R help - Nov 2006 - Generating a double-exponential jump diffusion process

Dear R Users,

Does anyone know of  a package which can generate random realisations of 
a double-exponential jump diffusion process with a drift ? Something 
where I can specify the likelihoods of an up or a down jump, the drift 
rate, and the mean size, and get back a vector of realisation of the 
process (for purposes of a Monte-Carlo).

Kind regards,
Tolga

Tolga Uzuner <tolga <at> coubros.com> writes:
> 
> Dear R Users,
> 
> Does anyone know of  a package which can generate random realisations of 
> a double-exponential jump diffusion process with a drift ? Something 
> where I can specify the likelihoods of an up or a down jump, the drift 
> rate, and the mean size, and get back a vector of realisation of the 
> process (for purposes of a Monte-Carlo).
> 
  Hmmm. I'm not exactly sure what you mean (this is one of the things
to remember about the R list -- there is a vast variety of readers,
most of whom aren't in your field, so the only things you can assume
we all know are basic statistics and stuff about R).  By "drift"
do you mean that likelihood of up is not equal to likelihood of down?
I can imagine that it would be easy to generate such a realisation
(without a particular package), if you could give us the precise definition.
As a starting example:

jumpdiff <- function(start=0,nt=100,up=0.5,scale=1) {
   res <- numeric(nt)
   res[1] <- start
   for (i in 2:nt) {
     dir <- if (runif(1)<up) 1 else -1
     res[i+1] <- res[i]+dir*rexp(1,scale=scale)
   }
}

  I wrote this out as a for loop to make it clearer: a much
more efficient way would be

dir <- ifelse(runif(nt)<up,1,-1)
jump <- rexp(nt,scale=scale)
cumsum(start+dir*jump)

  hope that helps

    Ben Bolker

On Sun, Nov 05, 2006 at 11:13:28AM +0000, Tolga Uzuner
wrote:
> Dear R Users,
> 
> Does anyone know of  a package which can generate random realisations of 
> a double-exponential jump diffusion process with a drift ? Something 
> where I can specify the likelihoods of an up or a down jump, the drift 
> rate, and the mean size, and get back a vector of realisation of the 
> process (for purposes of a Monte-Carlo).
Tolga,

I am not really an expert in this field, but AFAIK the likelihood of
Ito/Levy processes does not have closed form solutions apart from a
few well-known exceptions.  People use simulation (Euler method and
variants) to obtain simulated values and likelihoods.  For the latter,
a method called High Frequency Augmentation is used.  Some references
are Jones, C S (1998, 1999), Elerian (1998), Elerian, Chib and
Shephard (1998), Eraker (1997), most of them are about diffusions, but
I guess that they can be adapted to Levy processes too.

HTH,

Tamas