Martingales - Explaination.

• April 24th 2010, 01:40 AM
Redeemer_Pie
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 $X_{n}$ = $X_{0}$ + $\sum^n_{i=1}$ $Y_{i}$ be a random walk.

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

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

The property of martingale:

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

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

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

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

From that it becomes

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

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

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

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

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

Any explainations would be great. Thank you.

=)
• April 24th 2010, 02:25 AM
Moo
Hello,

Because $X_n=X_0+Y_1+\dots+Y_n$ ? :D

Quote:

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

From that it becomes

$E(e^{t^*X_{0}+t^*Y_{1}...+t^*Y_{\color{red}n+1}}|X _{1}...X_{n})$

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

$E(e^{t^*X_{n}}e^{{\color{red}t^*}Y_{\color{red}n+1 }}|X_{1}...X_{n})$

= $e^{t^*X_{n}}E(e^{{\color{red}t^*}Y_{\color{red}n+1 }}|X_{1}...X_{n})$
I think there are typos (in red)
• April 24th 2010, 02:34 AM
Redeemer_Pie
Oh k. Thanks. lol that was a bit embarassing but i forgot to add:

I thought via martingale property that it shouldnt we be solving for:

$E(e^{t^*X_{n+1}}|X_{1}...X_{n})$

rather than:

$E(e^{t^*X_{n}}|X_{1}...X_{n})$?
• April 24th 2010, 02:36 AM
Redeemer_Pie
Quote:

Originally Posted by Moo
Hello,

Because $X_n=X_0+Y_1+\dots+Y_n$ ? :D

I think there are typos (in red)

yep! sorry they meant to say $Y_{i}$

Sorry my first time using LaTeX
• April 24th 2010, 02:57 AM
Moo
I don't understand... Do you have any further question ? oO
• April 24th 2010, 07:58 PM
Redeemer_Pie
nope that'll be all thanks :)