Tyszui


Total Posts: 7 
Joined: Dec 2013 


Hi all,
very quick question: have you ever come across a simple jump diffusion model like Merton for which dW (the Brownian0 and dN (ie. the Poisson) are correlated?
Any paper reference is very much appreciated.
Thanks a lot 



pj


Total Posts: 3324 
Joined: Jun 2004 


Hi Tyszui, Brownian and Poisson processes are always uncorrelated.
check here I think there is a theorem about that in Revuz Yor book 
The older I grow, the more I distrust the familiar doctrine that age brings wisdom
Henry L. Mencken 


>Brownian and Poisson processes are always uncorrelated. Really? I don't think so. Finally, we can make be dependent on . Just by program code of a montecarlo simulation.
>have you ever come across a simple jump diffusion model like Merton for which dW >(the Brownian0 and dN (ie. the Poisson) are correlated?
Anyway, during long years of practice I learnt to start not from a mathematical model but from the data themselves. So why do you need them to be dependent? I would probably make Poisson part dependent on drift (not on noise). At least in equity markets it is a stylized fact that the stocks often drop after an exuberant growth and vice versa, a downtrend is often (partially) compensated by a jumpup. 
www.yetanotherquant.com  Knowledge rather than Hope: A Book for Retail Investors and Mathematical Finance Students 


deeds


Total Posts: 347 
Joined: Dec 2008 


" I learnt to start not from a mathematical model but from the data themselves."
finanzmaster, quote illuminates the different responses from you and pj.
There's a whole discussion there, but you do seem to be saying something with respect to a model assumption of 'Poisson process' (which has a definition that doesn't involve code)
Maybe refine the initial thought (of not so) to say that we are making a model assumption that is an extension of Poisson (compound, time dependent parameters, or...)?



pj


Total Posts: 3324 
Joined: Jun 2004 


"The important thing is not whether you are correct but whether you are right"
Jumps and Brownian part may be correlated, no problem with that. Just they won't be classical Poisson and Wiener.

The older I grow, the more I distrust the familiar doctrine that age brings wisdom
Henry L. Mencken 


Cheng


Total Posts: 2831 
Joined: Feb 2005 


The question has been answered on StackExchange (quoted below):
"In case (W,N) has independent increments, then W and N are also independent, since a Brownian Motion has no common jumps with the Poisson process. This of course doesn't say, there is no probability space, where (W,N) has dependent increments, but it gives you a hint, how it might be constructed." 
"He's man, he's a kid / Wanna bang with you / Headbanging man" (Grave Digger, Headbanging Man) 

 
deeds


Total Posts: 347 
Joined: Dec 2008 


cheng  agree...(and thanks for the assist! Doubling down on the pedant prize, we could distinguish correlation and dependence)
pj  agree, agree
finanzmaster  imho empiricism and experiments are essential, underlying assumptions can be very helpful in organizing results and for other related reasoning, too


