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Stochastic Processes and Simulation II

This course covers mainly Renewal theory, where we by removing the assumption that stochastic processes are memoryless consider models that are more complicated but more realistic than in previous courses, and Brownian motion, the random movement of a particles in a medium.

The course covers renewal theory, methods of stochastic simulation and the theory of Brownian motion.

Renewal Theory: A basic assumptions during previous courses is that stochastic processes are memoryless (Markovian). In the renewal theory we drop this assumption and study processes where the future advancement is not linked to the past. Therefore we lose some simplicity and elegance, but instead we obtain significantly more realistic results.

Brownian Motion: When a particle moves randomly, (like, for instance, a molecule in gas), its movement can often be viewed upon as a sum of a large number of impulses (collisions with other molecules in the gas). Due to the fact that the sums of stochastic variables are normally distributed, the particle's movements should approximately be normally distributed. Assuming that the time perspective of interest is a lot larger then the interval between two impulses, it follows that the particle's location is normally distributed. Then the particle describes Brownian motion. This mathematical model is frequently used, not only within physics, but also in many other areas of science and economy.