Suppose X1, X2,... are independent random variables with 1 1 P{X;= -1} = 2' P{X; =...

 

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Suppose X1, X2,... are independent random variables with 1 1 P{X;= -1} = 2' P{X; = 1} = | P{X; = 0} = P{X; = Let S 0 and Sn - 4 X++X. Let Fr denote the information contained in X1, ..., Xn. 1. Which of these is S: martingale, submartingale, supermartingale (more than one answer is possible)? 2. For which values of r is Yn := Sn rn a martingale? 3. For what u >1 is it true that if Mn := us, then Mn is a martingale? For the remainder of this exercise, use this value of u. 4. Is {M} a square integrable martingale? 5. Find 6. Find 20 (M; My-1)] . (; j=1 - Mj E[SF] E[M0 | F5]. Suppose X1, X2,... are independent random variables with 1 1 P{X;= -1} = 2' P{X; = 1} = | P{X; = 0} = P{X; = Let S 0 and Sn - 4 X++X. Let Fr denote the information contained in X1, ..., Xn. 1. Which of these is S: martingale, submartingale, supermartingale (more than one answer is possible)? 2. For which values of r is Yn := Sn rn a martingale? 3. For what u >1 is it true that if Mn := us, then Mn is a martingale? For the remainder of this exercise, use this value of u. 4. Is {M} a square integrable martingale? 5. Find 6. Find 20 (M; My-1)] . (; j=1 - Mj E[SF] E[M0 | F5].