Exercises of Stochastic Processes E-Book

Simone Malacrida

• Herausgeber: Simone Malacrida
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2022

Beschreibung

In this book, exercises are carried out regarding the following mathematical topics:
Markov chains and Markovian stochastic processes
time-dependent and time-independent stochastic processes
random walks and Brownian motion
Initial theoretical hints are also presented to make the performance of the exercises understood.

Leseprobe

Table of Contents

“Exercises of Stochastic Processes”

INTRODUCTION

THEORETICAL OUTLINE

EXERCISES

“Exercises of Stochastic Processes”

SIMONE MALACRIDA

In this book, exercises are carried out regarding the following mathematical topics:

Markov chains and Markovian stochastic processes

time-dependent and time-independent stochastic processes

random walks and Brownian motion

Initial theoretical hints are also presented to make the performance of the exercises understood.

Simone Malacrida (1977)

Engineer and writer, has worked on research, finance, energy policy and industrial plants.

ANALYTICAL INDEX

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INTRODUCTION

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I – THEORETICAL OUTLINE

Definitions

Markov chains and other processes

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II – EXERCISES

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9

Exercise 10

Exercise 11

Exercise 12

Exercise 13

Exercise 14

Exercise 15

Exercise 16

INTRODUCTION

In this workbook some examples of calculations relating to stochastic processes are carried out.

These processes represent a generalization of statistics applied to physical and technological phenomena, such as for example the interpretation of Brownian motion.

The importance of these processes in various disciplines (engineering, physics, economics, etc.) has increased over time, giving a precise configuration of its own to stochastic processes.

In order to understand in more detail what is presented in the resolution of the exercises, the theoretical reference context is recalled in the first chapter.

What is presented in this workbook is generally addressed in advanced statistics courses at university level.

I

THEORETICAL OUTLINE

Definitions

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A stochastic process represents a probabilistic dynamic system, i.e. a statistical evolution of a given system.

The variables of a stochastic process are obviously random variables, they are defined on a single finite sample space and assume values in a set called state space.

The characterization of a stochastic process takes place through the joint probability density function and thus it is possible to classify discrete and continuous stochastic processes.

If the transition probability between one state and the next depends on the previous states but not on the time, we speak of a homogeneous stochastic process ; cyclostationary stochastic processes, on the other hand, describe periodic phenomena and are particularly important in signal theory.

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A Gaussian stochastic process is a stochastic process whose random variables have joint probability distribution given by a Gaussian.

A Gaussian process is identified by its expected value and variance, as is a Gaussian function.

In signal theory, a Gaussian process defined over time is Gaussian noise (also called white noise).

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