Stochastic Processes and Simulation II

The two important parts of the course are the Renewal theory and the theory of Brownian motion. Renewal Theory: One of the most unrealistic assumptions you receive during the basic courses is that the stochastic processes are memoryless (Markovian, as it is called). In the Renewal Theory we gave goodbye to Markov, and study the processes where the future advancement is not linked to the past. Therefore we lose some of the simplicity and elegance, but instead receive significantly more realistic results. Brownian Motion: When a particle moves randomly, (like, for instance, a molecule in gas), its movement can often be seen as a sum of a large amount of impulses (collisions with other molecules in the gas). Due to the fact that the sums of stochastic variables are normally distributed the particles movements should during a certain time be normally distributed. If we assume that the time perspective which interests us is a lot larger then the interval between two impulses, we can pull out the normally distributed assumption to its outer consequence, and assume that the particles movement under as short of a period as we wish is normally distributed. Then the particle describes Brownian motion. This mathematical model has come to use not only within physics but also in many other areas within science and economy.