Semi-Markov Risk Models for Finance, Insurance and Reliability

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Semi-Markov Risk Models for Finance, Insurance and Reliability
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Semi-Markov Risk Models for Finance, Insurance and Reliability
出版社 Springer US
DOI 10.10070-387-70730-1
版权 2007
ISBN 978-0-387-70729-7 (Print) 978-0-387-70730-3 (Online)
学科分类 数学和统计学
学科 Mathematics, Probability Theory and Stochastic Processes, Quantitative Finance, Finance Banking, Financial Economics

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关键词：reliability Insurance liability Ability Finance Finance Finance models models Insurance Insurance Risk Risk reliability reliability

Contents
Preface     XV
 
1 Probability Tools for Stochastic Modelling 1
 1 The Sample Space 1
 2 Probability Space 2
 3 Random Variables 6
 4 Integrability, Expectation and Independence 8
 5 Main Distribution Probabilities 14
  5.1 The Binomial Distribution 15
  5.2 The Poisson Distribution 16
  5.3 The Normal (or Laplace-Gauss) Distribution 16
  5.4 The Log-Normal Distribution 19
  5.5 The Negative Exponential Distribution 20
  5.6 The Multidimensional Normal Distribution 20
 6 Conditioning (From Independence to Dependence) 22
  6.1 Conditioning: Introductory Case 22
  6.2 Conditioning: General Case 26
  6.3 Regular Conditional Probability 30
 7 Stochastic Processes 34
 8 Martingales 37
 9 Brownian Motion 40
 
2 Renewal Theory and Markov Chains 43
 1 Purpose of Renewal Theory 43
 2 Main Definitions 44
 3 Classification of Renewal Processes 45
 4 The Renewal Equation 50
 5 The Use of Laplace Transform 55
  5.1 The Laplace Transform 55
  5.2 The Laplace-Stieltjes (L-S) Transform 55
 6 Application of Wald!ˉs Identity 56
 7 Asymptotical Behaviour of the N(t)-Process 57
 8 Delayed and Stationary Renewal Processes 57
 9 Markov Chains 58
  9.1 Definitions 58
  9.2 Markov Chain State Classification 62
  9.3 Occupation Times 66
  9.4 Computations of Absorption Probabilities 67
  9.5 Asymptotic Behaviour 67

 
VI                                                                                                             Contents
  9.6 Examples 71
  9.7 A Case Study in Social Insurance (Janssen (1966)) 74
 
3 Markov Renewal Processes, Semi-Markov Processes And 
   Markov Random Walks  77
 1 Positive (J-X) Processes 77
 2 Semi-Markov and Extended Semi-Markov Chains 78
 3 Primary Properties 79
 4 Examples 83
 5 Markov Renewal Processes, Semi-Markov and
    Associated Counting Processes 85
 6 Markov Renewal Functions 87
 7 Classification of the States of an MRP 90
 8 The Markov Renewal Equation  91
 9 Asymptotic Behaviour of an MRP 92
  9.1 Asymptotic Behaviour of Markov Renewal Functions 92
  9.2 Asymptotic Behaviour of Solutions of Markov Renewal
       Equations 93
 10 Asymptotic Behaviour of SMP 94
  10.1 Irreducible Case 94
  10.2 Non-irreducible Case 96
   10.2.1 Uni-Reducible Case 96
   10.2.2 General Case 97
 11 Delayed and Stationary MRP 98
 12 Particular Cases of MRP  102
  12.1 Renewal Processes and Markov Chains 102
  12.2 MRP of Zero Order (PYKE (1962)) 102
   12.2.1 First Type of Zero Order MRP 102
   12.2.2 Second Type of Zero Order MRP 103
  12.3 Continuous Markov Processes 104
 13 A Case Study in Social Insurance (Janssen (1966)) 104
  13.1 The Semi-Markov Model 104
    13.2 Numerical Example  105
 14 (J-X) Processes 106
 15 Functionals of (J-X) Processes 107
 16 Functionals of Positive (J-X) Processes 111
 17 Classical Random Walks and Risk Theory 112
  17.1 Purpose 112
  17.2 Basic Notions on Random Walks 112
  17.3 Classification of Random Walks 115
 18 Defective Positive (J-X) Processes 117
 19 Semi-Markov Random Walks 121

