Stochastic processes for insurance and finance:
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| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
Chichester [u.a.]
Wiley
2008
|
| Ausgabe: | Paperback ed. |
| Schriftenreihe: | Wiley paperback series
|
| Schlagworte: | |
| Online-Zugang: | Kostenfrei Cover Inhaltsverzeichnis |
| Beschreibung: | XVIII, 654 S. graph. Darst. |
| ISBN: | 9780470743638 |
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| 245 | 1 | 0 | |a Stochastic processes for insurance and finance |c Tomasz Rolski ... |
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Datensatz im Suchindex
| _version_ | 1817681160153595904 |
|---|---|
| adam_text |
Contents
Preface
. xiii
List of Principal Notation
. xvii
1
Concepts from Insurance and Finance
. 1
1.1
Introduction
. 1
1.2
The Claim Number Process
. 2
1.2.1
Renewal Processes
. 3
1.2.2
Mixed
Poisson
Processes
. 4
1.2.3
Some Other Models
. 5
1.3
The Claim Size Process
. 7
1.3.1
Dangerous Risks
. 7
1.3.2
The Aggregate Claim Amount
. 8
1.3.3
Comparison of Risks
. 10
1.4
Solvability of the Portfolio
. 10
1.4.1
Premiums
. 10
1.4.2
The Risk Reserve
. 12
1.4.3
Economic Environment
. 12
1.5
Reinsurance
. 14
1.5.1
Need for Reinsurance
. 14
1.5.2
Types of Reinsurance
. 14
1.6
Ruin Problems
. 17
1.7
Related Financial Topics
. 20
1.7.1
Investment of Surplus
. 20
1.7.2
Diffusion Processes
. 20
1.7.3
Equity Linked Life Insurance
. 21
2
Probability Distributions
. 23
2.1
Random Variables and Their Characteristics
. 23
2.1.1
Distributions of Random Variables
. 23
2.1.2
Basic Characteristics
. 25
STOCHASTIC PROCESSES FOR INSURANCE AND FINANCE
2.1.3
Independence and Conditioning
. 26
2.1.4
Convolution
.
28
2.1.5
Transforms
.
28
2.2
Parametrized Families of Distributions
. 31
2.2.1
Discrete Distributions
. 31
2.2.2
Absolutely Continuous Distributions
. 32
2.2.3
Parametrized Distributions with Heavy Tail
. 33
2.2.4
Operations on Distributions
. 35
2.2.5
Some Special Functions
. 36
2.3
Associated Distributions
. 38
2.4
Distributions with Monotone Hazard Rates
. 43
2.4.1
Discrete Distributions
. 43
2.4.2
Absolutely Continuous Distributions
. 46
2.5
Heavy-Tailed Distributions
. 49
2.5.1
Definition and Basic Properties
. 49
2.5.2
Subexponential Distributions
. 49
2.5.3
Criteria for Subexponentiality and the Class S*
. . 54
2.5.4
Pareto Mixtures of Exponentials
. 62
2.6
Detection of Heavy-Tailed Distributions
. 65
2.6.1
Large Claims
. 65
2.6.2
Quantile Plots
. 68
2.6.3
Mean Residual Hazard Function
. 74
2.6.4
Extreme Value Statistics
. 76
Premiums and Ordering of Risks
. 79
3.1
Premium Calculation Principles
. 79
3.1.1
Desired Properties of "Good" Premiums
. 79
3.1.2
Basic Premium Principles
. 80
3.1.3
Quantile Function: Two More Premium Principles
. 82
3.2
Ordering of Distributions
. 83
3.2.1
Concepts of Utility Theory
. 83
3.2.2
Stochastic Order
. 86
3.2.3
Stop-Loss Order
. 87
3.2.4
The Zero Utility Principle
. 91
3.3
Some Aspects of Reinsurance
. 94
Distributions of Aggregate Claim Amount
. 99
4.1
Individual and Collective Model
. 99
4.2
Compound Distributions
. 100
4.2.1
Definition and Elementary Properties
. 100
4.2.2
Three Special Cases
. 103
4.2.3
Some Actuarial Applications
. 105
4.2.4
Ordering of Compounds
. 106
CONTENTS
vii
4.2.5
The Larger Claims in the Portfolio
.107
4.3
Claim Number Distributions
.110
4.3.1
Classical Examples; Panjer's Recurrence Relation
.
