Week 2.15: Compound Poisson processes-1

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来自 National Research University Higher School of Economics 的课程

Stochastic processes

40 个评分

National Research University Higher School of Economics

Stochastic processes

40 个评分

The purpose of this course is to equip students with theoretical knowledge and practical skills, which are necessary for the analysis of stochastic dynamical systems in economics, engineering and other fields. More precisely, the objectives are 1. study of the basic concepts of the theory of stochastic processes; 2. introduction of the most important types of stochastic processes; 3. study of various properties and characteristics of processes; 4. study of the methods for describing and analyzing complex stochastic models. Practical skills, acquired during the study process: 1. understanding the most important types of stochastic processes (Poisson, Markov, Gaussian, Wiener processes and others) and ability of finding the most appropriate process for modelling in particular situations arising in economics, engineering and other fields; 2. understanding the notions of ergodicity, stationarity, stochastic integration; application of these terms in context of financial mathematics; It is assumed that the students are familiar with the basics of probability theory. Knowledge of the basics of mathematical statistics is not required, but it simplifies the understanding of this course. The course provides a necessary theoretical basis for studying other courses in stochastics, such as financial mathematics, quantitative finance, stochastic modeling and the theory of jump - type processes.

从本节课中

Week 2: Poisson Processes

Upon completing this week, the learner will be able to understand the definitions and main properties of Poisson processes of different types and apply these processes to various real-life tasks, for instance, to model customer activity in marketing and to model aggregated claim sizes in insurance; understand a relation of this kind of models to Queueing Theory

Week 2.8: Non-homogeneous Poisson processes-14:24

Week 2.9: Non-homogeneous Poisson processes-24:23

Week 2.10: Relation between renewal theory and non-homogeneous Poisson processes-14:11

Week 2.11: Relation between renewal theory and non-homogeneous Poisson processes-27:10

Week 2.12: Relation between renewal theory and non-homogeneous Poisson processes-34:29

Week 2.13: Elements of the queueing theory. M/G/k systems-19:47

Week 2.14: Elements of the queueing theory. M/G/k systems-25:14

Week 2.15: Compound Poisson processes-16:05

Week 2.16: Compound Poisson processes-26:52

Week 2.17: Compound Poisson processes-33:18

与讲师见面

Vladimir Panov
Assistant Professor
Faculty of economic sciences, HSE

The last part of this lecture will be devoted to compound Poisson processes.

This is a very popular model which is

essentially based on what you call homogeneous Poisson processes.

And according to this model,

the process is defined as follows.

This is the sum by k from one to

some Poisson process Nt of eSk.

Here random variables eS1, eS2,

and so on are independent and identically distributed, not necessarily positive.

The process Nt is Poisson.

I mean homogeneous Poisson and I will denote its intensity parameter by lambda.

And also, assumes that eS1, eS2,

and so on and Nt are independent.

This model eventually arises in many applications,

but a lot of applications can be found in insurance.

In fact, Xt can be used for modeling

the aggregated claim amounts of one insurance company.

According to this interpretation eS1,

eS2, and so on are actually individual claim sizes.

And Nt expresses the amount of claims till

time t. Once more,

under this interpretation, Xt is an aggregated claim amount.

For analyzing of this processes,

we'll need to employ some functionals of this process.

So, here the main difficulty is the same as in the case of renewal processes.

The point is that the distribution of Xt can be written in closed form,

only in few cases which are rather trivial.

In general case, it is unknown and we should use some functionals and the point is,

that a kind of those functional definitely

depends on the properties of the random variables eS1,

eS2, and so on.

More precisely, if eS1, eS2,

and so on have only integer values which are non-negative.

Then Xt is also an integer non-negative random variable.

And in this case we can employ

the so called probability generating function.

So, the so-called BGF.

This function is defined as follows.

Actually, for a random variable eS which

has only integer non-negative values.

This probability generating function is defined as

a mathematical expectation of u in the power of eS.

Where this deterministic number,

u is smaller than one.

Absolutely is smaller than one.

Of course, if eS1 and eS2 are independent.

Then, the probability generating function of

their sum using the fact a product of their corresponding generating function.

In more general case,

when eS1 and eS2 and so on on the negative solves it can be non-integer,

but there no negative.

Xt is also non-negative.

And in this case one can imply the so called, moment generating function.

Which is very close to the so-called Laplace transform.

Let me define it as

mathematical expectation of the exponent minus u in the of power eS.

So, once more this definition is correct for NeS taking

non-negative values and the parameter u here is positive.

Sometimes, you can find in the books here plus without minus.

From my point of view it's a question of taste.

But here I defined it in this way just because to be sure it's object this.

In fact, if you look attentively as a probability generating function,

we get that this object exists for eS from this question for u from this range.

The same situation is for moment generating function.

Nevertheless, it seems very natural to think that eS can take any values.

And from this case,

we should apply the so-called characteristic function.

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