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.

Pierre Devolder, Université catholique de Louvain, Belgium.

Jacques Janssen, Solvay Business School, Brussels, Belgium.

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

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 Laplace–Gauss) 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 Markov renewal equation 162

CHAPTER 5. STOCHASTIC CALCULUS  165

5.1. Brownian motion 165

5.2. General definition of the stochastic integral 167

5.2.1. Problem of stochastic integration 167

5.2.2. Stochastic integration of simple predictable processes and semi-martingales 168

5.2.3. General definition of the stochastic integral 170

5.3. Itô’s formula 177

5.3.1. Quadratic variation of a semi-martingale 177

5.3.2. Itô’s formula 179

5.4. Stochastic integral with standard Brownian motion as an integrator process 180

5.4.1. Case of simple predictable processes 181

5.4.2. Extension to general integrator processes 183

5.5. Stochastic differentiation 184

5.5.1. Stochastic differential 184

5.5.2. Particular cases 184

5.5.3. Other forms of Itô’s formula 185

5.6. Stochastic differential equations 191

5.6.1. Existence and unicity general theorem 191

5.6.2. Solution of stochastic differential equations 195

5.6.3. Diffusion processes 199

5.7. Multidimensional diffusion processes 202

5.7.1. Definition of multidimensional Itô and diffusion processes 203

5.7.2. Properties of multidimensional diffusion processes 203

5.7.3. Kolmogorov equations 205

5.7.4. The Stroock–Varadhan martingale characterization of diffusion processes 208

5.8. Relation between the resolution of PDE and SDE problems. The Feynman–Kac formula 209

5.8.1. Terminal payoff 209

5.8.2. Discounted payoff function 210

5.8.3. Discounted payoff function and payoff rate 210

5.9. Application to option theory 213

5.9.1. Options 213

5.9.2. Black and Scholes model 216

5.9.3. The Black and Scholes partial differential equation (BSPDE) and the BS formula 216

5.9.4. Girsanov theorem 219

5.9.5. The risk-neutral measure and the martingale property 221

5.9.6. The risk-neutral measure and the evaluation of derivative products 224

CHAPTER 6. LÉVY PROCESSES 227

6.1. Notion of characteristic functions 227

6.2. Lévy processes 228

6.3. Lévy–Khintchine formula 230

6.4. Subordinators 234

6.5. Poisson measure for jumps 234

6.5.1. The Poisson random measure 234

6.5.2. The compensated Poisson process 235

6.5.3. Jump measure of a Lévy process 236

6.5.4. The Itô–Lévy decomposition 236

6.6. Markov and martingale properties of Lévy processes 237

6.6.1. Markov property 237

6.6.2. Martingale properties 239

6.6.3. Itô formula 240

6.7. Examples of Lévy processes 240

6.7.1. The lognormal process: Black and Scholes process 240

6.7.2. The Poisson process 241

6.7.3. Compensated Poisson process 242

6.7.4. The compound Poisson process 242

6.8. Variance gamma (VG) process 244

6.8.1. The gamma distribution 244

6.8.2. The VG distribution 245

6.8.3. The VG process 246

6.8.4. The Esscher transformation 247

6.8.5. The Carr–Madan formula for the European call 249

6.9. Hyperbolic Lévy processes 250

6.10. The Esscher transformation 252

6.10.1. Definition 252

6.10.2. Option theory with hyperbolic Lévy processes 253

6.10.3. Value of the European option call 255

6.11. The Brownian–Poisson model with jumps 256

6.11.1. Mixed arithmetic Brownian–Poisson and geometric Brownian–Poisson processes 256

6.11.2. Merton model with jumps 258

6.11.3. Stochastic differential equation (SDE) for mixed arithmetic Brownian–Poisson and geometric Brownian–Poisson processes 261

6.11.4. Value of a European call for the lognormal Merton model 264

6.12. Complete and incomplete markets 264

6.13. Conclusion 265

CHAPTER 7. ACTUARIAL EVALUATION, VAR AND STOCHASTIC INTEREST RATE MODELS 267

7.1. VaR technique 267

7.2. Conditional VaR value 271

7.3. Solvency II 276

7.3.1. The SCR indicator 276

7.3.2. Calculation of MCR 278

7.3.3. ORSA approach 279

7.4. Fair value 280

7.4.1. Definition 280

7.4.2. Market value of financial flows 281

7.4.3. Yield curve 281

7.4.4. Yield to maturity for a financial investment and a bond 283

7.5. Dynamic stochastic time continuous time model for instantaneous interest rate 284

7.5.1. Instantaneous deterministic interest rate 284

7.5.2. Yield curve associated with a deterministic instantaneous interest rate 285

7.5.3. Dynamic stochastic continuous time model for instantaneous interest rate 286

7.5.4. The OUV stochastic model 287

7.5.5. The CIR model 289

7.6. Zero-coupon pricing under the assumption of no arbitrage 292

7.6.1. Stochastic dynamics of zero-coupons 292

7.6.2. The CIR process as rate dynamic 295

7.7. Market evaluation of financial flows 298

BIBLIOGRAPHY 301

INDEX 309

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