lect0423 - IEOR 4106: Introduction to Operations Research:...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2009, Professor Whitt Topics for Discussion: Thursday, April 23 Introduction to Brownian Motion and Martingales In Ross, read Sections 10.1-10.3 and 10.6. (The total required reading is approximately 11 pages.) I. BM Basics. We discussed the material in the book in Sections 10.1-10.3. In particular, we started with the definition of Brownian motion, and then we elaborated upon the finite-dimensional distributions of Brownian motion, as given in (10.3) on page 628, the conditional density at time s given the value at a later time t , as given in the display in the middle of page 628 and equation (10.4) there, the reflection principle , as used to get the distribution of the hitting time T a in (10.6) on page 630, Brownian motion with drift , as in Section 10.3.1, and Brownian motion with three parameters: X ( t ) X (0) + μt + σB ( t ) , where B ≡ { B ( t ) : t 0 } is standard Brownian motion (with 0 drift and variance parameter 1), which has distribution N ( X (0) + μt,σ 2 t ) at time t . Then X (0) is the initial value, μ is the drift rate, and σ 2 is the variance parameter. The displayed stochastic process { X ( t ) : t 0 } might be used for an additive model of stock prices, approximating the initial random walk in the beginning of the last lecture notes. The general Geometric Brownian motion analog would be Y ( t ) = e X ( t ) , t 0 . It could be used to approximate the multiplicative random walk model, discussed below.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/26/2011 for the course IEOR 4106 taught by Professor Whitt during the Spring '08 term at Columbia.

Page1 / 3

lect0423 - IEOR 4106: Introduction to Operations Research:...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online