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lect0423 - IEOR 4106 Introduction to Operations Research...

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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 .
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