Martingales - Explaination.

Hi all!

I've got a problem with a question at the moment and i was wondering if you could clarify it? Thanks =)

Let $\displaystyle X_{n}$ = $\displaystyle X_{0}$ + $\displaystyle \sum^n_{i=1}$$\displaystyle Y_{i}$ be a random walk.

Suppose that $\displaystyle t^*$ > 0 such that $\displaystyle m(t^*) = 1$ where $\displaystyle m(t)=e^{tY_{i}}$ is the moment generating function of $\displaystyle Y_{i}$.

Show $\displaystyle e^{t^*X_{n}}$

The property of martingale:

$\displaystyle E|X_{n}| < \infty$

$\displaystyle E(X_{n+1}|X_{1},X_{1}...X_{n})$

so show that : $\displaystyle E(X_{n+1}|X_{1},X_{2}...X_{n})$ = $\displaystyle e^{t^*X_{n}}$

Now $\displaystyle E(e^{t^*(X_{0}+Y_{1}+...+Y_{n}} |X_{1}...X_{n})$

From that it becomes

$\displaystyle E(e^{t^*X_{0}+t^*Y_{1}...+t^*Y_{i}}|X_{1}...X_{n})$

Now the one thing i dont get is that the equation then becomes:

$\displaystyle E(e^{t^*X_{n}}e^{Y_{i}}|X_{1}...X_{n})$

=$\displaystyle e^{t^*X_{n}}E(e^{Y_{i}}|X_{1}...X_{n})$

why from suddenly $\displaystyle e^{t^*X_{0}}$ becomes $\displaystyle e^{t^*X_{n}}$?

Any explainations would be great. Thank you.

=)