Ill
4.3.2
Discrete Compound
Poisson
Distributions
. 112
4.3.3
Mixed
Poisson
Distributions
. 113
4.4
Recursive Computation Methods
. 115
4.4.1
The Individual Model:
De Priľs
Algorithm
.115
4.4.2
The Collective Model: Panjer's Algorithm
.118
4.4.3
A Continuous Version of Panjer's Algorithm
. 120
4.5
Lundberg
Bounds
.125
4.5.1
Geometric Compounds
. 125
4.5.2
More General Compound Distributions
. 128
4.5.3
Estimation of the Adjustment Coefficient
. 129
4.6
Approximation by Compound Distributions
. 131
4.6.1
The Total Variation Distance
.133
4.6.2
The Compound
Poisson
Approximation
.134
4.6.3
Homogeneous Portfolio
.135
4.6.4
Higher-Order Approximations
.138
4.7
Inverting the Fourier Transform
.141
5
Risk Processes
.147
5.1
Time-Dependent Risk Models
.147
5.1.1
The Ruin Problem
.147
5.1.2
Computation of the Ruin Function
.149
5.1.3
A Dual Queueing Model
.150
5.1.4
A Risk Model in Continuous Time
.151
5.2
Poisson
Arrival Processes
.156
5.2.1
Homogeneous
Poisson
Processes
.156
5.2.2
Compound
Poisson
Processes
.160
5.3
Ruin Probabilities: The Compound
Poisson
Model
.162
5.3.1
An Integro-Differential Equation
. 162
5.3.2
An Integral Equation
. 164
5.3.3
Laplace Transforms, Pollaczek-Khinchin Formula
. 165
5.3.4
Severity of Ruin
. 167
5.4
Bounds, Asymptotics and Approximations
. 169
5.4.1
Lundberg
Bounds
.170
5.4.2
The
Cramér-Lundberg
Approximation
.172
5.4.3
Subexponential Claim Sizes
.174
5.4.4
Approximation by Moment Fitting
.176
5.4.5
Ordering of Ruin Functions
.181
5.5
Numerical Evaluation of Ruin Functions
.183
5.6
Finite-Horizon Ruin Probabilities
.191
5.6.1
Deterministic Claim Sizes
.191
vm
STOCHASTIC PROCESSES FOR INSURANCE AND FINANCE
5.6.2
Seal's Formulae
.
193
5.6.3
Exponential Claim Sizes
.195
Renewal Processes and Random Walks
.205
6.1
Renewal Processes
.205
6.1.1
Definition and Elementary Properties
.205
6.1.2
The Renewal Function; Delayed Renewal Processes
208
6.1.3
Renewal Equations and Lorden's Inequality
. 213
6.1.4
Key Renewal Theorem
.216
6.1.5
Another Look at the Aggregate Claim Amount
. . 219
6.2
Extensions and Actuarial Applications
.221
6.2.1
Weighted Renewal Functions
.221
6.2.2
A Blackwell-Type Renewal Theorem
.225
6.2.3
Approximation to the Aggregate Claim Amount
. . 227
6.2.4
Lundberg
-Туре
Bounds
.231
6.3
Random Walks
.232
6.3.1
Ladder Epochs
.232
6.3.2
Random Walks with and without Drift
.233
6.3.3
Ladder Heights; Negative Drift
.235
6.4
The
Wiener-Hopf
Factorization
.237
6.4.1
General Representation Formulae
.237
6.4.2
An Analytical Factorization; Examples
.240
6.4.3
Ladder Height Distributions
.247
6.5
Ruin Probabilities:
Sparre
Andersen Model
.249
6.5.1
Formulae of Pollaczek-Khinchin Type
.249
6.5.2
Lundberg
Bounds
.255
6.5.3
The
Cramér-Lundberg
Approximation
.258
6.5.4
Compound
Poisson
Model with Aggregate Claims
. 260
6.5.5
Subexponential Claim Sizes
.263
Markov Chains
.269
7.1
Definition and Basic Properties
.269
7.1.1
Initial Distribution and Transition Probabilities
. . 269
7.1.2
Computation of the
η
-Step
Transition Matrix
. 272
7.1.3
Recursive Stochastic Equations
.275
7.1.4 Bonus-Malus
Systems
.277
7.2
Stationary Markov Chains
.280
7.2.1
Long-Run Behaviour
.280
7.2.2
Application of the Perron-Frobenius Theorem
. . . 283
7.2.3
Irreducibility and Aperiodicity
.285
7.2.4
Stationary Initial Distributions
.287
7.3
Markov Chains with Rewards
.290
7.3.1
Interest and Discounting
.290
CONTENTS ix
7.3.2
Discounted and Undiscounted Rewards
.291
7.3.3
Efficiency of
Bonus-Malus
Systems
.295
7.4
Monotonicity and Stochastic Ordering
