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Title: Some topics of topological measure theory with application in stochastic analysis Author: Pavel Kříž Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Josef Štěpán, DrSc., Department of Probability and Mathematical Statistics Abstract: This work studies identifications of values of probability limits based on trajectories of convergent (random) sequences. The key concept is the so called Probability Limit Identification Function (PLIF). The main concern is focused on the existence of PLIFs, mainly those, which are measurable and adapted. We also study in more detail special cases, when the convergence in probability and the convergence almost surely coincide. Furthermore, possible applications of the PLIF concept in stochastic analysis (path-wise representations of stochastic integrals and weak solutions of the stochastic differential equations), as well as in estimation theory (the existence of strongly consistent estimators) are outlined. The achieved results are based on analyses of the topologies on spaces of measures, spaces of random variables and spaces of real-valued functions. Keywords: Probability Limit, Identification, Almost-sure Convergence 1

>Magister Scientiae - MSc / HIV/AIDS is nowadays considered as the greatest public health disaster of modern time. Its progression has challenged the global population for decades. Through mathematical modelling, researchers have studied different interventions on the HIV pandemic, such as treatment, education, condom use, etc. Our research focuses on different compartmental models with emphasis on the effect of public health education. From the point of view of statistics, it is well known how the public health educational programs contribute towards the reduction of the spread of HIV/AIDS epidemic. Many models have been studied towards understanding the dynamics of the HIV/AIDS epidemic. The impact of ARV treatment have been observed and analysed by many researchers. Our research studies and investigates a compartmental model of HIV with treatment and education campaign. We study the existence of equilibrium points and their stability. Original contributions of this dissertation are the modifications on the model of Cai et al. [1], which enables us to use optimal control theory to identify optimal roll-out of strategies to control the HIV/AIDS. Furthermore, we introduce randomness into the model and we study the almost sure exponential stability of the disease free equilibrium. The randomness is regarded as environmental perturbations in the system. Another contribution is the global stability analysis on the model of Nyabadza et al. in [3]. The stability thresholds are compared for the HIV/AIDS in the absence of any intervention to assess the possible community benefit of public health educational campaigns. We illustrate the results by way simulation The following papers form the basis of much of the content of this dissertation, [1 ] L. Cai, Xuezhi Li, Mini Ghosh, Boazhu Guo. Stability analysis of an HIV/AIDS epidemic model with treatment, 229 (2009) 313-323. [2 ] C.P. Bhunu, S. Mushayabasa, H. Kojouharov, J.M. Tchuenche. Mathematical Analysis of an HIV/AIDS Model: Impact of Educational Programs and Abstinence in Sub-Saharan Africa. J Math Model Algor 10 (2011),31-55. [3 ] F. Nyabadza, C. Chiyaka, Z. Mukandavire, S.D. Hove-Musekwa. Analysis of an HIV/AIDS model with public-health information campaigns and individual with-drawal. Journal of Biological Systems, 18, 2 (2010) 357-375. Through this dissertation the author has contributed to two manuscripts [4] and [5], which are currently under review towards publication in journals, [4 ] G. Abiodun, S. Maku Vyambwera, N. Marcus, K. Okosun, P. Witbooi. Control and sensitivity of an HIV model with public health education (under submission). [5 ] P.Witbooi, M. Nsuami, S. Maku Vyambwera. Stability of a stochastic model of HIV population dynamics (under submission).

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In this study we are interested on the following 4-dimensional system of stochastic differential equations.dS=(-βSI+μ(K-S)+αI+ζR)dt-σ_1 SIF_1 (S,E,I,R)dW_1+σ_4 RF_4 (S,E,I,R)dW_4 dE=(βSI-(μ+η)E)dt+σ_1 SIF_1 (S,E,I,R)dW_1-σ_2 EF_2 (S,E,I,R)dW_2 dI=(ηE-(α+γ+μ)I)dt+σ_2 EF_2 (S,E,I,R)dW_2-σ_3 IF_3 (S,E,I,R)dW_3 dR=(γI-(μ+ζ)R)dt+σ_3 IF_3 (S,E,I,R)dW_3-σ_4 RF_4 (S,E,I,R)dW_4 with variance parameters σ_i≥0 and constants α,β,η,γ,μ ζ≥0. This system may be used to model the dynamics of susceptible, exposed, infected and recovering individuals subject to a present virus with state-dependent random transitions. Our main goal is to prove the existence of a bounded, unique, strong (pathwise), global solution to this system, and to discuss asymptotic stochastic and moment stability of the two equilibrium points, namely the disease free and the endemic equilibria. In this model, as suggested by our advisor, diffusion coefficients can be any local Lipschitz continuous functions on bounded domain D={(S,E,I,R)∈R_+^4:00 of maximum carrying capacity and W_i are independent and identical Wiener processes defined on a complete probability space (Ω,F,{F_t }_(t≥0),P). At the end we carry out some simulations to illustrate our results.

