Stochastic processes for insurance and finance

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Stochastic processes for insurance and finance / Tomasz Rolski ... [et al.]

資料種別:: 図書
出版情報:: Chichester : J. Wiley, c1999
形態:: xviii, 654 p. ; 24 cm
シリーズ名:: Wiley series in probability and mathematical statistics <BA00019013>
著者名:: Rolski, Tomasz <DA05456962>
ISBN:: 9780471959250 [0471959251]
書誌ID:: BA39782847

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注記:

Bibliography: p. [617]-638
Includes index

主題:

Finance--Mathematical models; Insurance--Mathematical models; Stochastic processes

言語:

英語

目次情報:

Preface

List of Principal Notation

Concepts from Insurance and Finance / 1：

Introduction / 1.1：

The Claim Number Process / 1.2：

Renewal Processes / 1.2.1：

Mixed Poisson Processes / 1.2.2：

Some Other Models / 1.2.3：

The Claim Size Process / 1.3：

Dangerous Risks / 1.3.1：

The Aggregate Claim Amount / 1.3.2：

Comparison of Risks / 1.3.3：

Solvability of the Portfolio / 1.4：

Premiums / 1.4.1：

The Risk Reserve / 1.4.2：

Economic Environment / 1.4.3：

Reinsurance / 1.5：

Need for Reinsurance / 1.5.1：

Types of Reinsurance / 1.5.2：

Ruin Problems / 1.6：

Related Financial Topics / 1.7：

Investment of Surplus / 1.7.1：

Diffusion Processes / 1.7.2：

Equity Linked Life Insurance / 1.7.3：

Probability Distributions / 2：

Random Variables and Their Characteristics / 2.1：

Distributions of Random Variables / 2.1.1：

Basic Characteristics / 2.1.2：

Independence and Conditioning / 2.1.3：

Convolution / 2.1.4：

Transforms / 2.1.5：

Parametrized Families of Distributions / 2.2：

Discrete Distributions / 2.2.1：

Absolutely Continuous Distributions / 2.2.2：

Parametrized Distributions with Heavy Tail / 2.2.3：

Operations on Distributions / 2.2.4：

Some Special Functions / 2.2.5：

Associated Distributions / 2.3：

Distributions with Monotone Hazard Rates / 2.4：

Heavy-Tailed Distributions / 2.4.1：

Definition and Basic Properties / 2.5.1：

Subexponential Distributions / 2.5.2：

Criteria for Subexponentiality and the Class S / 2.5.3：

Pareto Mixtures of Exponentials / 2.5.4：

Detection of Heavy-Tailed Distributions / 2.6：

Large Claims / 2.6.1：

Quantile Plots / 2.6.2：

Mean Residual Hazard Function / 2.6.3：

Extreme Value Statistics / 2.6.4：

Premiums and Ordering of Risks / 3：

Premium Calculation Principles / 3.1：

Desired Properties of "Good" Premiums / 3.1.1：

Basic Premium Principles / 3.1.2：

Quantile Function: Two More Premium Principles / 3.1.3：

Ordering of Distributions / 3.2：

Concepts of Utility Theory / 3.2.1：

Stochastic Order / 3.2.2：

Stop-Loss Order / 3.2.3：

The Zero Utility Principle / 3.2.4：

Some Aspects of Reinsurance / 3.3：

Distributions of Aggregate Claim Amount / 4：

Individual and Collective Model / 4.1：

Compound Distributions / 4.2：

Definition and Elementary Properties / 4.2.1：

Three Special Cases / 4.2.2：

Some Actuarial Applications / 4.2.3：

Ordering of Compounds / 4.2.4：

The Larger Claims in the Portfolio / 4.2.5：

Claim Number Distributions / 4.3：

Classical Examples; Panjer's Recurrence Relation / 4.3.1：

Discrete Compound Poisson Distributions / 4.3.2：

Mixed Poisson Distributions / 4.3.3：

Recursive Computation Methods / 4.4：

The Individual Model: De Pril's Algorithm / 4.4.1：

The Collective Model: Panjer's Algorithm / 4.4.2：

A Continuous Version of Panjer's Algorithm / 4.4.3：

Lundberg Bounds / 4.5：

Geometric Compounds / 4.5.1：

More General Compound Distributions / 4.5.2：

Estimation of the Adjustment Coefficient / 4.5.3：

Approximation by Compound Distributions / 4.6：

The Total Variation Distance / 4.6.1：

