Basic Stochastic Processes

Pierre Devolder, Jacques Janssen, Raimondo Manca

• Publisher: Wiley-ISTE
• Year: 2015
• Print ISBN: 9781848218826
• eBook ISBN: 9781119184546
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

This book presents basic stochastic processes, stochastic calculus including Lévy processes on one hand, and Markov and Semi Markov models on the other. From the financial point of view, essential concepts such as the Black and Scholes model, VaR indicators, actuarial evaluation, market values, fair pricing play a central role and will be presented.

The authors also present basic concepts so that this series is relatively self-contained for the main audience formed by actuaries and particularly with ERM (enterprise risk management) certificates, insurance risk managers, students in Master in mathematics or economics and people involved in Solvency II for insurance companies and in Basel II and III for banks.

1
Basic Probabilistic Tools for Stochastic Modeling

In this chapter, the readers will find a brief summary of the basic probability tools intensively used in this book. A more detailed version including proofs can be found in [JAN 06].

1.1. Probability space and random variables

Given a sample space Ω, the set of all possible events will be denoted by

, which is assumed to have the structure of a σ -field or a σ -algebra. P will represent a probability measure.

DEFINITION 1.1.– A random variable (r.v.) with values in a topological space (E,ψ) is an application X from Ω to E such that:

[1.1]

where X^-1(B) is called the inverse image of the set B defined by:

[1.2]

Particular cases:

a) If (E,ψ) = (

, β), X is called a real random variable.

b) If (E, ψ) =

, where

is the extended real line defined by

and

is the extended Borel σ -field of

, that is the minimal σ -field containing all the elements of β and the extended intervals:

[1.3]

X is called a real extended value random variable.

c) If E =

(n>1) with the product σ -field β⁽ⁿ⁾ of β, X is called an n-dimensional real random variable.

d) If E =

(n>1) with the product σ -field β⁽ⁿ⁾ of β, X is called a real extended n-dimensional real random variable.

A random variable...

Table of contents

• Cover
• Table of Contents
• Title
• Copyright
• Introduction
• 1: Basic Probabilistic Tools for Stochastic Modeling
• 2: Homogeneous and Non-homogeneous Renewal Models
• 3: Markov Chains
• 4: Homogeneous and Non-homogeneous Semi-Markov Models
• 5: Stochastic Calculus
• 6: Lévy Processes
• 7: Actuarial Evaluation, VaR and Stochastic Interest Rate Models
• Bibliography
• Index
• End User License Agreement