In this thesis we study the simple random walk. Firstly, we derive some basic probabilities, such as the probability of a walk finishing in a certain point and the probability of a walk reaching a certain point. We inspect the behaviour of a simple random walk when there is a special condition in a certain point that affects the probability of the next step and the amount of paths between two values under different conditions. We are also interested in the asymptotic characteristics of infinite random walks. In the second part of the thesis we introduce the concept of local time which notes the number of visits to a certain point. We determine the joint probabilities of local time and simple random walk with different modifications. In the last part we support our findings with simulations of the simple random walk with different modifications and uptick probabilities. We simulate multiple random walks and note their final value and local time.
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