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Unformatted text preview: 1.4. MARTINGALES 31 1.4 Martingales We first consider a simple example in analysis. Let f be an integrable function on [0 , 1], let P n = { 0 = 1 2 n ≤ ··· ≤ k 2 n ··· ≤ 1 } be a partition of [0 , 1] and let I n,k = [ k 2 n , k +1 2 n ). We define the average function f n of f on the partition P n : f n ( x ) = 2 n 1 X k =0 a n,k χ I n,k , x ∈ I n,k . (1.4.1) where a n,k = 1  I n,k  R I n,k f ( x ) dx. Then { f n } n converges to f in L 1 . Moreover { f n } n has the following consistency property: for m > n f n ( x ) = 1  I n,k  Z I n,k f m ( y ) dy x ∈ I n,k . (1.4.2) This property has been reformulated by Doob in the more general probability setting. Definition 1.4.1. Let { ( X n , F n ) } ∞ n =1 be a sequence of r.v. such that X n ∈ F n . It is called a martingale if (a) F n ⊂ F n +1 ; (b) E (  X n  ) < ∞ ; (c) X n = E ( X n +1 F ) . It is called a supermartingale (or submartingale) if ≥ (or ≤ respectively) in (c) holds. We will call { X n } n a smartingale if it is any one of the three cases. Condition (c) can be strengthened as X n = E ( X m F n ) for m > n . It follows from E ( X m F n ) = E ( E ( X m F m 1 ) F n ) = E ( X m 1 F n ) = ··· = E ( X n F n ) = X n . Martingale has its intuitive background in gambling. If X n is interpreted as the gambler’s capital at time n , then the defining property says that his 32 CHAPTER 1. BASIC PROBABILITY THEORY expected capital after next game, played with the knowledge of the entire past and present, is exactly equal to his current capital. In other words, his expected gain is zero, and is in this sense the game is said to be “fair”. The supermartingale and submartingale can be interpreted similarly. Example 1 . As a direct analog of the above function case, we let X be an integrable r.v. and let {F n } ∞ n =1 be an increasing sequence of sub σfields (e.g., take F n to be a partition). Let X n = E ( X F n ). Then { X n } ∞ n =1 is a martingale. Indeed we see that E (  X n  ) = E (  E ( X F n )  ) ≤ E ( E (  X F n )) = E (  X  ) < ∞ and (b) follows. For (c), we observe that E ( X n +1 F n ) = E ( E ( X F n +1 ) F n ) = E ( X F n ) = X n . Example 2 . Let { X n } ∞ n =1 be a sequence of independent integrable r.v. with mean zero. Let S n = ∑ n j =1 X n and F n = F ( X 1 , ··· ,X n ) . Then E ( S n +1 F n ) = E ( S n + X n +1 F n ) = S n + E ( X n +1 F n ) = S n + E ( X n +1 ) = S n . Hence { ( S n , F n ) } is a martingale. Proposition 1.4.2. If { ( X n , F n ) } ∞ n =1 is a submartingale, and ϕ is increas ing and convex in R . If { ϕ ( X n ) } is integrable, then { ( ϕ ( X n ) , F n ) } is also a submartingale....
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 Spring '08
 CHEUNG,CECILIA
 Calculus

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