 
 
Contents                                                                                                           
VII
20 Distribution of the Supremum for Semi-Markov 
     Random Walks 123
 21 Non-Homogeneous Markov and Semi-Markov Processes 124
  21.1 General Definitions 124
   21.1.1 Completely Non-Homogeneous Semi-Markov
    Processes 124
   21.1.2 Special Cases 128
 
4 Discrete Time and Reward SMP and their Numerical Treatment 131
 1 Discrete Time Semi-Markov Processes 131
  1.1 Purpose 131
  1.2 DTSMP Definition 131
 2 Numerical Treatment of SMP 133
 3 DTSMP and SMP Numerical Solutions  137
4 Solution of DTHSMP and DTNHSMP in the Transient Case: 
   a Transportation Example 142
  4.1. Principle of the Solution 142
  4.2. Semi-Markov Transportation Example 143
   4.2.1 Homogeneous Case 143
   4.2.2 Non-Homogeneous Case 147
 5 Continuous and Discrete Time Reward Processes 149
  5.1 Classification and Notation 150
   5.1.1 Classification of Reward Processes 150
   5.1.2 Financial Parameters 151
  5.2 Undiscounted SMRWP 153
   5.2.1 Fixed Permanence Rewards 153
   5.2.2 Variable Permanence and Transition Rewards 154
   5.2.3 Non-Homogeneous Permanence and Transition
            Rewards 155
  5.3 Discounted SMRWP 156
   5.3.1 Fixed Permanence and Interest Rate Cases 156
   5.3.2 Variable Interest Rates, Permanence 
            and Transition Cases 158
   5.3.3 Non-Homogeneous Interest Rate, Permanence 
            and Transition Case 159
 6 General Algorithms for DTSMRWP 159
 7 Numerical Treatment of SMRWP 161
  7.1 Undiscounted Case 161
  7.2 Discounted Case 163
 8 Relation Between DTSMRWP and SMRWP Numerical
    Solutions  165
  8.1 Undiscounted Case 166
  8.2 Discounted Case 168

 
 
VIII                                                                                                             Contents
5 Semi-Markov Extensions of the Black-Scholes Model 171
 1 Introduction to Option Theory 171
 2 The Cox-Ross-Rubinstein (CRR) or Binomial Model 174
  2.1 One-Period Model 175
   2.1.1 The Arbitrage Model 176
   2.1.2 Numerical Example 177
  2.2 Multi-Period Model 178
   2.2.1 Case of Two Periods 178
   2.2.2 Case of n Periods 179
   2.2.3 Numerical Example 180
 3 The Black-Scholes Formula as Limit of the Binomial
    Model   181
  3.1 The Log-Normality of the Underlying Asset 181
  3.2. The Black-Scholes Formula 183
 4 The Black-Scholes Continuous Time Model 184
  4.1 The Model 184
  4.2 The It? or Stochastic Calculus 184
  4.3 The Solution of the Black-Scholes-Samuelson
        Model 186
  4.4 Pricing the Call with the Black-Scholes-Samuelson
        Model 188
   4.4.1 The Hedging Portfolio 188
   4.4.2 The Risk Neutral Measure and the Martingale
            Property 190
   4.4.3 The Call-Put Parity Relation 191
 5 Exercise on Option Pricing 192
 6 The Greek Parameters 193
  6.1 Introduction 193
  6.2 Values of the Greek Parameters 195
  6.3 Exercises 196
 7 The Impact of Dividend Distribution 198
 8 Estimation of the Volatility 199
  8.1 Historic Method 199
  8.2 Implicit Volatility Method 200
 9 Black and Scholes on the Market 201
  9.1 Empirical Studies 201
  9.2 Smile Effect 201
 10 The Janssen-Manca Model 201
  10.1 The Markov Extension of the One-Period 
                       CRR Model 202
   10.1.1 The Model 202
   10.1.2 Computational Option Pricing Formula for the
              One-Period Model 206

