In article
<9ff20852-9505-4cb0-b54f-3af60306be99@[EMAIL PROTECTED]
>,
<excellentfeng@[EMAIL PROTECTED]
> wrote:
>On Mar 28, 10:23 am, hru...@[EMAIL PROTECTED]
(Herman Rubin) wrote:
>> In article
<95b06732-3d1a-4eb9-bd81-2a8c7e7bd...@[EMAIL PROTECTED]
>,
>> Luna Moon <lunamoonm...@[EMAIL PROTECTED]
> wrote:
>> >Hi all,
>> >I am wondering if the following process can exist in a suitable
>> >probability space P?
>> >It is a Poisson type process, but its intensity lambda_t is specified
>> >by
>> >d(lambda_t) = sigma*lambda_t*dBt,
>> >where Bt is a standard Brownian Motion.
>> >Basically, I want to see if it is okay to specify the intensity in
>> >terms of the Brownian Motion.
>> You can do this IN PRINCIPLE, but you would have to specify
>> what happens if you get something negative. The model above
>> does not exclude that.
>Hi Herman,
>No the model is a GBM with no drift, so lambda_t won't be negative.
I did not notice that lambda_t multiplies the Brownian term.
With a slightly different interpretation of the d(lambda_t)
statement above, the Rubin-Fish-Stratonovich interpretation,
the solution for lambda becomes
lambda_t = C*exp(int_0^t sigma*dB_t),
and if sigma is constant, this simplifies further.
There is no problem with such a process, and it can even
get more complicated. In this case, given the Brownian
motion, the process is generalized Poisson, but one can
even have the intensity change according to the process
itself, in which case almost anything can happen.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@[EMAIL PROTECTED]
Phone: (765)494-6054 FAX: (765)494-0558
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