Simulate AR(1) Poisson Jump Process

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Julian Williams 2014-2-20

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.

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