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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 ( ξ n ) n ≥ 1 be independent random variables 19 with E ξ n = 0 for all n ≥ 1 . Then the process ( M n ) n ≥ defined via M n def = M + ξ 1 + ··· + ξ n is a martingale as long as the random variable M is independent of ( ξ n ) n ≥ 1 and E  M  < ∞ . Solution. Indeed by the triangle inequality, E  M n  ≤ E  M  + n X j =1 E  ξ j  < ∞ for all n ≥ 0, whereas the independence property implies E ` M n +1 M n  M n ,...,M ´ ≡ E ` ξ n +1  M n ,...,M ´ = E ξ n +1 = 0 . Remark 4.4.1. Notice that if E ξ n ≥ 0 for all n ≥ 1, then ( M n ) n ≥ is a sub Z martingale, whereas if E ξ n ≤ 0 for all n ≥ 1, then ( M n ) n ≥ is a supermartingale. More generally, if ( ξ n ) n ≥ 1 are independent random variables with E  ξ n  < ∞ for all n ≥ 1, then the process M n = M + ( ξ 1 − E ξ 1 ) + ... ( ξ n − E ξ 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 ξ n have the same distribution! 23 O.H. Stochastic Processes III/IV – MATH 3251/4091 M09 Example 4.5. If ( ξ n ) n ≥ 1 are independent random variables with E ξ n = 0 and E ( ξ n ) 2 < ∞ for all n ≥ 1 , then the process ( T n ) n ≥ defined via T n = ( M n ) 2 , where M n def = M + ξ 1 + ··· + ξ 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 ξ n +1 + ( ξ n +1 ) 2 and E ` 2 M n ξ n +1  M n ,...,M ´ = 2 M n E ( ξ n +1 ) = 0 , E ` ( ξ n +1 ) 2  M n ,...,M ´ = E ` ( ξ 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
 DUR
 Math, Probability theory, Markov chain, Stochastic Processes III/IV

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