Week 1.5: Trajectories and finite-dimensional distributions

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

Stochastic processes

27 notes

National Research University Higher School of Economics

Stochastic processes

27 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

So, we have defined the stochastic process as a mapping from T times omega into R,

in the sense that T is typically equal to R plus.

We have said that this mapping should satisfy the following condition.

So actually, for any fixed small t,

Xt should be a random variable on omega Fp.

Okay good. This was a definition of a stochastic process.

And now, let me define,

what does the trajectory of the stochastic process mean.

So, trajectory of the same path

is a mapping from T to R which appears when we fix an element from omega.

So, we have stochastic process Xt and then, we fix omega.

And don't forget we get a function from T to

R. And this is exactly what we can observe in real life.

So, imagine that you analyze a stock price.

This is a curve, actually, from t,

I'm mapping from t to some St. And what you'll know is just a trajectory of this process.

So this is a complete analog of what will happen medical statistics.

We have some samples x1,

x2, and so on.

And we assume that this sample was

obtained as a realization of a set of random variables.

Well, sometimes you have one random variable,

that we assume that it's a random variable,

was observed many times.

So, what you have in real life is a set of real numbers.

Well, what we have here is just a curve which depends on t. Good.

This is a trajectory.

On the other side, there is a notion of finite dimensional distribution.

This is actually a distribution of a random vector Xt1,

Xt2, and so on Xtn.

Where t1, t2 and tn are just some time moments.

These are elements from R. And this is

exactly the main difference

between the probability theory and the theory of stochastic processes.

In the probability theory,

since elements would be independent.

And therefore, the final dimensional distribution is something which is trivial.

You can simply calculate the final dimensional distribution of each component,

and then just say that all of these compounds are independent.

And here, the theory of stochastic processes is essentially dependent.

And therefore, the calculation of

final dimensional distribution is sometimes rather challenging.

Let me provide one rather simple example.

Assumes as a process Xt is equal to xai multiplied by t,

while xai is equal to 1,

with probability one half and is equal to two with probability one half.

So, there is only one source of randomness.

This is as xai.

And this xai determines the process Xt.

What is the trajectory of this process?

Basically, there are two possible outcomes.

The first one when xai equal to one and in this case Xt is equal to

T. And now the outcome is when xai is equal to two.

And in this case, we have a trajectory 2t.

So, there are only two possible trajectories for the process Xt.

As for the final dimensional distributions,

let me calculate for instance the probabilities that Xt1 is

less or equal than X1 and Xt2 is less or equal than x2.

It is an easy exercise to shows that this probability is equal to 0.

If minimum between x1 divided by t1 and X2 divided by t2 is less than 1.

Moreover, it is equal to one half,

if this minimum is from 1 to 2.

And this is equal to 1,

if this minimum is larger than 2, larger or equal than 2.

I would like to leave the exact calculations as a simple exercise.

And basically, this topic is not difficult at all.

From many processes, one can easily plot a trajectory,

at least a typical trajectory of a process on a two-dimensional plot and basically,

the calculation of final dimensional distributions in many cases,

also possible at least for two-dimensional case.

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