Steven Shreve Bk2 Pg 468 Topic 11.3 Compound Poisson Process

[kdmfe]

Dear experts,

I have below put a brief discussion of my understanding, but could not manage to get a complete overview. Kindly help me understand.

Below I have put my understanding in my own words based on the content from the book mentioned above.

Below I would like to discuss and seek clarifications primarily on Compound Poisson Process.

Before that, I have below, put briefly my understating on Simple Poisson process and then move to Compound Poisson Process.

Simple Poisson Process:

In the below diagram, the jumps are of uniform size one (or one step). Here jumps appear to be strictly positive.

Here RV are termed as S1, S2,…Sn (so countably finite)

11.3.1 Construction of Compound Poisson Process:

Compound Poisson processes have random jump sizes

N(t) be a Poisson process with intensity λ

Let Y1, Y2… be a sequence of identically distributed random variables with mean with β = E[Yi]

 

[PepeQuant]

How is it possible each RV Yi has the same mean. Can this be explained with the help of an example?
They are i.i.d

Question: What does it mean, when we say RV Y1, Y2… are independent of Poisson process N(t)?
Yes

From the definition of Q(t), can we assume there are only N(t) number RV i.e., Y1, Y2…YN(t) and not countable infinite?
N(t) itself a RV no? Ofc it can be infinite. If assume poisson jump, then Pr(N(t)=x) where x is defined on integers across [0, \infty)

From the chart it is clear that the size can either be positive or negative. How can they be negative size if the simple Poisson process are uniform positive and of size one? Does this compound Poisson process not build up from Simple Poisson process.
Poisson Jump is positive. When jump, it is 1. If you put Y_i = 1, then your compound poisson process is just a poisson process. But Y_i can be either positive or negative. Bottom line is, N(t) is the rate of jump (positive only). Y_i is the jump size (also can be negative).

A quick example is the Merton jump process. The jump rate is poisson while the jump size is Gaussian.

https://www.csie.ntu.edu.tw/~lyuu/finance1/2015/20150513.pdf

 

[kdmfe]

How is it possible each RV Yi has the same mean. Can this be explained with the help of an example?
They are i.i.d

Question: What does it mean, when we say RV Y1, Y2… are independent of Poisson process N(t)?
Yes

From the definition of Q(t), can we assume there are only N(t) number RV i.e., Y1, Y2…YN(t) and not countable infinite?
N(t) itself a RV no? Ofc it can be infinite. If assume poisson jump, then Pr(N(t)=x) where x is defined on integers across [0, \infty)

From the chart it is clear that the size can either be positive or negative. How can they be negative size if the simple Poisson process are uniform positive and of size one? Does this compound Poisson process not build up from Simple Poisson process.
Poisson Jump is positive. When jump, it is 1. If you put Y_i = 1, then your compound poisson process is just a poisson process. But Y_i can be either positive or negative. Bottom line is, N(t) is the rate of jump (positive only). Y_i is the jump size (also can be negative).

A quick example is the Merton jump process. The jump rate is poisson while the jump size is Gaussian.

https://www.csie.ntu.edu.tw/~lyuu/finance1/2015/20150513.pdf

Pepe Quant, Thank you for your time and advice.

I fully understand that stochastic is very exhaustive subject.

As things stand, My PhD sir and myself have tried to understand the concept of RV construction for Poission process in Chapter 11 ( Stochastic Calculus book 2 by Steven Shreve), but obviously we have not got any far in our understanding. But on the other hand, my sir has been able to give me notes for almost all sections/theorem baring on or two.

I am happy about it.

Can I please request you to suggest a book or material which covers the topic on the construction of the random variables in the Poisson and compound Poisson process?

I think you/experts guidance will help me putting at least a decent finish to the chapter.

Thank you in advance.

 

[PepeQuant]

@kdmfe have you learned any probability courses? I am specifically talking about combination, permutation, discrete distribution, continuous distribution, independence. If not please study those in depth.

 

[kdmfe]

@kdmfe have you learned any probability courses? I am specifically talking about combination, permutation, discrete distribution, continuous distribution, independence. If not please study those in depth.

Thank you for your prompt response.

I have read and written my notes from Sheldon Ross' book ' A first course in Probability'.

I have not gone through 'Introduction to Probability models' by the same author.

Kindly suggest, if I am missing anything.
Thank you

 

[PepeQuant]

Thank you for your prompt response.

I have read and written my notes from Sheldon Ross' book ' A first course in Probability'.

I have not gone through 'Introduction to Probability models' by the same author.

Kindly suggest, if I am missing anything.
Thank you

Ok. Do you know basic probability theory? It is not about what you read. Do you actually know those stuff by heart? Think of it as going into a final exam of probability can you pass? What if I ask you the relationship between a Poisson dist and binomial, can you prove it (with some help)?