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Stochastic Processes

by: Mafalda Ebert

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Stochastic Processes OR 645

Mafalda Ebert
Mason
GPA 3.9

John Shortle

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COURSE
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John Shortle
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PAGES
5
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KARMA
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Popular in Operations And Info Mgmt

This 5 page Class Notes was uploaded by Mafalda Ebert on Monday September 28, 2015. The Class Notes belongs to OR 645 at George Mason University taught by John Shortle in Fall. Since its upload, it has received 43 views. For similar materials see /class/215267/or-645-george-mason-university in Operations And Info Mgmt at George Mason University.

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Date Created: 09/28/15
OR 645 Stochastic Models 11 Lecture 11 Brownian Motion Given 1123 2004 Brownian Motion Symmetric Random Walk which in each time step is equally likelytotake a unit step tothe le or to the right It is a recurrent Markov Chain with 01 a o o jumps occur of size x at time Increments of 1 Let 1 If imjump is to the right or up wprobability 12 I 1 If imjump is tothe le or down wprobability 12 M Al 1 BmwnianMotjon Basic idea is to let xand Iget smaller and smaller We get Axt16Att16 BmwnianMO39iO M 64 At 164 Brownian Motion xtAx lxi x2quotl quotquot xtAtJ where x1 are thejumps Ax is the scaling factor and IAl isthe greatest integer that is f tAl aka oor of lAlquot Calculating Lecture by John Shortle transcribed by James LaBeIle based on the class textbook Ross 8 M 2003 Introduction to Probability Models 8 Academic Press Stochastic Models Ex 0 Varx Ex2k Ex2 1 Ext 0 Ex1 Ex2 ExtAt Varxt VarAx x1 xwm VarxtAc2 Varx1 xwm Varixw M Varixi Ax2t Varxt N Al Now suppose that we let Ax 0 and Al 0 then Varxl m M214 0 AI xt gt o w probabilityl This is the noninteresting trivial limit Now suppose an alternative Ax am where Ax 0 and At 0 Now we get 02At Al We are letting the step duration and the step size get smaller and smaller We started with Ax1 and At 1 Now our scaling 2 Varxtm I 0392t becomes AI Ax c 2 Atiand nally Axl and Ali 16 8 64 where we start with c 2 1 Then Ax1 and Atl Then further towardsthe limit of Axl and 2 4 4 M 8 AL M34 Brownian Motion xlAxxxllx Given that we have i 2 Mn in its limit xt Nmmazbl oyar 52 Brownian Motion Properties DEFN A stochastic process is said to be a x ii xll 2 0 has independent and stationary increments Independent increments what happens in two disjointed intervals is independent of each other ie jumps in one interval are independent ofjumps in another interval xt2 xt1 is independent of x5 x14 In short Axl is independent of Ax for all ij i j next gure Stationary increments no dependence on time This says that xt2 xt1 has same distr bution as 9512 t1 950 xl Normally 0Var 024 NOTE the variance increases with time NOTE xt is continuous everywhere and nondifferentiable everywhere Lecture by John Shortle transcribed by James LaBeIle based on the class textbook Ross 8 M 2003 Introduction to 21 Probability Modes 8m Academic Press Stochastic Models Brownian Mou39on AX 116At164 2 to VarYli2o2ll Therefore Yl is Standard Brownian Motion C Joint Distribution ofXt1 ampXtz Assume r1 lt 12 Time Ordering Prob Density XI1 x ltgt ME x Prob XIzx2 XIZXItx2 x1 where X11 Normal0l Xt2 Xt1 is independent of Xt1 by independent increments Xt2 Xt1 Normal0tz II by stationarity PXt1 xlet239 X01 x2 39 xi ii fry 2 e 2 Z W 27115 0 7 dwnt 2 8 27tht2 tl It s just the product of two Normal Distributions Density Function of X5 xlXz B for 5 lt2 Brownian M otion AX 116At164 4 y aLoA Given X1019375 X500625 Then EX9 B 193751744 I General Formulae for Brownian Motion EXsXz B 13 Lecture by John Shortle transcribed by James LaBeIle based on the class textbook Ross 8 M 2003 Introduction to 22 Probability Modes 8m Academic Press Stochastic Models VarXsXl B 51 s XSXl B Normal Example Have a stock follows Brownian Motion xl Normal 0021 for 4 If stock is up 10 a er 3 hours what is probability you will make a profit a er 6 hours Let xz measure incremental stock price change from time 0 First converting from 8M symbol is Xt to Standard BM symbol is Yt xt xl Y I C 4 Second Px6 gt 0x3 10 Writing this as an increment Px6 x3 gt 10x3 10 Then by independent increments Px6 x3gt 10 Variance of N03 is involves the duration according to a2t 123 standard deviation is P ampgt E PY6 Y3gt 4 4 4 2 5 N0 3 5 PNO3 gt P gt 2 J5 2 ii PN01 gt 2 Example In a bicycle race between two competitors let Yz denote the amount of time in seconds by which the racerthat started in the inside position is ahead when 100t percent of the race has been completed and suppose that Yt0 l 1 can be effectively modeled as a Brownian Motion process with variance parameter c 2 a If the inside racer is leading by c seconds at the midpoint ofthe race what is the probability that she is the winner PY1gt 0Y12 039 PY1 Y12 gt o Y12 039 PY1 Y12gt c by independent increments PY12gt c by stationary increments pgt z09213 where xPN01 x is the Standard Normal Distr bution Function b lfthe inside racer wins the race by a margin of c seconds what is the probability that she was ahead at the midpoint PY12gt 0Y1 039 PlNo392o3924gt 0 1z08413 Find fXSXtXlB density of fX5 given You x two end points at 0 amp at t What happens in the middle Lecture by John Shortle transcribed by James LaBeIle based on the class textbook Ross 8 M 2003 Introduction to 23 Probability Modes 8m Academic Press Stochastic Models Brownian M otion XSXt N2 t t Variance is zero at 0 origin and at I and greatest in the middle area Proof fXSlXtX n E f ltXBgt 2 Mr I 13403 fqum XIB Wm happened up W710 happened to time er time These two time intervals are independent N0S Nl0 15 fXg XtXB Does not depend on x T his is the normalizing constant K e l l 2 exp 205 K1 exp 712 18702 K1 exp 2 2275 2 l 1 Br 7 82 2K1 explf thk 2075 1 Br K2 epr L252tis 2 275 i K2 explli 25tistis 2 L i K2 exp 2 275 Lecture by John Shortle transcribed by James LaBeIle based on the class textbook Ross 8 M 2003 Introduction to 24 Probability Modes 8m Academic Press

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