Stochastic Processes and Stochastic Calculus II

Department: 
MATH
Course Number: 
7245
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Every spring semester

An introduction to the Ito stochastic calculus and stochastic differential equations through a development of continuous-time martingales and Markov processes. (2nd of two courses in sequence)

Prerequisites: 

MATH 7244 or equivalent

Course Text: 

At the level of Karatzas and Shreve, Brownian Motion and Stochastic Calculus

Topic Outline: 
  • Additional examples and applications for Ito stochastic integrals and Ito's rule for change-of-variables
  • Representation theorems for continuous martingales in terms of Brownian motion
  • Girsanov transformations, theory and applications
  • Local time, including definitions and elementary properties, and connections to reflected Brownian motion, generalized Ito rules, and extensions
  • Stochastic differential equations -- strong solutions, including definitions and basic existence, uniqueness, comparison, and approximation results
  • Stochastic differential equations -- weak solutions, including definitions, basic existence and uniqueness, and connections to the martingale problem of Stroock and Varadhan