Synopsis for Practical Stochastic Calculus

Introduction to Brownian motion and its properties. Informal derivation of Itos formula. Connection with PDE through Ito and Feynman-Kac. Properties of diusions, calculation of transition densities via Kolmogorov equations, hitting times, mean rst passage times.

The course aims to introduce stochastic calculus from an applied viewpoint. The emphasis will be on doing calculations to determine the properties of particular models without discussion of the mathematical underpinings as covered in the SDEs course.

This material is covered in many books at different mathematical levels: The following will be at roughly the level of the course

1. Karlin and Taylor, A second course in stochastic processes
   
2. Grimmett and Stirzaker, Probability and random processes
   
3. Cox and Miller, The theory of stochastic processes
   

In the financial context some material is in:

1. Bjork, Arbitrage theory in continuous time
   
2. Etheridge, A course in financial calculus