6.262.Lec25

# 6.262.Lec25 - 6.262 Lecture 25 Martingales Friday, May 7,...

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6.262 Lecture 25 Martingales Friday, May 7, 2010 1

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Review: Martingales Def.: A discrete-time stochastic process { Z n | n N } is a martingale if it satisﬁes: 1) E( | Z n | ) < for all n N . 2) E( Z n | Z n - 1 ,Z n - 2 ,...,Z 1 ) = Z n - 1 More generally: a discrete-time stochastic process { Z n | n N } is a martingale with respect to the process X 1 ,X 2 ,... if 2) is replaced by 2 0 ) E( Z n | X n - 1 n - 2 ,...,X 1 ) = Z n - 1 . Even more generally: { Z t | t 0 } is a martingale if . . . 2
A few easy properties If { Z n | z N } is a martingale, then E( Z n | Z i ,Z i - 1 ,...,Z 1 ) = Z i for i n . E( Z n | Z i 1 i 2 i k ) = Z i 1 for n i 1 i 2 ... i k 1. E( Z n | Z i ) = Z i for i n . E( Z n ) = E( Z 1 ). Proof. Iterated expectations. 3

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Review: Supermartingales and Submartingales Def.: A discrete-time stochastic process { Z n | n N } with E( Z n ) < for all n N is a: Submartingale if Z n E( Z n +1 | Z n ,Z n - 1 ,...,Z 1 ). Supermartingale if Z n E( Z n +1 | Z n n - 1 1 ). 4
Review: Convex Functions Def.: A function f : R R is convex if for every x,y R and t [0 , 1], f ( tx + (1 - t ) y ) tf ( x ) + (1 - t ) f ( y ) . xy tx + (1-t)y f f( tx + (1-t)y ) tf(x) + (1-t)f(y) 5

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If f is continuously diﬀerentiable, f is convex if it lies “on or above all its tangents”. f 6
Review: Jensen’s Inequality For a random varable X with expectation E( X ) and f convex, E( f ( X )) f (E( X )) . Mnemonic: var( X ) = E( X 2 ) - ( E ( X )) 2 0 = E( X 2 ) ( E ( X )) 2 . Corollary If { Z n | n N } is a martingale/submartingale, f is convex, and E( | f ( Z n ) | ) < , then { f ( Z n ) | n N } is a submartingale .

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## This note was uploaded on 10/21/2010 for the course EE 5581 taught by Professor Moon,j during the Spring '08 term at Minnesota.

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6.262.Lec25 - 6.262 Lecture 25 Martingales Friday, May 7,...

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