Hello everyone.
I always seem to have troubles when it comes to proving an integrability condition of a particular process... somehow I don't have the right method.
I have the following stochastic process:
M(t) = N(t) - kt
Where N(t) is a poisson process of parameter kt.
M is called a compensated poisson process.
I want to prove that: E[ | M(t) | ] < +infinity (in other words, that it is finite).
I have searched for a while, and all my attempts have been infructuous.
Thank you for your help.
Jack
Hi Laurent,
Thank you very much.
Stupid question: since k > 0 and t >= 0, why is kt < infinity?
The weird thing is that I found the second question easier than this one (the second question was obviously proving the martingale characterization).
When it comes to integrability condition, I am always a bit unsure about how to tackle the thing.
Jack
Hello Laurent,
Okay I got it. k and t are fixed so the product is obviously finite.
I guess this is the correct reasoning when you try to prove that some expression is finite.
Would you have an example in which E[ | some stochastic process | ] is infinite? (so that I can picture it in my mind).
Thanks a bunch
Jack
The next question asked to show that is a martingale, which means :
1)
2) for any ,...
The assumption 1) is the one that makes assumption 2) make sense. That's probably why the first question was to show 1).
Don't be impressed by the "stochastic process" thing: since is fixed, you only deal with real random variables, no more no less.
So what you're asking for is just an example of distribution with infinite mean. For instance, take a Cauchy distribution (density ). If you want a "stochastic process", take the constant process for all , where is Cauchy distributed... A more natural example would be a so-called Cauchy process (or other stable processes), but it would be unnecessarily complicated to try to define it here.