Basic Stochastic Processes

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.

Autorentext

Pierre Devolder, Université catholique de Louvain, Belgium.

Jacques Janssen, Solvay Business School, Brussels, Belgium.

Raimondo Manca, University "La Sapienza", Rome, Italy.

Inhalt

INTRODUCTION xi

CHAPTER 1. BASIC PROBABILISTIC TOOLS FOR STOCHASTIC MODELING 1

1.1. Probability space and random variables 1

1.2. Expectation and independence 4

1.3. Main distribution probabilities 7

1.3.1. Binomial distribution 7

1.3.2. Negative exponential distribution 8

1.3.3. Normal (or LaplaceGauss) distribution 8

1.3.4. Poisson distribution 11

1.3.5. Lognormal distribution 11

1.3.6. Gamma distribution 12

1.3.7. Pareto distribution 13

1.3.8. Uniform distribution 16

1.3.9. Gumbel distribution 16

1.3.10. Weibull distribution 16

1.3.11. Multi-dimensional normal distribution 17

1.3.12. Extreme value distribution 19

1.4. The normal power (NP) approximation 28

1.5. Conditioning 31

1.6. Stochastic processes 39

1.7. Martingales 43

CHAPTER 2. HOMOGENEOUS AND NON-HOMOGENEOUS RENEWAL MODELS 47

2.1. Introduction 47

2.2. Continuous time non-homogeneous convolutions 49

2.2.1. Non-homogeneous convolution product 49

2.3. Homogeneous and non-homogeneous renewal processes 53

2.4. Counting processes and renewal functions 56

2.5. Asymptotical results in the homogeneous case 61

2.6. Recurrence times in the homogeneous case 63

2.7. Particular case: the Poisson process 66

2.7.1. Homogeneous case 66

2.7.2. Non-homogeneous case 68

2.8. Homogeneous alternating renewal processes 69

2.9. Solution of non-homogeneous discrete timevevolution equation 71

2.9.1. General method 71

2.9.2. Some particular formulas 73

2.9.3. Relations between discrete time and continuous time renewal equations 74

CHAPTER 3. MARKOV CHAINS 77

3.1. Definitions 77

3.2. Homogeneous case 78

3.2.1. Basic definitions 78

3.2.2. Markov chain state classification 81

3.2.3. Computation of absorption probabilities 87

3.2.4. Asymptotic behavior 88

3.2.5. Example: a management problem in an insurance company 93

3.3. Non-homogeneous Markov chains 95

3.3.1. Definitions 95

3.3.2. Asymptotical results 98

3.4. Markov reward processes 99

3.4.1. Classification and notation 99

3.5. Discrete time Markov reward processes (DTMRWPs) 102

3.5.1. Undiscounted case 102

3.5.2. Discounted case 105

3.6. General algorithms for the DTMRWP 111

3.6.1. Homogeneous MRWP 112

3.6.2. Non-homogeneous MRWP 112

CHAPTER 4. HOMOGENEOUS AND NON-HOMOGENEOUS SEMI-MARKOV MODELS 113

4.1. Continuous time semi-Markov processes 113

4.2. The embedded Markov chain 117

4.3. The counting processes and the associated semi-Markov process 118

4.4. Initial backward recurrence times 120

4.5. Particular cases of MRP 122

4.5.1. Renewal processes and Markov chains 122

4.5.2. MRP of zero-order (PYKE (1962)) 122

4.5.3. Continuous Markov processes 124

4.6. Examples 124

4.7. Discrete time homogeneous and non-homogeneous semi-Markov processes 127

4.8. Semi-Markov backward processes in discrete time 129

4.8.1. Definition in the homogeneous case 129

4.8.2. Semi-Markov backward processes in discrete time for the non-homogeneous case 130

4.8.3. DTSMP numerical solutions 133

4.9. Discrete time reward processes 137

4.9.1. Undiscounted SMRWP 137

4.9.2. Discounted SMRWP 141

4.9.3. General algorithms for DTSMRWP 144

4.10. Markov renewal functions in the homogeneous case 146

4.10.1. Entrance times 146

4.10.2. The Markov renewal equation 150

4.10.3. Asymptotic behavior of an MRP 151

4.10.4. Asymptotic behavior of SMP 153

4.11. Markov renewal equations for the non-homogeneous case 158

4.11.1. Entrance time 158 4.11.2. The ...