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CrashedMint
Topic Author
Posts: 2591
Joined: January 25th, 2008, 9:12 pm

### Easy Integral

What is the solution? I don't get it.

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Easy Integral

w(T)-w(0) ?

list
Posts: 2041
Joined: October 26th, 2005, 2:08 pm

### Easy Integral

it is a normal r.v. having m=0 the second moment could be calculated if you take into account that Ew(t) w(s) = min(s,t)

Alan
Posts: 10270
Joined: December 19th, 2001, 4:01 am
Location: California
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### Easy Integral

Yes, list is correct.This is a FAQ that never was ... ahem

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Easy Integral

so w_t doesn't mean (dw/dt) here ?

Alan
Posts: 10270
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Easy Integral

No, the OP will amost certainly say it means W(t), a Brownian motion.

Cuchulainn
Posts: 62626
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Easy Integral

dbl
Last edited by Cuchulainn on November 11th, 2008, 11:00 pm, edited 1 time in total.
Step over the gap, not into it. Watch the space between platform and train.
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Cuchulainn
Posts: 62626
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Easy Integral

QuoteOriginally posted by: ppauperso w_t doesn't mean (dw/dt) here ?When I first say SDEs I thought the same. It is very confusing indeed. They mean w(t). I went to a training once and it was into day 2 that I realised what it was (trainer assumed everyone's background was the same).I reckon X% of writers use the first notation and (100-X)% use the other notation.
Last edited by Cuchulainn on November 11th, 2008, 11:00 pm, edited 1 time in total.
Step over the gap, not into it. Watch the space between platform and train.
http://www.datasimfinancial.com
http://www.datasim.nl

CrashedMint
Topic Author
Posts: 2591
Joined: January 25th, 2008, 9:12 pm

### Easy Integral

Umm, actually i meant to ask for the integral of a standard wiener process with respect to time. I found this as a question in Crack's book, but his answer is like 2 pages long. Can that be? should this be something very simple with a very simple solution like w(T)?

list
Posts: 2041
Joined: October 26th, 2005, 2:08 pm

### Easy Integral

integrate by parts and you get T W(T) - int from 0 to T " tdw(t)" if tis you like it better.

CrashedMint
Topic Author
Posts: 2591
Joined: January 25th, 2008, 9:12 pm

### Easy Integral

like this?

mj
Posts: 3449
Joined: December 20th, 2001, 12:32 pm

### Easy Integral

it's a linear combination of normals with mean zero so it's a normal with mean zero. You just have to compute the variance. One easy way to do it is just to approximate by a sum of N pieces and let N go to infinity.The answer is not W_T.
Last edited by mj on November 12th, 2008, 11:00 pm, edited 1 time in total.

phil451
Posts: 83
Joined: December 7th, 2007, 8:21 am

### Easy Integral

This is a Stochastic Integral which cannot be solved by standard deterministic methods. If you want to get serious about financial engineering then you really need to learn how this works. Unfortunately, there is no easy solution and you really have to thinkThe approach to solving it is similar to solving a deterministic integral using Rienmann sums. So, you have to understand the concept of summing with letting n tend to infinity.The chapter on Stochastic Integration in Paul Wilmott's Introduction to Quantitative Finance will help to get you started with Brownian Motion, particularly the coin tossing example and then i would suggest 'Kwok' which has just been re-issued.Good Luck!

Jeriot
Posts: 97
Joined: January 16th, 2007, 11:54 am

### Easy Integral

Thanks mint, that made me feel a lot better. I don't get it either. Didn't get it when I took that module, still don't get it now.If I do it as the gurus suggested, thenAlas, that's the limit of my understanding. I presume since we do not have knowledge of w_t values past t=0, we would not be able to come up with a value for the integral but we know the expected value would be zero.I'd like to know what the solution is tho, if you could scan that 2 page answer or something. Don't wanna remain ditzy all my life

mj
Posts: 3449
Joined: December 20th, 2001, 12:32 pm

### Easy Integral

it's a sum of normals so it's normal. We know it's mean so all we need is the variance. This is not hard to compute if we first compute the covariance of w_i and w_j for i \neq j.

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