Random Processes

First-Passage and Escape

Jaume Masoliver

• Publisher: WSPC
• Year: 2018
• eBook ISBN: 9789813225336
• Topic: Mathematics
• Subtopic: Finance
• Index: Mathematics

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Random processes are one of the most powerful tools in the study and understanding of countless phenomena in natural and social sciences.

The book is a complete medium-level introduction to the subject. The book is written in a clear and pedagogical manner but with enough rigor and scope that can appeal to both students and researchers.

This book is addressed to advanced students and professional researchers in many branches of science where level crossings and extremes appear but with some particular emphasis on some applications in socio-economic systems.

--> Contents:

• Random Processes:
  • Fundamentals of Probability
  • Random Processes: Definitions and General Properties
  • Markov Processes
  • Diffusion Processes
  • Fokker–Planck Equations in Several Dimensions
  • Linear Response Theory
• Stochastic Calculus with Some Applications:
  • Introduction to Stochastic Calculus
  • Stochastic Differential Equations
  • Some Financial Applications
• The Level Crossing Problem:
  • First-Passage, Escape and Extremes. General Settings
  • The Level Crossing Problem for Diffusion Processes
  • First-Passage and Extremes in Socio-Economic Systems

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--> Readership: Advanced students and professional researchers. -->
Random Processes;First-Passage Times;Stochastic Calculus;Extremes0 Key Features:

• The book gathers some aspects that are often disseminated in several publications (books and technical reports)
• The treatment of topics is rather unique (specially those of first-passages)

PART 1

Random processes

Chapter 1

Fundamentals of probability

In this first chapter we will review the basic concepts of probability which constitute the pillar of the theory of random processes and all its subsequent developments and applications.¹ The reader familiar with probability theory may skip this introductory material and go directly to the second chapter.

1.1 Spaces of probability

In many physical phenomena, as well as in biological, technical and socioeconomic matters, just to name a few, the occurrence of a given result cannot be told in advance with certainty. One may give, at most, some numerical estimate, usually a percentage –in fact, a number between 0 and 1– of the likelihood of that result to occur. This numerical estimate is called probability and higher probability implies higher likelihood. Different observations of the same phenomenon and under the same circumstances will give different results.

We call experiment to any observation that can be repeated under the same conditions an unlimited number of times. The set of all possible outcomes of an experiment is called the sample space Ω. It is an abstract set that may or may not contain numerical values. Thus, for instance, the sample space of the experiment consisting in tossing a fair coin consists in two elements: head and tail, i.e., Ω = {H, T}, while the sample space of rolling a fair die consists in six (numerical) elements Ω = {1, 2, 3, 4, 5, 6}.

These are examples of finite sample spaces. There are, however, infinite sample spaces. For example, randomly choosing a positive integer number form the set of all integers or measuring a given voltage in a noisy electric circuit are examples of infinite sample spaces. In the first case the sample space is infinite but countable, the set of all natural numbers Ω = {1, 2, 3, . . . , n, . . . } while in the second case, Ω is uncountable.²

Any subset A ⊂ Ω of the sample space is called event and the elements of Ω are elementary events. The whole sample space is also an event, often called the sure event, and the empty set

is the impossible event. For example, in the experiment of throwing two coins the set of all possible results, the sample space, is Ω = {(H, H); (H, T); (T, H); (T, T)} and the event “the appearance of at least one head” is A = {(H, H); (H, T); (T, H)}.

Events are subjected to set operations such as unions and intersections. We thus have the union and intersection of two sets A and B defined as

and the difference of two sets an...

Table of contents

• Cover Page
• Title Page
• Copyright
• Dedication
• Preface
• Contents
• Random processes
• Stochastic calculus with some applications
• The level crossing problem
• Bibliography
• Index