Browsing by Subject "Continuous-time Markov Chains"

A continuous-time Markov chain is a stochastic process which has the Markov property. The Markov property states that the transition to a next state of the process only depends on the current state, that is, it does not depend on the process’ preceding states. Continuous-time Markov Chains are fundamental tools to model stochastic systems in finance and insurance such as option pricing and modelling insurance claim processes. This thesis examines continuous-time Markov chains and their most important concepts and typical properties. For instance, we introduce and investigate the Kolmogorov forward and backward equations, which are essential for continuous-time systems. However, the main aim of the thesis is to present a method and proof for constructing a Markov process from continuous transition intensity matrix. This is achieved by generating a transition probability matrix from given transition intensity matrix. When the transition intensities are known, the challenge is to determine the transition probabilities since the calculations can easily become difficult to solve analytically. Through the introduced theorem it becomes possible to simplify the calculations by approximations. In this thesis, we also make applications of the theory. We demonstrate how determining transition probabilities using Kolmogorov’s forward equations can become challenging in a simple setup. Furthermore, we will compare the approximations of transition probabilities derived from the main theorem to the actual transition probabilities. We make observations about the theorem’s transition probability function; the approximations derived from the main theorem provides quite satisfactory estimates of the actual transition probabilities.