.297
7.4.1
Monotone Transition Matrices
.297
7.4.2
Comparison of Markov Chains
.300
7.4.3
Application to
Bonus-Malus
Systems
.300
7.5
An Actuarial Application of Branching Processes
.302
8
Continuous-Time Markov Models
.309
8.1
Homogeneous Markov Processes
.309
8.1.1
Matrix Transition Function
.309
8.1.2
Kolmogorov Differential Equations
.313
8.1.3
An Algorithmic Approach
.317
8.1.4
Monotonicity of Markov Processes
.320
8.1.5
Stationary Initial Distributions
.323
8.2
Phase-Type Distributions
.324
8.2.1
Some Matrix Algebra and Calculus
.324
8.2.2
Absorption Time
.329
8.2.3
Operations on Phase-Type Distributions
.333
8.3
Risk Processes with Phase-Type Distributions
.337
8.3.1
The Compound
Poisson
Model
.337
8.3.2
Numerical Issues
.340
8.4
Nonhomogeneous Markov Processes
.344
8.4.1
Definition and Basic Properties
.344
8.4.2
Construction of Nonhomogeneous Markov Processes
349
8.4.3
Application to Life and Pension Insurance
.351
8.5
Mixed
Poisson
Processes
.358
8.5.1
Definition and Elementary Properties
.358
8.5.2
Markov Processes with Infinite State Space
.361
8.5.3
Mixed
Poisson
Processes as Pure Birth Processes
. 363
8.5.4
The Claim Arrival Epochs
.365
8.5.5
The Inter-Occurrence Times
.368
8.5.6
Examples
.370
9
Martingale Techniques I
.375
9.1
Discrete-Time Martingales
.375
9.1.1
Fair Games
.375
9.1.2
Filtrations and Stopping Times
.377
9.1.3
Martingales, Sub- and
Supermartingales
.379
9.1.4
Life-insurance Model with Multiple Decrements
. . 382
9.1.5
Convergence Results
.385
9.1.6
Optional Sampling Theorems
.387
9.1.7
Doob's Inequality
.391
;
STOCHASTIC PROCESSES
POR
INSURANCE AND FINANCE
9.1.8
The Doob-Meyer Decomposition
.392
9.2
Change of the Probability Measure
.393
9.2.1
The Likelihood Ratio Martingale
.394
9.2.2
Kolmogorov's Extension Theorem
.395
9.2.3
Exponential Martingales for Random Walks
. 397
9.2.4
Finite-Horizon Ruin Probabilities
.399
9.2.5
Simulation of Ruin Probabilities
.400
10
Martingale Techniques II
.403
10.1
Continuous-Time Martingales
.403
10.1.1
Stochastic Processes and Filtrations
.403
10.1.2
Stopping Times
.404
10.1.3
Martingales, Sub- and
Supermartingales
.406
10.1.4
Brownian Motion and Related Processes
.410
10.1.5
Uniform Integrability
.412
10.2
Some Fundamental Results
.415
10.2.1
Doob's Inequality
.415
10.2.2
Convergence Results
.416
10.2.3
Optional Sampling Theorems
.417
10.2.4
The Doob-Meyer Decomposition
.420
10.2.5
Kolmogorov's Extension Theorem
.421
10.2.6
Change of the Probability Measure
.422
10.3
Ruin Probabilities and Martingales
.425
10.3.1
Ruin Probabilities for Additive Processes
.425
10.3.2
Finite-Horizon Ruin Probabilities
.428
10.3.3
Law of Large Numbers for Additive Processes
. . . 431
10.3.4
An Identity for Finite-Horizon Ruin Probabilities
. 433
11
Piecewise Deterministic Markov Processes
.437
11.1
Markov Processes with Continuous State Space
.437
11.1.1
Transition Kernels
. 437
11.1.2
The Infinitesimal Generator
. 439
11.1.3
Dynkin's Formula
. 442
11.1.4
The Full Generator
. 443
11.2
Construction and Properties of PDMP
. 444
11.2.1
Behaviour between Jumps
.445
11.2.2
The Jump Mechanism
.447
11.2.3
The Generator of a PDMP
.448
11.2.4
An Application to Health Insurance
.455
11.3
The Compound
Poisson
Model Revisited
.459
11.3.1
Exponential Martingales via PDMP
.459
11.3.2
Change of the Probability Measure
.461
11.3.3
Cramér-Lundberg
Approximation
.464
CONTENTS xi
11.3.4
A Stopped Risk Reserve Process
.465
11.3.5
Characteristics of the Ruin Time
.467
11.4
Compound
Poisson
Model in an Economic Environment
