# Simulate AR(1) Poisson Jump Process

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Julian Williams on 20 Feb 2014
I am being a bit thick, so apologies if this is a trivial question.
I would like to simulate a mixed frequency (intra-day versus daily) discrete time AR(1) jump process.
So the data generating process of interest is:
N(t) is the counting process.
with intensity eta(t)
I would like the intensity to have the following simple dynamics:
eta(t+1) = rho*eta(t) + gamma*noise!
Now the problem I have is the noise! bit.
when writing out the density function I follow the textbook treatment and write:
E(Nt) - eta(t) = sum j*Probability(Nt = j) - eta(t)
Probability(N(t+1) = j) = (1/factorial(j))*exp(eta(t+1))*(eta(t+1))^j;
So I use the law of iterated expectations to recover noise!
Sad thing is that for a Hawks process I know how to generate this, but I just seem to be clueless how to do it for an independent noise process.
If anyone can help, it would be much appreciated, I am sure it is something trivial and I am just missing something practical.
Thanks.