1. Let (Q.3,P) be a probability space and let the two sequences (Xn)NEN and (y,)

Question

1. Let (Q.3,P) be a probability space and let the two sequences (Xn)NEN and (y,)neN be independent. Define Zn XnYn for all neN Let If (Xn, Sn)meN and (Yn,MnN are martingales, is (Zn, Fn)nEN a martingale? Provide the details of your argument 2. Let (Q, ,P) be a probability space. Define Zn XnYn for all neN Let If (Xn, Sn)meN and (Yn,MnN are martingales, is (Zn, Fn)nEN a martingale? Provide the details of your argument 3. Let (S2,F, Fn, P) be a filtered probability space and let the two sequences (Xn) nen and (Yn)neN be independent. Define Zn XnYn for all neN If (Xn, Fn)neN and (Yn, F)neN are martingales, is (Zn, n)nEN a martingale? Provide the details of your argument

Explanation / Answer

The sequences Xn and Yn are martingales.

Zn=Xn* Yn may or may not be martingale and thatdepneds on the hypothesis. It can be considered martingale if they are orthogonal.

But to answer the question

1) In general is not a martingale.

2) this is the same question as a.

3) After applying Filtration also, Zn may not be a matingale.

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