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?

Transcribed Image Text:

Pt i=1 +1 Head 1 Tail