Dear all,

i would like to implement two compound Poisson processes that simulate
upwards and downwards jumps respectively. Thereby, the up jump magnitudes
are Pareto distributed and the down jump magnitudes are Beta distributed.
These jump processes as well as a Brownian motion are a part of a model
describing security prices.
My goal is to simulate the two processes (independently of each other) with
known values of the parameters - the Poisson lambdas and the Beta / Pareto
parameters. Afterwards, I want to take the simulated data and try to
estimate the parameters using Maximum likelihood estimation (MLE) to see how
well an estimation algorithm that I implemnted is working.

My questions:

1) Is it right to simulate exponentially distributed waiting times between
the jumps and then just "jump" with a Beta / Pareto distributed magnitude?

2) When I simulate the down jumps and take zero as a starting point, then I
get negative values. The MLE does not like it, neither do I, because it
means, that I simulate negative prices. Can I take another starting value
for the process (for example one) ?

I am very grateful for a concise answer.

Cheers,
Zoe

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