Change of Time and Change of Measure

Ole E Barndorff-Nielsen, Albert Shiryaev

• Publisher: WSPC
• Year: 2015
• ISBN:
  9789814678605
• Edition: 2
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Change of Time and Change of Measure provides a comprehensive account of two topics that are of particular significance in both theoretical and applied stochastics: random change of time and change of probability law.

Random change of time is key to understanding the nature of various stochastic processes, and gives rise to interesting mathematical results and insights of importance for the modeling and interpretation of empirically observed dynamic processes. Change of probability law is a technique for solving central questions in mathematical finance, and also has a considerable role in insurance mathematics, large deviation theory, and other fields.

The book comprehensively collects and integrates results from a number of scattered sources in the literature and discusses the importance of the results relative to the existing literature, particularly with regard to mathematical finance.

In this Second Edition a Chapter 13 entitled 'A Wider View' has been added. This outlines some of the developments that have taken place in the area of Change of Time and Change of Measure since the publication of the First Edition. Most of these developments have their root in the study of the Statistical Theory of Turbulence rather than in Financial Mathematics and Econometrics, and they form part of the new research area termed 'Ambit Stochastics'.

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Contents:

• Random Change of Time
• Integral Representations and Change of Time in Stochastic Integrals
• Semimartingales: Basic Notions, Structures, Elements of Stochastic Analysis
• Stochastic Exponential and Stochastic Logarithm. Cumulant Processes
• Processes with Independent Increments. Lévy Processes
• Change of Measure. General Facts
• Change of Measure in Models Based on Lévy Processes
• Change of Time in Semimartingale Models and Models Based on Brownian Motion and Lévy Processes
• Conditionally Gaussian Distributions and Stochastic Volatility Models for the Discrete-time Case
• Martingale Measures in the Stochastic Theory of Arbitrage
• Change of Measure in Option Pricing
• Conditionally Brownian and Lévy Processes. Stochastic Volatility Models
• A Wider View. Ambit Processes and Fields, and Volatility/Intermittency

Readership: Mathematical researchers, graduate students and practitioners interested in application of probabilistic theories & stochastic processes to economics & finance, and to turbulence.
Key Features:

• There is no other book on the market with such a specific and in-depth focus on these two main issues of change of time and change of probability law and Lévy measure
• It is invaluable as a textbook for graduate-level courses and students, or a handy reference for researchers and practitioners in financial mathematics and econometrics
• Provides a strong and comprehensive account by drawing together sources from a wide range of books and research papers in the fields of theoretical and applied probability

Chapter 1

Random Change of Time

1.1Basic Definitions

1. Let (Ω,

, P) be a Kolmogorov’s probability space, where Ω is the space of elementary events ω,

is some σ-algebra of subsets of Ω, and P is a probability measure. In all our considerations, a crucial role is played by an additional structure (

_t)_t≥0(called a filtration) which is a nondecreasing right-continuous family of sub-σ-algebras of

(in other words,

_s ⊆

_t for all 0 ≤ s ≤ t and

_t =

_t+, where

_t+ = ∩_s>t

_s.

The collection (Ω,

, (

_t)_t≥0, P) is called a filtered probability space or stochastic basis[128]. (As a rule we assume a stochastic basis to satisfy the usual conditions, namely, the σ-algebra

is P-complete and every

_t contains all P-null sets of

; see [128].) Sometimes it is convenient to consider

_t as the “information” available on the time interval [0, t].

2. As was mentioned in the Introduction, it is convenient, when defining the notion of “change of time”, to distinguish between the “old” (physical, calendar)t-time and a “new” (operational, business) θ-time.

The following definition is useful if we need to construct, starting from the initial process X = (X_t)_t≥0(adapted to the filtration (

_t)_t≥0), a new process

= (

_θ)_θ≥0 evolving in θ-time and having certain desired properties.

Definition 1.1. A family of random variables

= (

(θ))_θ≥0 is said to be a random change of time (or rather, a change of θ-time into t-time in accordance with the map θ

t =

(θ)), if

(a)	( (θ))θ≥0 is a nondecreasing (in the terminology of stochastic analysis—increasing), right-continuous family...

Table of contents

• Cover Page
• Title page
• Copyright
• Foreword
• Foreword1
• Contents
• Introduction
• 1. Random Change of Time
• 2. Integral Representations and Change of Time in Stochastic Integrals
• 3. Semimartingales: Basic Notions, Structures, Elements of Stochastic Analysis
• 4. Stochastic Exponential and Stochastic Logarithm. Cumulant Processes
• 5. Processes with Independent Increments. Lévy Processes
• 6. Change of Measure. General Facts
• 7. Change of Measure in Models Based on Lévy Processes
• 8. Change of Time in Semimartingale Models and Models Based on Brownian Motion and Lévy Processes
• 9. Conditionally Gaussian Distributions and Stochastic Volatility Models for the Discrete-time Case
• 10. Martingale Measures in the Stochastic Theory of Arbitrage
• 11. Change of Measure in Option Pricing
• 12. Conditionally Brownian and Lévy Processes. Stochastic Volatility Models
• 13. A Wider View. Ambit Processes and Fields, and Volatility/Intermittency
• Afterword
• Afterword to the Second Edition
• Bibliography
• Index