Week 1.2: Difference between various fields of stochastics

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En provenance du cours de National Research University Higher School of Economics

Stochastic processes

100 notes

National Research University Higher School of Economics

Stochastic processes

100 notes

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.

À partir de la leçon

Week 1: Introduction & Renewal processes

Upon completing this week, the learner will be able to understand the basic notions of probability theory, give a definition of a stochastic process; plot a trajectory and find finite-dimensional distributions for simple stochastic processes. Moreover, the learner will be able to apply Renewal Theory to marketing, both calculate the mathematical expectation of a countable process for any renewal process

Week 1.1: Difference between deterministic and stochastic world4:49

Week 1.2: Difference between various fields of stochastics6:45

Week 1.3: Probability space8:48

Week 1.4: Definition of a stochastic function. Types of stochastic functions.4:32

Week 1.5: Trajectories and finite-dimensional distributions5:36

Rencontrer les enseignants

Vladimir Panov
Assistant Professor
Faculty of economic sciences, HSE

Theory of Stochastic Processes belongs to the field of mathematics known as Stochastics.

Basically, there are three disciplines which are taught in

almost all universities and which are related to Stochastics.

The first one is the probability theory,

the second is mathematical statistics,

and the third one is the theory of the stochastic processes.

And if you are familiar with all of these disciplines you can take

much more difficult parts of Stochastics like jump processes,

stochastic calculus, stochastic differential equations,

also population dynamics, jump processes and many others.

Well, we will somehow discuss many more difficult ideas within this course,

but now let me explain what is the difference between

these three parts of Stochastic world.

Well, let me start with probability theory.

Assume that we have a pond with fishes,

and when we speak about probability theory,

the most typical task in the context of this example will be to

analyze the amount of fishes in this pond at some given time moment.

And the most typical situation is when you consider distribution of this amount,

and if you know the distribution,

you can further analyze this amount,

for instance, calculate mathematical expectation,

variance or even to find

a limit law for the variable capital M. Now,

let me move to the mathematical statistics.

Mathematical statistics solves problems which are upper side to the probability theory.

Namely, one can ask how to estimate

the amount of fishes n by providing some statistical experiments.

I would like to show one approach which can be

used to solve this rather practical problem.

For instance, we can catch some amount of fishes.

Say capital M fishes and mark them,

and then we put this amount of fishes back in the pond,

so the moment we have capital M,

marked fishes and capital N minus capital M, unlabeled fishes.

Next, we catch some amount of fishes once more.

Let me say small n fishes.

From this amount, there are some marks and some unlabeled fishes.

Let me assume that we have a small m,

marked, and n minus m, unlabeled fishes.

The question which we should take into account is what is

the probability that the amount of

marked fishes is exactly

small n. This can be

calculated using some very simple considerations from the probability theory.

Namely, this probability is equal to

C capital M small m

multiplied by C capital N minus capital M,

n minus m and divided by C capital N capital M. And afterwards,

we can repeat this experiment many times,

namely we get the values m1,

m2 and so on, mq,

and for any m,

we can calculate the [inaudible] probabilities.

And afterwards, we can consider as a log-likelihood function which

is equal to the sum K from 1 to q,

are probability that the amount of marked fishes is equal to mK.

And here, if you look attentively at this formula,

you will get the following conclusion.

That almost all elements of this formula are exactly known.

For instance, you know capital M because you know

how many fishes you'll catch at the first time.

Then you know also small m and small n because these are the results of your experiment,

and the only value here which is unknown is exactly capital N. And therefore,

you can deal with this parameter as

an unknown parameter and you can just maximize log-Iikelihood malfunction

with respect to capital M. This means that you will get an estimator of

capital M and this estimator is known as maximum likelihood estimator.

Okay, this was a practical solution for a very simple task,

and in this context,

we can also introduce the notion of stochastic process

because this m amount of fishes in the pond,

definitely depends on T events on time, and therefore,

we can ask the same questions,

but taking into account that the amount of

fishes changes and this is exactly the objective of the theory of stochastic processes.

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