Statistical Inference in Financial and Insurance Mathematics with R

By Alexandre Brouste

Finance and insurance companies are facing a wide range of parametric statistical problems. Statistical experiments generated by a sample of independent and identically distributed random variables are frequent and well understood, especially those consisting of probability measures of an exponential type. However, the aforementioned applications also offer non-classical experiments implying observation samples of independent but not identically distributed random variables or even dependent random variables.

Three examples of such experiments are treated in this book. First, the Generalized Linear Models are studied. They extend the standard regression model to non-Gaussian distributions. Statistical experiments with Markov chains are considered next. Finally, various statistical experiments generated by fractional Gaussian noise are also described.

In this book, asymptotic properties of several sequences of estimators are detailed. The notion of asymptotical efficiency is discussed for the different statistical experiments considered in order to give the proper sense of estimation risk. Eighty examples and computations with R software are given throughout the text.

• Examines a range of statistical inference methods in the context of finance and insurance applications
• Presents the LAN (local asymptotic normality) property of likelihoods
• Combines the proofs of LAN property for different statistical experiments that appears in financial and insurance mathematics
• Provides the proper description of such statistical experiments and invites readers to seek optimal estimators (performed in R) for such statistical experiments

• Language: English
• Publisher: Elsevier Science
• Release date: Nov 22, 2017
• ISBN: 9780081012611

Alexandre Brouste

Alexandre Brouste is a Professor in Statistics with the Institute of Risk and Insurance at Le Mans University, France

Book preview

Statistical Inference in Financial and Insurance Mathematics with R

Alexandre Brouste

Optimization in Insurance and Finance Set

coordinated by

Nikolaos Limnios and Yuliya Mishura

Table of Contents

Cover

Title page

Dedication

Copyright

Preface

List of Notations

Introduction

Part 1: Inference in Parametric Statistical Experiments

1: Statistical Experiments

Abstract

1.1 Dominated and homogeneous statistical experiments

1.2 Experiments generated by a sample of independent and identically distributed random variables

1.3 Probability measures of exponential type

2: Statistical Inference

Abstract

2.1 Asymptotic properties of sequences of estimators

2.2 Examples of sequences of estimators

2.3 Asymptotic normality

3: Asymptotic Efficiency

Abstract

3.1 Likelihood ratio, local asymptotic properties of the likelihoods and the van Trees inequality

3.2 LAN property for different statistical experiments

3.3 Asymptotic efficiency of some sequence of estimators

Part 2: Statistical Inference for Insurance

4: Statistical Experiments in Insurance

Abstract

4.1 Statistical inference in generalized linear models

4.2 Score and Fisher information of GLM statistical experiments

4.3 Asymptotic properties of the sequence of maximum likelihood estimators

4.4 Numerical approximations of the sequence of maximum likelihood estimators

Part 3: Statistical Inference for Finance

5: Homogeneous Diffusion Processes

Abstract

5.1 Examples of pricing in finance

5.2 Examples of closed-form transition probability density functions

5.3 Simulation of diffusions

5.4 General classes of diffusion processes

6: Statistical Experiments in Finance

Abstract

6.1 Large-sample convergence scheme

6.2 Mixed convergence scheme

6.3 High-frequency convergence scheme

Appendix 1: Cholesky Method

Appendix 2: L²(ν)-Differentiable Family of Probability Measures

A2.1 Differentiability in quadratic mean

A2.2 More regular models

A2.3 Classical examples

Appendix 3: Stochastic Calculus

Itô’s integral and Itô’s formula

Itô’s formula for diffusion processes

Bibliography

Index

Dedication

To Sophie, Aurèle and Lucie

Copyright

First published 2018 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

For information on all our publications visit our website at http://store.elsevier.com/

© ISTE Press Ltd 2018

The rights of Alexandre Brouste to be identified as the authors of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

Library of Congress Cataloging in Publication Data

A catalog record for this book is available from the Library of Congress

ISBN 978-1-78548-083-6

Printed and bound in the UK and US

Preface

Alexandre Brouste October 2017

This book summarizes lectures which I have given for eight years in a Master course in Statistics for Finance and Insurance at Le Mans University and the Institute of Risk and Insurance. This book is thus dedicated to students, engineers and practitioners interested in mathematical statistics and parametric estimation in general.

Our statistics team is historically focused on statistics for stochastic processes and their applications. Statistical inference for continuously observed stochastic processes has mainly been developed here by Yury Kutoyants for Markov processes and Marina Kleptsyna for long-memory processes. I hope this book can contribute to this local story.

I would also like to thank Alison Bates, Christophe Dutang, Didier Loiseau, Mathieu Rosenbaum and Alexandre Popier for taking some of their precious time to carry out a careful revision. I am equally grateful to all the other people who support me in my everyday life.

When I was young, my grandfather Charles Brouste offered me a book that I had chosen randomly on the little mathematical shelf of a small bookshop in Pau, France. This book was a lecture on mathematical statistics by Alain Monfort and it has never left me. A love of mathematics (more especially statistics) and fond memories of the pleasure felt when I managed to solve the problems given to me by my grandfather in summer times also never left me. If one person reading this book chooses to work harder in mathematical statistics, it will not have been written in vain.

List of Notations

is the indicator function of A, or 0 accordingly as x ∈ A or x ∉ A.

– * is the transposition.

is the σ.

– ∘ is the composition of two applications.

.

– δa .

– ∇ is the gradient (nabla symbol).

– h·ν: the probability measure h·ν is defined by

– Ip is the p × p identity matrix.

is the set of positive integers.

is the set of strictly positive integers.

is the distribution of a Gaussian random variable of mean μ and variance σ².

is the real line.

is the set of positive real numbers.

is the set of strictly positive real numbers.

is the (d-dimensional Euclidean space.

– σ (Yu, 0 ≤ u ≤ s) is the σ-algebra generated by the process Yu.

Introduction

Finance and insurance companies are facing a wide range of mathematical problems. The former consider, for instance, the valuation of common derivative securities (call and put options, forward and future contracts, swaps) in complete or incomplete financial markets as well as the economic consumption and investment problem; the latter consider the tarification of insurance premia and claims reserving. The present volume is dedicated to the statistical estimation of the key parameters in some of the aforementioned considerations in finance and insurance.

The notion of statistical experiment generated by an observation sample is introduced in the first part of this book with the notion of statistical inference of the unknown parameter (and the notion of sequence of estimators). In order to evaluate a sequence of estimators when the size of the observation sample increases and then to compare it with others, several asymptotic properties of sequences of estimators are defined, such as consistency, asymptotic normality and asymptotic efficiency. Different sequences of estimators of the unknown parameter (maximum likelihood estimators and Bayesian estimators) are also presented.

Statistical experiments generated by a sample of independent and identically distributed random variables are relatively common in the aforementioned applications. Such classical experiments are well understood theoretically, especially those with probability measures of exponential type. They are fully described in this book, and we pay special attention to the so-called Gaussian shift statistical experiment. Indeed, in this setting, the sequence of maximum likelihood estimators is consistent, asymptotically normal and asymptotically efficient.

However, finance and insurance applications also offer non-classical statistical experiments. Three examples of non-classical statistical experiments are treated in this book.

First, the generalized linear models are studied. They extend the standard regression model to non-Gaussian distributions. In this case, the random variables of the observation sample are neither identically distributed nor Gaussian. These models are famous for the tarification of insurance premia and are described in the second part of this book. In these statistical experiments, the sequence of maximum likelihood estimators of the unknown parameter is shown to be consistent and asymptotically normal under proper assumptions. This sequence of estimators is generally not in