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montecastello
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Compound Poisson Process

January 29th, 2018, 3:48 pm

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Hi,
please find attached the definition of compound Poisson process in "A Factor Model Approach to Derivative Pricing" by Primbs.
I have two questions (maybe trivial but I'm quite new to these topics):
How do you justify the second equality sign in equation 1.22?
How do you get equation 1.23?

Many thanks in advance
 
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gatarek
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Re: Compound Poisson Process

January 29th, 2018, 4:30 pm

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Last edited by gatarek on March 11th, 2019, 3:02 pm, edited 1 time in total.
 
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Alan
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Re: Compound Poisson Process

January 29th, 2018, 5:53 pm

Well, I think the second equality in (1.22) can be "justified" by picturing a sample chart of [$]\pi(s)[$] vs. s. So, formally [$]d \pi(s) = \delta(s - t_i) \, ds[$] 
in the vicinity of each [$]t_i[$], using Dirac deltas. Now integrate.  

Then  [$]d \pi^Y(s) = Y_i \,\delta(s - t_i) \, ds[$]  in the same vicinity for the same reason, which was the other question.

Is it just notation? Probably. In general, jump-process notation is pretty awful and you just get used to it. (Should be in Student forum).
 
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gatarek
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Re: Compound Poisson Process

January 29th, 2018, 6:39 pm

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Last edited by gatarek on March 11th, 2019, 3:03 pm, edited 1 time in total.
 
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Edophokles
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Re: Compound Poisson Process

January 30th, 2018, 8:10 am

The second equation (1.23) is simply the differential notation of the integral form. 

The first one (1.22) is due to the fact that a Poisson process only changes when it jumps, hence dπ(sis zero everywhere except when the jump occurs, then it is 1. Thus, the integral simply becomes the sum of jumps. This is more of an intuition behind the equality, but you can also derive it a more rigorous way by using indicator functions.