Peter Kempthorne continues MIT's 18.642, Topics in Mathematics with Applications in Finance, with a lecture on martingales and their use in solving stochastic processes problems, including random walks, stopping times, and gambler's ruin probabilities. He shows how martingale properties simplify otherwise difficult calculations before introducing Markov processes and chains, explaining the memoryless property and its use in finance applications such as credit rating transitions and stock price modeling. The lecture then shifts to regression analysis, covering multiple linear regression, the assumptions underlying the model, and estimation techniques. Delivered as a standard chalkboard lecture at MIT, it is aimed at students with a working grasp of probability who want to see how these tools are applied directly to financial modeling problems.