The stochastic SEIR(S) model with random total population is given by the system of stochastic differential equations:dS=(-βSI+μ(K-S)+αI+ζR)dt-σ_1 SIF_1 (S,E,I,R)dW_1+σ_4 RF_4 (S,E,I,R)dW_4+σ_5 S(K-N)dW_5\\ dE=(βSI-(μ+η)E)dt+σ_1 SIF_1 (S,E,I,R)dW_1-σ_2 EF_2 (S,E,I,R)dW_2+σ_5 E(K-N)dW_5 \\ dI=(ηE-(α+γ+μ)I)dt+σ_2 EF_2 (S,E,I,R)dW_2-σ_3 IF_3 (S,E,I,R)dW_3+σ_5 I(K-N)dW_5 \\ dR=(γI-(μ+ζ)R)dt+σ_3 IF_3 (S,E,I,R)dW_3-σ_4 RF_4 (S,E,I,R)dW_4+σ_5 R(K-N)dW_5, where σ_i>0 and constants α, β, η, γ, ζ, μ≥0. K represents the maximum carrying capacity for the total population and W_k=(W_k (t))_(t≥0) are independent, standard Wiener processes on a complete probability space (Ω,F,(F_t )_(t≥0),P). The SDE for the total population N=S+E+I+R has the form dN(t)=μ(K-N)dt+σ_5 N(K-N)dW_5 on D_0=(0,K). The goal of our study is to prove the existence of unique, Markovian, continuous time solutions on the 4D prism D={(S,E,I,R)∈R_+^4:0≤S, E,I,R≤K, S+E+I+R≤K}. Then using the method of Lyapunov functions we prove the asymptotic stochastic and moment stability of disease-free and endemic equilibria. Finally, we use numerical simulations to illustrate our results.

Dans cette thèse nous étudions les théorèmes limites dans l’analyse statistique dessystèmes dynamiques. Le premier chapitre est consacré aux notions des bases des systèmesdynamiques ainsi que la théorie ergodique. Dans le deuxième chapitre nous introduisonsun cadre fonctionnel abstrait pour lequel la version quenched du théorème de la limitecentrale (TLC) en dimension 1 pour les systèmes dynamiques uniformément dilatantsest satisfaite sous une condition de validité nécessaire et suffisante. Le troisième chapitreest consacré au principe d’invariance presque sûr (PIPS) pour les application aléatoiresdilatantes par morceaux. Nous présentons certaines hypothèses sous lesquelles le (PIPS)est vérifié en utilisant la méthode d’approximation des martingales de Cuny et Merlèvede.Nous étudions aussi le théorème de Sprindzuk et ses conséquences. Nous établissons dansle chapitre quatre la décroissance des corrélations pour les systèmes dynamiques aléatoiresuniformément dilatants par la méthode de couplage en dimension 1. Nous terminons cetravail par une présentation des concepts de base de la théorie des mesures et probabilitéset une présentation de l’espace des fonctions à variation bornée. / In this thesis we study the limit theorems in the statistical analysis of dynamicalsystems. The first chapter is devoted to the basic notions in dynamical systems as well asthe ergodic theory. In the second chapter we introduce an abstract functional frameworkunder which the quenched version of the central limit theorem (CLT) in dimension 1for uniformly expanding dynamic systems is satisfied under a necessary and sufficientcondition validity. The third chapter is devoted to the almost sure invariance principle(ASIP) for random piecewise expanding maps. We present some hypotheses under whichthe (ASIP) is verified using the method of approximation of the martingales of Cuny andMerlèvede. We also study the Sprindzuk theorem and its consequences. In chapter four,we define the decay of correlations for the random dynamical systems uniformly expandingby the coupling method in dimension 1. We finish this work with a presentation of thebasic concepts of the theory of measures and probabilities and a presentation of the spaceof functions with bounded variation.

對樣本資料之機率密度函數f(x)的無母數估計方法，一直是統計推論領域的研究重點之一，而且在通訊理論與圖形辨別上有非常重要的地位。傳統的文獻對密度函數的估計方法大部分著重於獨立樣本的情形。對於時間數列的相關樣本（例如：經濟指標或加權股票指數資料）比較少提到。本文針對具有弱相關性的穩定時間數列樣本，嘗試提出一個核密度估計的方法並探討其性質。 / For a sample data, the nonparametric estimation of a probability density f(x) is always one point of research problem in statistical inference and plays an important role in communication theory and pattern recognition. Traditionally, the literature dealing with density estimation when the observations are independent is extensive. Time series sample with weak dependence, (for example, an economic indicator or a stock market index data), less in this aspect of discussion. Our main purpose is concerned with the estimation of the probability density function f(x) of a stationary time series sample and discusses some properties of this kernel density.

This thesis is concerned with identifying the best decision among a set of possible decisions in the presence of uncertainty. We are primarily interested in situations where the objective function value at any feasible solution needs to be estimated, for example via a ``black-box' simulation procedure. We develop adaptive random search methods for solving such simulation optimization problems. The methods are adaptive in the sense that they use information gathered during previous iterations to decide how simulation effort is expended in the current iteration. We consider random search because such methods assume very little about the structure of the underlying problem, and hence can be applied to solve complex simulation optimization problems with little expertise required from an end-user. Consequently, such methods are suitable for inclusion in simulation software. We first identify desirable features that algorithms for discrete simulation optimization need to possess to exhibit attractive empirical performance. Our approach emphasizes maintaining an appropriate balance between exploration, exploitation, and estimation. We also present two new and almost surely convergent random search methods that possess these desirable features and demonstrate their empirical attractiveness. Second, we develop two frameworks for designing adaptive and almost surely convergent random search methods for discrete simulation optimization. Our frameworks involve averaging, in that all decisions that require estimates of the objective function values at various feasible solutions are based on the averages of all observations collected at these solutions so far. We present two new and almost surely convergent variants of simulated annealing and demonstrate the empirical effectiveness of averaging and adaptivity in the context of simulated annealing. Finally, we present three random search methods for solving simulation optimization problems with uncountable feasible regions. One of the approaches is adaptive, while the other two are based on pure random search. We provide conditions under which the three methods are convergent, both in probability and almost surely. Lastly, we include a computational study that demonstrates the effectiveness of the methods when compared to some other approaches available in the literature.

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