize expected utility function in different market setups. One can solve this problem in its generality in terms of a function of a likelihood ratio between the subjective measure of an agent P and a risk neutral measure Q. Such payoffs should be transformed to the function of the terminal stock price. The question is what measure P should be chosen, the natural candidates would correspond to either the frequentist or the Bayesian choice of the parameters. The thesis should provide a link to the Kelly Criterion in the binomial evolution of the stock price and to the Merton's Portfolio Problem in the geometric Brownian motion exam ple showing the possible extensions of these well known problems in the novel Bayesian setup. The thesis should discuss pricing and hedging of these contracts together with their asymptotic behavior. 1


Malliavin operators for realvalued Gaussian random variables and their applications
Kubát, Martin ; Kříž, Pavel (advisor) ; Čoupek, Petr (referee)
In this thesis, we introduce Malliavin Operators. We will focus on derivative, di vergence and OrnsteinUhlenbeck operator to study properties of transformed Gaussian random variables. We will explain all concepts in detail and add some typical examples. Then we will use Malliavin operators in the proofs of famous Poincaré inequality and variance expansions. The technique of the last proofs provides a good general approach how to solve similar problems with understanding Malliavin Operators. 1


Chainladder method as maximum likelihood estimator in Poisson model
Wagner, Vojtěch ; Kříž, Pavel (advisor) ; Pešta, Michal (referee)
First, the distributionfree chainladder is introduced. Then, the Poisson model is in troduced. It is proven that the total reserves for one accident year given by the maximum likelihood estimation applied to the Poisson model lead to the identical reserves as the reserves derived from the distributionfree chainladder used in the Poisson model. Later, inadequacies of the Poisson model are discussed. Hessian matrices of the loglikelihood evaluated at the Poisson estimators are analyzed. The question whether the inverse of the Fisher information matrix approximates the real covariance matrix of the Poisson esti mators is explored. Comparing the variance of the total reserves derived from the inverse of the Fisher information and the real covariance matrix leads to negative conclusion, that the former does not approximate the latter well. 1


Stein's method for normal approximation of random variables
Strnad, Martin ; Kříž, Pavel (advisor) ; Nagy, Stanislav (referee)
The Stein's method is a collection of probabilistic techniques for answering the ques tion as to how far the probability distributions of random variables are from each other. This thesis only concerns the basics of the approach. We use the Kolmogorov distance and the total variation distance to formalize the concept of the distance between mea sures. We focus on the normal distribution for which we first find a suitable differential operator, often called Stein operator, that bears much information. Not only does it charactize the Gaussian measure, it also gives us a means to quantify the distance from another random variable's distribution. Finally, we apply the method to prove the clas sic BerryEsseen inequality for a sum of independent and identically distributed random variables. 1


Difference and differential equations in life insurance
Kirešová, Katarína ; Kříž, Pavel (advisor) ; Pešta, Michal (referee)
The diploma thesis deals with the calculation of life insurance reserves, higher mo ments and the distribution function of future payments of reserves using difference and differential equations. In the beginning, the basic theory of a stochastic process, insu rance model, cash flow, and reserve is summarized. After that, equations themselves are derived; first in general and then for four specific types of insurance. Subsequently, a cal culation of premiums is presented for each type of insurance. The next two chapters deal with the calculation of higher moments and the distribution function. After deriving the formulas for four types of insurance, the reserves, standard deviations, and distribution functions are calculated for specific values and then they are compared with the Monte Carlo simulation. The conclusion contains pros and cons of the method compared to the simulation. 1


Quantifying Mortality and Longevity Risk by Means of Stochastic Models
Plotnikova, Valeriya ; Mazurová, Lucie (advisor) ; Kříž, Pavel (referee)
In this thesis we investigate the structure of the generalized ageperiodcohort mortality model and we comment on the key components of its structure. As an example of the generalized age periodcohort model we take a closer look at the widely used LeeCarter mortality model. We further construct mortality models for the Czech male and female populations, by using a certain procedure that involves expert judgment. To project mortality rates we choose the most suitable time series processes for the selected parameters in the model. Finally, we describe and implement the value at risk framework for the longevity risk, which is one of the possible applications where the obtained mortality models can be used in practice. In particular, we investigate how much a temporary life annuity liability might change based on new information over the course of one year.

 

Strong stationary times and convergence of Markov chains
Suk, Luboš ; Prokešová, Michaela (advisor) ; Kříž, Pavel (referee)
In this thesis we study the estimation of speed of convergence of Markov chains to their stacionary distributions. For that purpose we will use the method of strong stationary times. We focus on irreducible and aperiodic chains only since in that case the existence of exactly one stationary distribution is guaranteed. We introduce the mixing time for a Markov chain as the time needed for the marginal distribution of the chain to be sufficiently close to the stationary dis tribution. The distance between two distributions is measured by the total variation distance. The main goal of this thesis is to construct an appropriate strong stationary time for selected chains and then use it for obtaining an upper bound for the mixing time.


The identification function for the convergence in probability with an application in the estimation theory
Kříž, Pavel ; Štěpán, Josef (advisor) ; Hlubinka, Daniel (referee)
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 01 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 01 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.
