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Chapter 6 Question 6.10 - Query

B

bonoloz

Member
Hi

I understand why we can subtract drift in order to arrive at a martingale in Solution 6.10, but am not clear as to why specifically 23t was subtracted from both sides of the equation stated.

Kindly assist me with this.
 
It was subtracted to make it a martingale. I hope you mean 2t also....

I don't have the notes infront of me as I am at work but the reason is something like this.

E[A_t | F_s ] = A_s + 2(t-s) (I think?) this is clearly not a martingale by definition. (We need E[A_t | F_s] = A_s for A_t to be a martingale which is not possible - so we need to do something to it.)

This is achieved by subtracting 2t from both sides, which can be seen by inspection or by working it out again using the definition of whatever distribution it followed again (uniform I think?).

So, E[A_t - 2t | F_s] = A_s - 2s ... which is clearly a martingale by definition.
 
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