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How to check these sequences generated by i.i.d. random variables are martingales?

Dhamnekar Winod

Active member
Nov 17, 2018
100
How to answer this question $\rightarrow$https://stats.stackexchange.com/q/398321/72126
Hello, I am reproducing the question in the hyperlink given in #1 of this thread so that the viewers of this MHB will conveniently read it.

Let $\{Y_n\}_{n\geq 1}$ be a sequence of independent, identically distributed random variables.

$P(Y_i=1)=P(Y_i=-1)=\frac12$

Set $S_0=0$ and $S_n=Y_1+...Y_n$ if $n\geq 1$

I want to check if the following sequences are Martingales.

$$M_n^{(1)}=\frac{e^{\theta S_n}}{(\cosh{\theta})^n}$$


$$M_n^{(2)}=\displaystyle\sum_{k=1}^n sign(S_{k-1})Y_k, n\geq 1,M_0^{(2)}=0$$


$$M_n^{(3)}=S_n^2 -n$$

I have no idea how to answer these questions. If any member knows the correct answers to these questions may reply with correct answers.
 

steep

Member
Dec 17, 2018
51
I'd like to see some work from you on these. The third one in particular is extremely common and easy. The first is pretty straightforward if you know how to manipulate MGFs to get a martingale. As for the second one, since $S_n$ is even iff n is even, it should be relatively straight forward to attempt as well.

If I were in your shoes, I would start by writing out definitions and applying them to the third one. My guess is that you have foundational issues related to conditional expectations that are holding you back.
 

Dhamnekar Winod

Active member
Nov 17, 2018
100
I'd like to see some work from you on these. The third one in particular is extremely common and easy. The first is pretty straightforward if you know how to manipulate MGFs to get a martingale. As for the second one, since $S_n$ is even iff n is even, it should be relatively straight forward to attempt as well.

If I were in your shoes, I would start by writing out definitions and applying them to the third one. My guess is that you have foundational issues related to conditional expectations that are holding you back.
Hello,
I am confident that you will reply to this thread because you are the knowledgable (expert) person in 'Advanced probability and statistics'.

Right now, what happened, I am studying Mathematics, Statistics, Quantitative Finance, R and Octave programming, MS Office (Online) etc. So I don't get any idea to answer these questions.

Please suggest me any reading material , educational videos useful for solving such types of questions.

I found these questions in 'Introduction to Quantitative Finance'. PDF. But the author didn't provide answers to these questions in that PDF.
 

steep

Member
Dec 17, 2018
51
Part of the challenge is that I don't know of an easy introduction to the subject. If I were in your shoes, I would first go through the conditional expectations chapter in the Blitzstein and Hwang book -- that will be useful for martingales and many other things

https://projects.iq.harvard.edu/stat110/home

I think Karlin and Taylor's "A First Course in Stochastic Processes" (try 2nd edition) has a decent chapter on martingales and some extremely good exercises at the end (in particular the 'elementary problems'). Ross's "Stochastic Processes" (2nd edition) has a decent chapter on them. If you want to go the long route, Williams' "Probability with Martingales" is recommended. All 3 of these are difficult books and the first two are typically used in graduate courses on stochastics, while Williams is sometimes grad, sometimes undergrad.

Learning martingales from any of those books are going to take an enormous amount of effort and time. It may well be that your time is better spent elsewhere, for the time being at least.

It certainly is possible that there is an accessible video collection on e.g. youtube that teaches martingales -- I just don't know of any such thing.