The Compound Poisson Approximation / 4.6.2：

Homogeneous Portfolio / 4.6.3：

Higher-Order Approximations / 4.6.4：

Inverting the Fourier Transform / 4.7：

Risk Processes / 5：

Time-Dependent Risk Models / 5.1：

The Ruin Problem / 5.1.1：

Computation of the Ruin Function / 5.1.2：

A Dual Queueing Model / 5.1.3：

A Risk Model in Continuous Time / 5.1.4：

Poisson Arrival Processes / 5.2：

Homogeneous Poisson Processes / 5.2.1：

Compound Poisson Processes / 5.2.2：

Ruin Probabilities: The Compound Poisson Model / 5.3：

An Integro-Differential Equation / 5.3.1：

An Integral Equation / 5.3.2：

Laplace Transforms, Pollaczek-Khinchin Formula / 5.3.3：

Severity of Ruin / 5.3.4：

Bounds, Asymptotics and Approximations / 5.4：

The Cramer-Lundberg Approximation / 5.4.1：

Subexponential Claim Sizes / 5.4.3：

Approximation by Moment Fitting / 5.4.4：

Ordering of Ruin Functions / 5.4.5：

Numerical Evaluation of Ruin Functions / 5.5：

Finite-Horizon Ruin Probabilities / 5.6：

Deterministic Claim Sizes / 5.6.1：

Seal's Formulae / 5.6.2：

Exponential Claim Sizes / 5.6.3：

Renewal Processes and Random Walks / 6：

The Renewal Function; Delayed Renewal Processes / 6.1：

Renewal Equations and Lorden's Inequality / 6.1.3：

Key Renewal Theorem / 6.1.4：

Another Look at the Aggregate Claim Amount / 6.1.5：

Extensions and Actuarial Applications / 6.2：

Weighted Renewal Functions / 6.2.1：

A Blackwell-Type Renewal Theorem / 6.2.2：

Approximation to the Aggregate Claim Amount / 6.2.3：

Lundberg-Type Bounds / 6.2.4：

Random Walks / 6.3：

Ladder Epochs / 6.3.1：

Random Walks with and without Drift / 6.3.2：

Ladder Heights; Negative Drift / 6.3.3：

The Wiener-Hopf Factorization / 6.4：

General Representation Formulae / 6.4.1：

An Analytical Factorization; Examples / 6.4.2：

Ladder Height Distributions / 6.4.3：

Ruin Probabilities: Sparre Andersen Model / 6.5：

Formulae of Pollaczek-Khinchin Type / 6.5.1：

Compound Poisson Model with Aggregate Claims / 6.5.2：

Markov Chains / 6.5.5：

Initial Distribution and Transition Probabilities / 7.1：

Computation of the n-Step Transition Matrix / 7.1.2：

Recursive Stochastic Equations / 7.1.3：

Bonus-Malus Systems / 7.1.4：

Stationary Markov Chains / 7.2：

Long-Run Behaviour / 7.2.1：

Application of the Perron-Frobenius Theorem / 7.2.2：

Irreducibility and Aperiodicity / 7.2.3：

Stationary Initial Distributions / 7.2.4：

Markov Chains with Rewards / 7.3：

Interest and Discounting / 7.3.1：

Discounted and Undiscounted Rewards / 7.3.2：

Efficiency of Bonus-Malus Systems / 7.3.3：

Monotonicity and Stochastic Ordering / 7.4：

Monotone Transition Matrices / 7.4.1：

Comparison of Markov Chains / 7.4.2：

Application to Bonus-Malus Systems / 7.4.3：

An Actuarial Application of Branching Processes / 7.5：

Continuous-Time Markov Models / 8：

Homogeneous Markov Processes / 8.1：

Matrix Transition Function / 8.1.1：

Kolmogorov Differential Equations / 8.1.2：

An Algorithmic Approach / 8.1.3：

Monotonicity of Markov Processes / 8.1.4：

Phase-Type Distributions / 8.1.5：

Some Matrix Algebra and Calculus / 8.2.1：

Absorption Time / 8.2.2：

Operations on Phase-Type Distributions / 8.2.3：

Risk Processes with Phase-Type Distributions / 8.3：

The Compound Poisson Model / 8.3.1：

Numerical Issues / 8.3.2：

Nonhomogeneous Markov Processes / 8.4：

Construction of Nonhomogeneous Markov Processes / 8.4.1：

Application to Life and Pension Insurance / 8.4.3：

Markov Processes with Infinite State Space / 8.5：

Mixed Poisson Processes as Pure Birth Processes / 8.5.3：

The Claim Arrival Epochs / 8.5.4：

The Inter-Occurrence Times / 8.5.5：

Examples / 8.5.6：

Martingale Techniques I / 9：

Discrete-Time Martingales / 9.1：

