Admistrivia

• Readings: Chapter 3 up to page 126
• Questions from the homework?

Chapter III.4: First step analysis

• Suppose we start in state 0. What is probability of ever leaving?
• Suppose we start in state 2. What is probability of ever leaving?
• Called absorbing.
• Suppose we start in state 1. What is probability of entering state 0 or state 2?

Where do we go?

• T = stopping time (absorbtion time)
• u = P(X_T= 0|X₀ = 1)
• (Martingale approach:)
  • u = P(X_infinity= 0|X₀ = 1)
  • E(I(X_infinity)X₀)) = E(E(I(X_infinity)|X₁,X₀)))) = ...
  • Called first step analysis
• Induction: Suppose we knew it from time 1 on. Concrete first step analsyis.

When do we get there?

• T = stopping time (absorbtion time)
• v = E(T|X₀ = 1)
• same methodology

Harder: 4 states

• What if there are two transiant states?
• There are now two probabilities of absorbing in state 0
• There are now two waiting times
• Same recursion:

  1	0	0	0
  .1	.8	.1	0
  0	.3	.6	.1
  0	0	0	1

Which do we come to first?