# Lectur25 - Lecture XXV Brownian Motion Itos Lemma and...

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Lecture XXV Brownian Motion, Ito’s Lemma and Stochastic Optimal Control I. Introduction to Stochastic Process. A. Most of the material in the section originates from Dixit, Avinash The Art of Smooth Pasting (Buffalo, NY: Gordon and Breach Publishing Co., 1993). The publisher can be reached on the network at http://www.gbhap.com/. B. Brownian Motion 1. Brownian motion refers to a continuous time stochastic process where x t evolves over time according to some stochastic differential equation: dx dt dw =+ μσ where dt represents an increment in time and dw denotes a random component. The expected value of x t given an initial condition x 0 becomes x 0 + μ t with a variance of σ 2 t. 2. Random Walk Representation: To make the formulation more concrete, we will begin by depicting x as a random walk with a probability, p, that the value of x will increase h at any given time period and a probability (1-p) that the value of x will decrease by h in the same time period. A graphical representation for the value of x can be depicted as: p p 2 p 3 p 4 (1-p) (1-p) 2 2p(1-p) 3p(1-p) 2 3p 2 (1-p) 4p 3 (1-p) 6p 2 (1-p) 2 4p(1-p) 3 (1-p) 3 (1-p) 4 x 0 x 0 + h x 0 +2 h x 0 +4 h x 0 +3 h x 0 - h x 0 -2 h x 0 -3 h x 0 -4 h a. The expected value of x is: [] E x ph p h ph qh p q h q p ∆∆ - - =-= - = - () ( ) ( ) 11 such that Similarly, the variance of x is:

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## This note was uploaded on 11/08/2011 for the course AEB 6533 taught by Professor Moss during the Fall '08 term at University of Florida.

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Lectur25 - Lecture XXV Brownian Motion Itos Lemma and...

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