 
Contents                                                                                                           
IX
s
      10.1.3 Example 207
  10.2 The Multi-Period Discrete Markov Chain Model 209
    10.3 The Multi-Period Discrete Markov Chain Limit
             Model   211
  10.4 The Extension of the Black-Scholes Pricing 
      Formula with Markov Environment: 
             The Janssen-Manca Formula 213
 11 The Extension of the Black-Scholes Pricing Formula
      with Markov Environment: The Semi-Markovian
     Janssen-Manca-Volpe formula 216
  11.1 Introduction 216
  11.2 The Janssen-Manca-?inlar Model 216
   11.2.1 The JMC (Janssen-Manca-?inlar) Semi-
              Markov Model (1995, 1998) 217
   11.2.2 The Explicit Expression of S(t) 218
  11.3 Call Option Pricing 219
  11.4 Stationary Option Pricing Formula 221
 12 Markov and Semi-Markov Option Pricing Models with
      Arbitrage Possibility 222
  12.1 Introduction to the Janssen-Manca-Di Biase 
       Models 222
  12.2 The Homogeneous Markov JMD (Janssen-Manca-
    Di Biase) Model for the Underlying Asset 223
  12.3 Particular Cases 224
  12.4 Numerical Example for the JMD Markov Model 225
    12.5 The Continuous Time Homogeneous Semi-Markov
             JMD Model for the Underlying Asset  227
  12.6 Numerical Example for the Semi-Markov 
          JMD Model 228
  12.7 Conclusion 229
 
6 Other Semi-Markov Models in Finance and Insurance 231
 1 Exchange of Dated Sums in a Stochastic Homogeneous
    Environment 231
  1.1 Introduction 231
  1.2 Deterministic Axiomatic Approach to Financial Choices 232
  1.3 The Homogeneous Stochastic Approach 234
  1.4 Continuous Time Models with Finite State Space  235
  1.5 Discrete Time Model with Finite State Space 236
  1.6 An Example of Asset Evaluation 237
  1.7 Two Transient Case Examples 238
  1.8 Financial Application of Asymptotic Results 244
 2 Discrete Time Markov and Semi-Markov Reward Processes
    and Generalised Annuities 245

 
 
X                                                                                                             Contents
  2.1 Annuities and Markov Reward Processes 246
  2.2 HSMRWP and Stochastic Annuities Generalization 248
 3 Semi-Markov Model for Interest Rate Structure 251
  3.1 The Deterministic Environment 251
  3.2 The Homogeneous Stochastic Interest Rate Approach 252
  3.3 Discount Factors 253
  3.4 An Applied Example in the Homogeneous Case 255
  3.5 A Factor Discount Example in the Non-Homogeneous
        Case  257
 4 Future Pricing Model 259
  4.1 Description of Data 260
  4.2 The Input Model 261
  4.3 The Results 262
 5 A Social Security Application with Real Data 265
  5.1 The Transient Case Study 265
  5.2 The Asymptotic Case 267
 6 Semi-Markov Reward Multiple-Life Insurance Models 269
 7 Insurance Model with Stochastic Interest Rates 276
  7.1 Introduction 276
  7.2 The Actuarial Problem 276
  7.3 A Semi-Markov Reward Stochastic Interest Rate Model 277
 
7 Insurance Risk Models 281
 1 Classical Stochastic Models for Risk Theory and Ruin 
    Probability 281
 1.1 The G/G or E.S. Andersen Risk Model 282
   1.1.1 The Model 282
   1.1.2 The Premium 282
   1.1.3 Three Basic Processes 284
   1.1.4 The Ruin Problem 285
  1.2 The P/G or Cramer-Lundberg Risk Model 287
   1.2.1 The Model 287
   1.2.2 The Ruin Probability 288
   1.2.3 Risk Management Using Ruin Probability 293
   1.2.4 Cramer!ˉs Estimator 294
 2 Diffusion Models for Risk Theory and Ruin Probability 301
  2.1 The Simple Diffusion Risk Model 301
  2.2 The ALM-Like Risk Model (Janssen (1991), (1993)) 302
  2.3 Comparison of ALM-Like and Cramer-Lundberg Risk
        Models 304
  2.4 The Second ALM-Like Risk Model 305
 3 Semi-Markov Risk Models 309

 
 