Fair Games / 9.1.1：

Filtrations and Stopping Times / 9.1.2：

Martingales, Sub- and Supermartingales / 9.1.3：

Life-Insurance Model with Multiple Decrements / 9.1.4：

Convergence Results / 9.1.5：

Optional Sampling Theorems / 9.1.6：

Doob's Inequality / 9.1.7：

The Doob-Meyer Decomposition / 9.1.8：

Change of the Probability Measure / 9.2：

The Likelihood Ratio Martingale / 9.2.1：

Kolmogorov's Extension Theorem / 9.2.2：

Exponential Martingales for Random Walks / 9.2.3：

Simulation of Ruin Probabilities / 9.2.4：

Martingale Techniques II / 10：

Continuous-Time Martingales / 10.1：

Stochastic Processes and Filtrations / 10.1.1：

Stopping Times / 10.1.2：

Brownian Motion and Related Processes / 10.1.3：

Uniform Integrability / 10.1.5：

Some Fundamental Results / 10.2：

Ruin Probabilities and Martingales / 10.2.1：

Ruin Probabilities for Additive Processes / 10.3.1：

Law of Large Numbers for Additive Processes / 10.3.2：

An Identity for Finite-Horizon Ruin Probabilities / 10.3.4：

Piecewise Deterministic Markov Processes / 11：

Markov Processes with Continuous State Space / 11.1：

Transition Kernels / 11.1.1：

The Infinitesimal Generator / 11.1.2：

Dynkin's Formula / 11.1.3：

The Full Generator / 11.1.4：

Construction and Properties of PDMP / 11.2：

Behaviour between Jumps / 11.2.1：

The Jump Mechanism / 11.2.2：

The Generator of a PDMP / 11.2.3：

An Application to Health Insurance / 11.2.4：

The Compound Poisson Model Revisited / 11.3：

Exponential Martingales via PDMP / 11.3.1：

Cramer-Lundberg Approximation / 11.3.2：

A Stopped Risk Reserve Process / 11.3.4：

Characteristics of the Ruin Time / 11.3.5：

Compound Poisson Model in an Economic Environment / 11.4：

A Discounted Risk Reserve Process / 11.4.1：

The Adjustment Coefficient / 11.4.3：

Decreasing Economic Factor / 11.4.4：

Exponential Martingales: the Sparre Andersen Model / 11.5：

Backward Markovization Technique / 11.5.1：

Forward Markovization Technique / 11.5.3：

Point Processes / 12：

Stationary Point Processes / 12.1：

Palm Distributions and Campbell's Formula / 12.1.1：

Ergodic Theorems / 12.1.3：

Marked Point Processes / 12.1.4：

Ruin Probabilities in the Time-Stationary Model / 12.1.5：

Mixtures and Compounds of Point Processes / 12.2：

Nonhomogeneous Poisson Processes / 12.2.1：

Cox Processes / 12.2.2：

Compounds of Point Processes / 12.2.3：

Comparison of Ruin Probabilities / 12.2.4：

The Markov-Modulated Risk Model via PDMP / 12.3：

A System of Integro-Differential Equations / 12.3.1：

Law of Large Numbers / 12.3.2：

The Generator and Exponential Martingales / 12.3.3：

Periodic Risk Model / 12.3.4：

The Bjork-Grandell Model via PDMP / 12.5：

General Results / 12.5.1：

Poisson Cluster Arrival Processes / 12.6.2：

Superposition of Renewal Processes / 12.6.3：

The Markov-Modulated Risk Model / 12.6.4：

The Bjork-Grandell Risk Model / 12.6.5：

Diffusion Models / 13：

Stochastic Differential Equations / 13.1：

Stochastic Integrals and Ito's Formula / 13.1.1：

Levy's Characterization Theorem / 13.1.2：

Perturbed Risk Processes / 13.2：

Modified Ladder Heights / 13.2.1：

Other Applications to Insurance and Finance / 13.2.3：

The Black-Scholes Model / 13.3.1：

Stochastic Interest Rates in Life Insurance / 13.3.2：

Simple Interest Rate Models / 13.4：

Zero-Coupon Bonds / 13.4.1：

The Vasicek Model / 13.4.2：

The Cox-Ingersoll-Ross Model / 13.4.3：

Distribution Tables

References

Index

Preface

List of Principal Notation

Concepts from Insurance and Finance / 1：

Introduction / 1.1：

The Claim Number Process / 1.2：

Renewal Processes / 1.2.1：

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