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 |