Contents                                                                                                           
XI
  3.1 The Semi-Markov Risk Model (or SMRM) 309
   3.1.1 The General SMR Model 309
   3.1.2 The Counting Claim Process 312
   3.1.3 The Accumulated Claim Amount Process 314
   3.1.4 The Premium Process 315
   3.1.5 The Risk and Risk Reserve Processes 316
  3.2 The Stationary Semi-Markov Risk Model 316
  3.3 Particular SMRM with Conditional
        Independence 316
   3.3.1 The SM/G Model 317
   3.3.2 The G/SM Model 317
   3.3.3 The P/SM Model 317
   3.3.4 The M/SM Model 318
   3.3.5 The M
'/SM Model 318
   3.3.6 The SM(0)/SM(0) Model 318
   3.3.7 The SM
'(0)/SM'(0) Model 318
   3.3.8 The Mixed Zero Order SM
' (0)/SM(0) and 
            SM(0)/SM
' (0) Models 319
  3.4 The Ruin Problem for the General SMRM 320
   3.4.1 Ruin and Non-Ruin Probabilities 320
   3.4.2 Change of Premium Rate 321
   3.4.3 General Solution of the Asymptotic Ruin 
            Probability Problem for a General SMRM 322
  3.5 The Ruin Problem for Particular SMRM 324
   3.5.1 The Zero Order Model SM(0)/SM(0) 324
   3.5.2 The Zero Order Model SM
'(0)/SM'(0) 325
   3.5.3 The Model M/SM 325
   3.5.4 The Zero Order Models as Special Case 
            of the Model M/SM 328
  3.6 The M
'/SM Model 329
   3.6.1 General Solution 329
   3.6.2 Particular Cases: the M/M and M
'/M Models 332
 
8 Reliability and Credit Risk Models 335
 1 Classical Reliability Theory 335
  1.1 Basic Concepts 335
  1.2 Classification of Failure Rates 336
  1.3 Main Distributions Used in Reliability 338
  1.4 Basic Indicators of Reliability 339
  1.5 Complex and Coherent Structures 340
 2 Stochastic Modelling in Reliability Theory 343
  2.1 Maintenance Systems 343
  2.2 The Semi-Markov Model for Maintenance Systems 346
  2.3 A Classical Example 348

 
 
XII                                                                                                             Contents
 3 Stochastic Modelling for Credit Risk Management 351
  3.1 The Problem of Credit Risk 351
  3.2 Construction of a Rating Using the Merton Model
         for the Firm 352
  3.3 Time Dynamic Evolution of a Rating 355
   3.3.1 Time Continuous Model 355
   3.3.2 Discrete Continuous Model 356
   3.3.3 Example 358
   3.3.4 Rating and Spreads on Zero Bonds 360
 4 Credit Risk as a Reliability Model 361
  4.1 The Semi-Markov Reliability Credit Risk Model 361
  4.2 A Homogeneous Case Example 362
  4.3 A Non-Homogeneous Case Example 365
 
9 Generalised Non-Homogeneous Models for Pension Funds and
     Manpower Management 373
 1 Application to Pension Funds Evolution 373
  1.1 Introduction 374
  1.2 The Non-homogeneous Semi-Markov Pension Fund
        Model 375
   1.2.1 The DTNHSM Model 376
   1.2.2 The States of DTNHSMPFM 379
   1.2.3 The Concept of Seniority in the DTNHSPFM 379
  1.3 The Reserve Structure 382
  1.4 The Impact of Inflation and Interest Variability 383
  1.5 Solving Evolution Equations 385
  1.6 The Dynamic Population Evolution of the Pension
        Funds 389
  1.7 Financial Equilibrium of the Pension Funds 392
  1.8 Scenario and Data 395
   1.8.1 Internal Scenario 396
   1.8.2 Historical Data 396
   1.8.3 Economic Scenario 397
  1.9 Usefulness of the NHSMPFM 398
 2 Generalized Non-Homogeneous Semi-Markov Model for
             Manpower Management 399
  2.1 Introduction 399
  2.2 GDTNHSMP for the Evolution of Salary Lines 400
  2.3 The GDTNHSMRWP for Reserve Structure 402
  2.4 Reserve Structure Stochastic Interest Rate 403
  2.5 The Dynamics of Population Evolution 404
  2.6 The Computation of Salary Cost Present Value 405

 
 
 
Contents                                                                                                           
XIII
References    407
Author index   423
Subject index   425