Consider the random variable where each Bi is obtained as a result of the toss of a fair coin We let n 4 and consider X4 (a) Calculate the E X4 I1 ,E X4 I2 , E X4 I4 (b) Let Zi E X4 Ii Is Zi, i 1, , 4 a martingale (c) Now define Vi Bi i and Is Vi a martingale (d) Can you convert Vi into a martingale by an appropriate transformation (e) Can you convert Vi into a martingale by changing the probabilities associated with a coin toss Pt i 1 1 Head 1 Tail

Question: Consider the random variable: where each Bi is obtained as a result of the toss of a fair coin: We let n = 4 and

Consider the random variable:

where each Bi is obtained as a result of the toss of a fair coin:

We let n = 4 and consider X4.
(a) Calculate the E[X4| I1] ,E[X4| I2] , E[X4| I4].
(b) Let
Zi = E[X4| Ii]
Is Zi, i = 1, . . . , 4 a martingale?
(c) Now define:
Vi = Bi + ˆši
and

Is Vi a martingale?
(d) Can you convert Vi into a martingale by an appropriate transformation?
(e) Can you convert Vi into a martingale by changing the probabilities associated with a coin toss?

Pt i=1 +1 Head 1 Tail

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