Part 7 of 7
Stochastic Calculus
Ito processes, dynamic programming in continuous time, and applications.
This part of the course develops the mathematical tools necessary to study how random variables affect optimization problems. We start with an overview of the most relevant stochastic processes that we will encounter in the applications to come. The most important result is Ito's Lemma, which defines the way in which we can take derivatives of functions that depend on diffusions. Then we can apply Ito's Lemma to problems of dynamic optimization, with special attention to stopping time problems. Finally we apply it to the characterization of the distribution of a random variable. This is done by means of the Kolmogorov forward equation.
All these sections follow closely Dixit and Pindyck (1994), with some portions adapted from Stokey (2009).