. 471
11.4.1
Interest and Discounting
.471
11.4.2
A Discounted Risk Reserve Process
.472
11.4.3
The Adjustment Coefficient
.474
11.4.4
Decreasing Economic Factor
.475
11.5
Exponential Martingales: the
Sparre
Andersen Model
. . . 478
11.5.1
An Integral Equation
.479
11.5.2
Backward Markovization Technique
.480
11.5.3
Forward Markovization Technique
.481
12
Point Processes
.483
12.1
Stationary Point Processes
.484
12.1.1
Definition and Elementary Properties
.484
12.1.2
Palm Distributions and Campbell's Formula
. 486
12.1.3
Ergodic Theorems
.490
12.1.4
Marked Point Processes
.493
12.1.5
Ruin Probabilities in the Time-Stationary Model
.496
12.2
Mixtures and Compounds of Point Processes
.501
12.2.1
Nonhomogeneous
Poisson
Processes
.501
12.2.2
Cox Processes
.503
12.2.3
Compounds of Point Processes
.507
12.2.4
Comparison of Ruin Probabilities
.509
12.3
The Markov-Modulated Risk Model via PDMP
.514
12.3.1
A System of Integro-Differential Equations
.514
12.3.2
Law of Large Numbers
.515
12.3.3
The Generator and Exponential Martingales
. 516
12.3.4
Lundberg
Bounds
.520
12.3.5
Cramér-Lundberg
Approximation
.522
12.3.6
Finite-Horizon Ruin Probabilities
.524
12.4
Periodic Risk Model
.526
12.5
The
Björk-Grandell
Model via PDMP
.529
12.5.1
Law of Large Numbers
.530
12.5.2
The Generator and Exponential Martingales
. 531
12.5.3
Lundberg
Bounds
.534
12.5.4
Cramér-Lundberg
Approximation
.538
12.5.5
Finite-Horizon Ruin Probabilities
.539
12.6
Subexponential Claim Sizes
.540
12.6.1
General Results
.541
12.6.2
Poisson
Cluster Arrival Processes
.546
12.6.3
Superposition of Renewal Processes
.548
12.6.4
The Markov-Modulated Risk Model
.549
xii
STOCHASTIC PROCESSES FOR INSURANCE AND FINANCE
12.6.5
The
Björk-Grandell
Risk Model
.551
13
Diffusion Models
.553
13.1
Stochastic Differential Equations
.553
13.1.1
Stochastic Integrals and
Itô's
Formula
.553
13.1.2
Diffusion Processes
.561
13.1.3
Levy's Characterization Theorem
.565
13.2
Perturbed Risk Processes
.568
13.2.1
Lundberg
Bounds
.568
13.2.2
Modified Ladder Heights
.574
13.2.3
Cramér-Lundberg
Approximation
.578
13.2.4
Subexponential Claim Sizes
.580
13.3
Other Applications to Insurance and Finance
.582
13.3.1
The Black-Scholes Model
.582
13.3.2
Equity Linked Life Insurance
.588
13.3.3
Stochastic Interest Rates in Life Insurance
.593
13.4
Simple Interest Rate Models
.598
13.4.1
Zero-Coupon Bonds
.598
13.4.2
The Vasicek Model
.600
13.4.3
The Cox-Ingersoll-Ross Model
.602
Distribution Tables
.609
References
.617
Index
.639 |
| any_adam_object | 1 |
| building | Verbundindex |
| bvnumber | BV037237629 |
| classification_rvk | QP 826 QP 890 |
| ctrlnum | (OCoLC)699147780 (DE-599)HBZHT015795396 |
| discipline | Wirtschaftswissenschaften |
| edition | Paperback ed. |
| format | Book |
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| genre | 1\p (DE-588)4113937-9 Hochschulschrift gnd-content |
| genre_facet | Hochschulschrift |
| id | DE-604.BV037237629 |
| illustrated | Illustrated |
| indexdate | 2024-12-06T09:03:57Z |
| institution | BVB |
| isbn | 9780470743638 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-021151183 |
| oclc_num | 699147780 |
| open_access_boolean | 1 |
| owner | DE-634 DE-355 DE-BY-UBR DE-523 |
| owner_facet | DE-634 DE-355 DE-BY-UBR DE-523 |
| physical | XVIII, 654 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Wiley |
| record_format | marc |
| series2 | Wiley paperback series |
