# Applied Stochastic Methods -- 2940

## Textbook

An Introduction to Stochastic Processes with Applications to Biology.

Linda Allen

The grades are based on the homework which will be given roughly weekly.

## Overview

1. Review of Probability Theory and Introductions (Chapter 1)
2. Discrete time Markov Chains and applications (Chapt 2,3)
3. Continuous Time Markov Chains (Chapt 5)
4. Continuous time Birth and Death processes and applications (Chapt 6,7)
5. Diffusion processes and stochastic differential equations (8,9)

## Detailed Syllabus

• Week 1 Jan 8-10. 1.2-1.4 HW#1: 1.7:1,5,9,10,21,27 and also this Extra stuff
• Week 2 Jan 17 (No class Jan 15 MLK day) 1.5-1.7,some 2.2 maybe
• Week 3 Jan 22-24 2.2-2.6 HOMEWORK #1 Due Jan 24
• Week 4 Jan 29-32 more section 2 HW#2 Chapter 2 p84-92, 5,12,13,17,18,26, and this additional problem:

Consider a random walk on the vertices of a triangle. (a) let the moves be from one vertex to another with prob 1/2 (p12=p21=p13=p23=p31=p32=1/2) Find the prob that in n steps wou return to the vertex you started from; is every state recurrent? ; if so, compute the stationary distribution. (b) Same thing but now, p12=p23=p31=2/3, p21=p32=p13=1/3. (Hint: The transition matrix P is a circulant matrix, so it can be diagonalized by the matrix S:

S = (1/sqrt(3)) [1 1 1; 1 w wbar;1 wbar,w]

where w=exp(2 pi i/3). Note that S^(-1)=S^*. P^n = S D^n S^* where D is the matrix of eigenvalues.)

• Homework 2 due on Wed Feb 14
• Week of Feb 12-14 - Finish Chapter 2, do Chapter 5 Homework number 3 Due on Feb 28
• Weeks of Feb 19-28 Chapter 5, 6, maybe some Chapter 7
• Class cancelled March 12
• March 14 Chapter 6, 7.3, 7.4
• Homework 4 Due Mar 28 Problems 1,2,8,16,21,22 in Chapter 6 of the book
• Mar 19 Finish chapter 7 Wednesday Mar 21 Class cancelled - due to special panel discussion with Martin Nowak - you dhould all go. Its 2 PM-4 PM in the Ohara student center
• Mar 26-28 Chapters 8.1-8.4 Allen
• April 2-4 Chapter 8 finish
• HOMEWORK 5 Due April 18