SERVING THE QUANTITATIVE FINANCE COMMUNITY

JSHellen
Topic Author
Posts: 13
Joined: July 10th, 2013, 6:42 am

### The Mean Square Limit

Hello everyone,This is officially my first post here!I'm going to start my master studies later this year but I decided to take a head start and read some interesting text books in summer. So it has been almost two years since I completed my calculus courses. I started to skim through and study Wilmott s. QF Vol 1-3 2nd Ed. but it seems that I've forgotten/missed some fundamental properties of finite series. On the page 77 (Chapter 4 (4.9)) the mean square limit is being expanded. I don't quite understand what happens there because this is how I do the expansion and it does not look quite right: my manipulation does not have a double summation.$E[(\sum_{j=1}^{n}(X(t_j)-X(t_{j-1}))^2 - t)^2]$ (1)$\sum_{j=1}^{n}(X(t_j)-X(t_{j-1}))^2 = a,t=b$Now if I substitute a and b back to equation (1) I get this:$E[(a-b)^2]=E[a^2-2ab+b^2]=E[\color{blue} {(\sum_{j=1}^{n}(\Delta X)^2)^2} \color{black} -2*t*\sum_{j=1}^{n}(\Delta X)^2+t^2]$This is what it should be:$E[\color{purple}{(\sum_{j=1}^{n}(\Delta X)^4 + \sum_{i=1}^{n}\sum_{j<i}^{ }(X(t_i)-X(t_{i-1})^2(X(t_j)-X(t_{j-1}))^2}-2*t*\sum_{j=1}^{n}(\Delta X)^2+t^2]$ Thanks
Last edited by JSHellen on July 9th, 2013, 10:00 pm, edited 1 time in total.

rolandograndi
Posts: 105
Joined: December 21st, 2012, 7:25 pm

### The Mean Square Limit

Hi guys,I'm also having trouble with the expansion.Can anyone help us?

Ultraviolet
Posts: 1655
Joined: August 15th, 2012, 9:46 am

### The Mean Square Limit

QuoteOriginally posted by: JSHellenHello everyone,This is officially my first post here!I'm going to start my master studies later this year but I decided to take a head start and read some interesting text books in summer. So it has been almost two years since I completed my calculus courses. I started to skim through and study Wilmott s. QF Vol 1-3 2nd Ed. but it seems that I've forgotten/missed some fundamental properties of finite series. On the page 77 (Chapter 4 (4.9)) the mean square limit is being expanded. I don't quite understand what happens there because this is how I do the expansion and it does not look quite right: my manipulation does not have a double summation.$E[(\sum_{j=1}^{n}(X(t_j)-X(t_{j-1}))^2 - t)^2]$ (1)$\sum_{j=1}^{n}(X(t_j)-X(t_{j-1}))^2 = a,t=b$Now if I substitute a and b back to equation (1) I get this:$E[(a-b)^2]=E[a^2-2ab+b^2]=E[\color{blue} {(\sum_{j=1}^{n}(\Delta X)^2)^2} \color{black} -2*t*\sum_{j=1}^{n}(\Delta X)^2+t^2]$This is what it should be:$E[\color{purple}{(\sum_{j=1}^{n}(\Delta X)^4 + \sum_{i=1}^{n}\sum_{j<i}^{ }(X(t_i)-X(t_{i-1})^2(X(t_j)-X(t_{j-1}))^2}-2*t*\sum_{j=1}^{n}(\Delta X)^2+t^2]$ Thanks$a^2 = (\sum_{j=1}^{n}(X(t_j)-X(t_{j-1}))^2)^2 = \sum_{j=1}^{n}(X(t_j)-X(t_{j-1}))^2 \sum_{i=1}^{n}(X(t_i)-X(t_{i-1}))^2$$=\sum_{j=1 (i=j)}^{n}(X(t_j)-X(t_{j-1}))^4 + \sum_{j=1,i\neq j}^{n}(X(t_j)-X(t_{j-1}))^2 (X(t_i)-X(t_{i-1}))^2$$= \sum_{j=1 (i=j)}^{n}(X(t_j)-X(t_{j-1}))^4 + 2 \sum_{j=1,i< j}^{n}(X(t_j)-X(t_{j-1}))^2 (X(t_i)-X(t_{i-1}))^2$I'm guessing that you incorrectly assumed that $a^2 = \sum_{j=1 (i=j)}^{n}(X(t_j)-X(t_{j-1}))^4$ forgetting about $i\neq j$.

rolandograndi
Posts: 105
Joined: December 21st, 2012, 7:25 pm

### The Mean Square Limit

Hi Ultraviolet!Thanks for the help, it is clear for me now! Have a nice sunday!

EZNeo
Posts: 1
Joined: September 7th, 2014, 3:33 pm

### The Mean Square Limit

I've also got an issue with the second part when taking the expectation of delat X ^4. Can anyone elaborate?