| spelling | Stochastic processes for insurance and finance Tomasz Rolski ... Paperback ed. Chichester [u.a.] Wiley 2008 XVIII, 654 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Wiley paperback series Versicherung (DE-588)4063173-4 gnd rswk-swf Versicherungsmathematik (DE-588)4063194-1 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Finanzmathematik (DE-588)4017195-4 gnd rswk-swf Versicherungswirtschaft (DE-588)4063206-4 gnd rswk-swf Finanzwirtschaft (DE-588)4017214-4 gnd rswk-swf Statistisches Modell (DE-588)4121722-6 gnd rswk-swf 1\p (DE-588)4113937-9 Hochschulschrift gnd-content Versicherungsmathematik (DE-588)4063194-1 s Mathematisches Modell (DE-588)4114528-8 s DE-604 Finanzmathematik (DE-588)4017195-4 s Stochastischer Prozess (DE-588)4057630-9 s 2\p DE-604 Versicherungswirtschaft (DE-588)4063206-4 s 3\p DE-604 4\p DE-604 Finanzwirtschaft (DE-588)4017214-4 s 5\p DE-604 Versicherung (DE-588)4063173-4 s 6\p DE-604 7\p DE-604 Statistisches Modell (DE-588)4121722-6 s 8\p DE-604 Rolski, Tomasz Sonstige oth DE-601 pdf/application http://www.zentralblatt-math.org/zmath/en/search/?an=1152.60006 Zentralblatt MATH Kostenfrei DE-576;wiley image/jpeg http://swbplus.bsz-bw.de/bsz307632202cov.htm 20090618192320 Cover Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021151183&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 4\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 5\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 6\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 7\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 8\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Stochastic processes for insurance and finance Versicherung (DE-588)4063173-4 gnd Versicherungsmathematik (DE-588)4063194-1 gnd Mathematisches Modell (DE-588)4114528-8 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Finanzmathematik (DE-588)4017195-4 gnd Versicherungswirtschaft (DE-588)4063206-4 gnd Finanzwirtschaft (DE-588)4017214-4 gnd Statistisches Modell (DE-588)4121722-6 gnd |
| subject_GND | (DE-588)4063173-4 (DE-588)4063194-1 (DE-588)4114528-8 (DE-588)4057630-9 (DE-588)4017195-4 (DE-588)4063206-4 (DE-588)4017214-4 (DE-588)4121722-6 (DE-588)4113937-9 |
| title | Stochastic processes for insurance and finance |
| title_auth | Stochastic processes for insurance and finance |
| title_exact_search | Stochastic processes for insurance and finance |
| title_full | Stochastic processes for insurance and finance Tomasz Rolski ... |
| title_fullStr | Stochastic processes for insurance and finance Tomasz Rolski ... |
| title_full_unstemmed | Stochastic processes for insurance and finance Tomasz Rolski ... |
| title_short | Stochastic processes for insurance and finance |
| title_sort | stochastic processes for insurance and finance |
| topic | Versicherung (DE-588)4063173-4 gnd Versicherungsmathematik (DE-588)4063194-1 gnd Mathematisches Modell (DE-588)4114528-8 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Finanzmathematik (DE-588)4017195-4 gnd Versicherungswirtschaft (DE-588)4063206-4 gnd Finanzwirtschaft (DE-588)4017214-4 gnd Statistisches Modell (DE-588)4121722-6 gnd |
| topic_facet | Versicherung Versicherungsmathematik Mathematisches Modell Stochastischer Prozess Finanzmathematik Versicherungswirtschaft Finanzwirtschaft Statistisches Modell Hochschulschrift |
| url | http://www.zentralblatt-math.org/zmath/en/search/?an=1152.60006 http://swbplus.bsz-bw.de/bsz307632202cov.htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021151183&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT rolskitomasz stochasticprocessesforinsuranceandfinance |