1. ## martingale proof

I need to do the proof for the second condition but I'm not getting the right answer.

$\displaystyle \{S_n , n\geq 1\}$} is a martingale if

1) E[|$\displaystyle S_n$|]< $\displaystyle \infty \forall n$
2) E[$\displaystyle S_{n+1}|S_1,...,S_n$] = $\displaystyle S_n \forall n$

$\displaystyle S_n$=$\displaystyle X_1X_2...X_n$

Suppose E[|$\displaystyle X_n$|] < $\displaystyle \infty \forall n$ and E[$\displaystyle X_n$]=1

Prove that $\displaystyle \{S_n , n\geq 1\}$} is a martingale

I can prove the first condition but for the second one I keep getting E[$\displaystyle S_{n+1}|S_1,...,S_n$] = $\displaystyle S_n+1$ which is wrong. Can anyone shine a light on this problem? Thanks

(ps sorry for any bad use of LaTex, this is the first time I have used it!)

2. Originally Posted by willowtree
I need to do the proof for the second condition but I'm not getting the right answer.

$\displaystyle \{S_n , n\geq 1\}$} is a martingale if

1) E[|$\displaystyle S_n$|]< $\displaystyle \infty \forall n$
2) E[$\displaystyle S_{n+1}|S_1,...,S_n$] = $\displaystyle S_n \forall n$

$\displaystyle S_n$=$\displaystyle X_1X_2...X_n$

Suppose E[|$\displaystyle X_n$|] < $\displaystyle \infty \forall n$ and E[$\displaystyle X_n$]=1

Prove that $\displaystyle \{S_n , n\geq 1\}$} is a martingale

I can prove the first condition but for the second one I keep getting E[$\displaystyle S_{n+1}|S_1,...,S_n$] = $\displaystyle S_n+1$ which is wrong. Can anyone shine a light on this problem? Thanks

(ps sorry for any bad use of LaTex, this is the first time I have used it!)
If this is a product and not a sum, then

E[$\displaystyle S_{n+1}|S_1,...,S_n] = E( X_1X_2\cdots X_{n+1}|S_1,...,S_n)=X_1X_2\cdots X_nE(X_{n+1}|S_n)$

and I'm guessing you need $\displaystyle E(X_1)=1$

YUP, I see it there.
I found your $\displaystyle E(X_1)=1$

$\displaystyle =X_1X_2\cdots X_nE(X_{n+1}) =S_n (1)=S_n$

3. ahh i seee, thank you, i was doing something a bit backwards with the stuff inside the brackets!