Stochastic control in insurance / Hanspeter Schmidli.

Format:

Book

Author/Creator:

Schmidli, Hanspeter.

Contributor:

Simon Nelson Patten Fund.

Series:

Probability and its applications

Language:

English

Subjects (All):

Insurance--Mathematics.

Insurance.

Stochastic processes.

Physical Description:

xv, 254 pages : illustrations ; 24 cm.

Place of Publication:

London : Springer, [2008]

Summary:

Stochastic control is one of the methods being used to find optimal decision-making strategies in fields such as operations research and mathematical finance. In recent years, stochastic control techniques have been applied to non-life insurance problems, and in life insurance the theory has been further developed.

This book provides a systematic treatment of optimal control methods applied to problems from insurance and investment, complete with detailed proofs. The theory is discussed and illustrated by way of examples, using concrete simple optimisation problems that occur in the actuarial sciences. The problems come from non-life insurance as well as life and pension insurance and also cover the famous Merton problem from mathematical finance. Wherever possible, the proofs are probabilistic but in some cases well-established analytical methods are used.

The book is directed towards graduate students and researchers in actuarial science and mathematical finance who want to learn stochastic control within an insurance setting, but it will also appeal to applied probabilists interested in the insurance applications and to practitioners who want to learn more about how the method works.

Readers should be familiar with basic probability theory and have a working knowledge of Brownian motion, Markov processes, martingales and stochastic calculus. Some knowledge of measure theory will also be useful for following the proofs.

Contents:

1 Stochastic Control in Discrete Time 1

1.1 Dynamic Programming 1

1.1.2 Dynamic Programming 2

1.1.3 The Optimal Strategy 4

1.1.4 Numerical Solutions for T = [infinity] 6

1.2 Optimal Dividend Strategies in Risk Theory 9

1.2.1 The Model 9

1.2.2 The Optimal Strategy 12

1.2.3 Premia of Size One 16

1.3 Minimising Ruin Probabilities 20

1.3.1 Optimal Reinsurance 20

1.3.2 Optimal Investment 24

2 Stochastic Control in Continuous Time 27

2.1 The Hamilton-Jacobi-Bellman Approach 28

2.2 Minimising Ruin Probabilities for a Diffusion Approximation 34

2.2.1 Optimal Reinsurance 34

2.2.2 Optimal Investment 39

2.2.3 Optimal Investment and Reinsurance 42

2.3 Minimising Ruin Probabilities for a Classical Risk Model 43

2.3.1 Optimal Reinsurance 44

2.3.2 Optimal Investment 54

2.3.3 Optimal Reinsurance and Investment 64

2.4 Optimal Dividends in the Classical Risk Model 69

2.4.1 Restricted Dividend Payments 70

2.4.2 Unrestricted Dividend Payments 79

2.5 Optimal Dividends for a Diffusion Approximation 97

2.5.1 Restricted Dividend Payments 97

2.5.2 Unrestricted Dividend Payments 102

2.5.3 A Note on Viscosity Solutions 104

3 Problems in Life Insurance 113

3.1 Merton's Problem for Life Insurers 114

3.1.1 The Classical Merton Problem 114

3.1.2 Single Life Insurance Contract 122

3.2 Optimal Dividends and Bonus Payments 127

3.2.1 Utility Maximisation of Dividends 127

3.2.2 Utility Maximisation of Bonus 132

3.3 Optimal Control of a Pension Fund 135

3.3.1 No Constraints 136

3.3.2 Fixed [theta] 141

3.3.3 Fixed c 142

3.3.4 Power Loss Function and [sigma subscript B] = 0 143

4 Asymptotics of Controlled Risk Processes 147

4.1 Maximising the Adjustment Coefficient 147

4.1.1 Optimal Reinsurance 148

4.1.2 Optimal Investment 152

4.1.3 Optimal Reinsurance and Investment 153

4.2 Cramer-Lundberg Approximations for Controlled Classical Risk Models 154

4.2.1 Optimal Proportional Reinsurance 154

4.2.2 Optimal Excess of Loss Reinsurance 163

4.2.3 Optimal Investment 165

4.2.4 Optimal Proportional Reinsurance and Investment 171

4.3 The Heavy-Tailed Case 174

4.3.1 Proportional Reinsurance 174

4.3.2 Excess of Loss Reinsurance 179

4.3.3 Optimal Investment 181

4.3.4 Optimal Proportional Reinsurance and Investment 194

A Stochastic Processes and Martingales 201

A.1 Stochastic Processes 201

A.2 Filtration and Stopping Times 201

A.3 Martingales 202

A.4 Poisson Processes 203

A.5 Brownian Motion 205

A.6 Stochastic Integrals and Ito's Formula 206

A.7 Some Tail Asymptotics 209

B Markov Processes and Generators 211

B.1 Definition of Markov Processes 211

B.2 The Generator 211

C Change of Measure Techniques 215

C.2 The Brownian Motion 216

C.3 The Classical Risk Model 217

D Risk Theory 219

D.1 The Classical Risk Model 220

D.1.2 Small Claims 221

D.1.3 Large Claims 223

D.2 Perturbed Risk Models 225

D.3 Diffusion Approximations 226

D.4 Premium Calculation Principles 227

D.5 Reinsurance 228

E The Black-Scholes Model 231

F Life Insurance 235

F.1 Classical Life Insurance 235

F.2 Bonus Schemes 237

F.3 Unit-Linked Insurance Contracts 238.

Notes:

Includes bibliographical references (pages [241]-249) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Simon Nelson Patten Fund.

ISBN:

9781848000025

1848000022

1848000030

9781848000032

OCLC:

166357917