Random Walks

We start with the random walk on the 1-d lattice of the integers of the real line. From this simple model we derive the equations for the process of diffusion and from that, Brownian Motion. We examine the question of recurrence for random walks and in the process present Polya’s Theorem and Donsker’s Invariance Principle. Applications discussed include the derivation of option pricing in finance, self-avoiding walks, gambler’s ruin, the Kelly Criterion for risk, and Kinetic Monte Carlo. An in depth study of the random walk for analyzing electrical networks leads to developing formulas for calculating the hitting time to a goal for a Markov Chain and its relationship with the fundamental matrix of the chain and to reversible Markov Chains.