This thesis consists of three parts. In the first part, we recall some background theory that will be used throughout the thesis. In the second part, we studied the existence and uniqueness of solutions of the stochastic heat equations with Markovian switching. In the third part, we investigate the properties of solutions, such as Feller property, strong Feller property and stability.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-11T20:52:36Z No. of bitstreams: 1 ANTONIO LUIZ SOARES SANTOS - DISSERTAÇÃO PPGMAT 2007..pdf: 419776 bytes, checksum: 648bbdd93c2d2fdd67f2dd99256a1ff7 (MD5) / Made available in DSpace on 2018-07-11T20:52:36Z (GMT). No. of bitstreams: 1 ANTONIO LUIZ SOARES SANTOS - DISSERTAÇÃO PPGMAT 2007..pdf: 419776 bytes, checksum: 648bbdd93c2d2fdd67f2dd99256a1ff7 (MD5) Previous issue date: 2007-02 / Neste trabalho estudamos o processo de risco a tempo discreto, considerado modelo clássico na teoria do risco, com variantes propostas por Jun Cai e David Dickson (2004). Serão incluídas taxas de juros, as quais seguem uma Cadeia de Markov, e seus efeitos, em relação à probabilidade de ruína serão analisados. O conhecido limitante superior proposto por Lundberg para essa probabilidade fica reduzido em virtude dessa nova abordagem e a desigualdade clássica é generalizada. / In this work we study discrete time risk process considered classical model, with variants proposed by Jun Cai and David Dickson (2004). Rates of interest, which follows a Markov chain will be introduced and their effect on the ruin probabilities will be analysed. Generalized Lundberg inequalities will be obtained and shown how the classical bounds for the ruin probability can be derived.

This thesis deals with the systems reliability. At First, there is discussed the concept of reliability itself and its indicators, which can specifically express reliability. The second chapter describes the different kinds of reliability models for simple and complex systems. It further describes the basic methods for construction of reliability models. The fourth chapter is devoted to a very important Markov models. Markov models are very powerful and complex model for calculating the reliability of advanced systems. Their suitability is explained here for recovered systems, which may contain absorption states. The next chapter describes the standby redundancy. Discusses the advantages and disadvantages of static, dynamic and hybrid standby. There is described the influence of different load levels on the service life. The sixth chapter is devoted to the implementation, description of the application and description of the input file in XML format. There are discussed the results obtaining in experimental calculations.

In this thesis we focus on new methods for probabilistic model checking (PMC) with linear temporal logic (LTL). The standard approach translates an LTL formula into a deterministic ω-automaton with a double-exponential blow up. There are approaches for Markov chain analysis against LTL with exponential runtime, which motivates the search for non-deterministic automata with restricted forms of non-determinism that make them suitable for PMC. For MDPs, the approach via deterministic automata matches the double-exponential lower bound, but a practical application might benefit from approaches via non-deterministic automata. We first investigate good-for-games (GFG) automata. In GFG automata one can resolve the non-determinism for a finite prefix without knowing the infinite suffix and still obtain an accepting run for an accepted word. We explain that GFG automata are well-suited for MDP analysis on a theoretic level, but our experiments show that GFG automata cannot compete with deterministic automata. We have also researched another form of pseudo-determinism, namely unambiguity, where for every accepted word there is exactly one accepting run. We present a polynomial-time approach for PMC of Markov chains against specifications given by an unambiguous Büchi automaton (UBA). Its two key elements are the identification whether the induced probability is positive, and if so, the identification of a state set inducing probability 1. Additionally, we examine the new symbolic Muller acceptance described in the Hanoi Omega Automata Format, which we call Emerson-Lei acceptance. It is a positive Boolean formula over unconditional fairness constraints. We present a construction of small deterministic automata using Emerson-Lei acceptance. Deciding, whether an MDP has a positive maximal probability to satisfy an Emerson-Lei acceptance, is NP-complete. This fact has triggered a DPLL-based algorithm for deciding positiveness.