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Unformatted text preview: O.H. Stochastic Processes III/IV MATH 3251/4091 M09 4 Martingales 4.1 Definition and some examples A martingale is a generalized version of a fair game. Definition 4.1. A process ( M n ) n is a martingale if b for every n the expectation E M n is finite, equivalently, E  M n  < ; for every n and all m n , m n 1 , . . . , m we have E ( M n +1  M n = m n ,...,M = m ) = m n . (4.1) Of course, (4.2) can just be written as E ( M n +1  M n ,...,M ) = M n . Definition 4.2. We say that ( M n ) n is a supermartingale 18 if the equality in b (4.2) holds with , ie., E ( M n +1  M n ,...,M ) M n ; and we say that ( M n ) n is a submartingale , if (4.2) holds with , ie., E ( M n +1  M n ,...,M ) M n . Definition 4.3. A process ( M n ) n is a martingale w.r.t. a sequence ( X n ) n b of random variables, if for every n the expectation E M n is finite, equivalently, E  M n  < ; for every n and all x n , x n 1 , . . . , x we have E ( M n +1 M n  X n = x n ,...,X = x ) E ( M n +1 M n  X n ,...,X ) = 0 . (4.2) Example 4.4. Let ( n ) n 1 be independent random variables 19 with E n = 0 for all n 1 . Then the process ( M n ) n defined via M n def = M + 1 + + n is a martingale as long as the random variable M is independent of ( n ) n 1 and E  M  < . Solution. Indeed by the triangle inequality, E  M n  E  M  + n X j =1 E  j  < for all n 0, whereas the independence property implies E ` M n +1 M n  M n ,...,M E ` n +1  M n ,...,M = E n +1 = 0 . Remark 4.4.1. Notice that if E n 0 for all n 1, then ( M n ) n is a sub Z martingale, whereas if E n 0 for all n 1, then ( M n ) n is a supermartingale. More generally, if ( n ) n 1 are independent random variables with E  n  < for all n 1, then the process M n = M + ( 1 E 1 ) + ... ( n E n ), n 0, is a martingale. 18 If M n traces your fortune, then there is nothing super about a supermartingale . 19 Notice that we do not assume that all n have the same distribution! 23 O.H. Stochastic Processes III/IV MATH 3251/4091 M09 Example 4.5. If ( n ) n 1 are independent random variables with E n = 0 and E ( n ) 2 < for all n 1 , then the process ( T n ) n defined via T n = ( M n ) 2 , where M n def = M + 1 + + n , is a martingale w.r.t. ( M n ) n . Solution. By the Cauchy inequality `P n j =1 a j 2 n P n j =1 ( a j ) 2 , we obviously have E T n < for all n 0. Now, because T n +1 T n = 2 M n n +1 + ( n +1 ) 2 and E ` 2 M n n +1  M n ,...,M = 2 M n E ( n +1 ) = 0 , E ` ( n +1 ) 2  M n ,...,M = E ` ( n +1 ) 2 , we get E ` T n +1 T n  M n ,...,M 0, ie., the process ( T n ) n is a submartingale....
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 Fall '07
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