Application of Markov Chain in Actuarial modelling

Abstract

In this research Markov processes are used for modelling a phenomenon in insurance, which changes over time of a random variable comprise a sequence of values in the future. Each of values depends only on the immediately preceding state, not on other past states. A Markov process is completely characterized by specifying the finite set S of possible states and the stationary probabilities of transition between these states. Random processes are of interest for describing the behaviour of a system evolving over period of time, hence they were greatly applied in the actuarial mathematics and enabled us to deal with very complicated actuarial problems. Markov processes can be observed in dicreate-time and a continuous time. A discrete-time random process involves a system which is in a certain state at each step, with the state changing randomly between steps. The Markov property states that the conditional probability distribution for the system at the next step (and in fact at all future steps) depends only on the current state of the system, and not additionally on the state of the system at previous steps. Since the system changes randomly, it is generally impossible to predict with certainty the state of a Markov chain at a given point in the future. A continuous-time Markov chain is a mathematical model which takes values in some finite or countable set. The time spent in each state takes non-negative real values and has an exponential distribution. It is a random process with the Markov property which means that future behavior of the model depends only on the current state of the model and not on historical behavior. The model is a continuous-time version of the Markov chain model, named because the output from such a process is a sequence (or chain) of states. This research illustrates how the mathematics of Markov Processes can be used, in the actuarial modelling and calculation. First the possibility of application of the Markov chain was shown on the Belgrade Stock Exchange (BSE) in order to forecast stock prices and return of stocks, as an important part of the investment strategies. Markov chains are a simple non-parametric method with application in the stock market analysis, however, insufficiently researched in the area of return modelling. In a review of lot of articles published on this subject, it was found that no one provides the possibility to apply it on the BSE. Special attention in this research was brought to a bonus-malus system, as it can be considered as a special case of Markov processes. The nature of life assurance indicates to application of Markov chains. Examples show the modelling method for term life assurance as well as for the disability model.