Stochastic Processes for Insurance and Finance

by Tomasz Rolski, Hanspeter Schmidli, Volker Schmidt & Jozef Teugels
      John Wiley & Sons, Chichester, 1999
       ISBN 0-471-95925-1

Table of contents

1. Concepts from Insurance and Finance
   1. Introduction
   2. The Claim Number Process
      1. Renewal Processes
      2. Mixed Poisson Processes
      3. Some Other Models
   3. The Claim Size Process
      1. Dangerous Risks
      2. The Aggregate Claim Amount
      3. Comparison of Risks
   4. Solvability of the Portfolio
      1. Premiums
      2. The Risk Reserve
      3. Economic Environment
   5. Reinsurance
      1. Need for Reinsurance
      2. Types of Reinsurance
   6. Ruin Problems
   7. Related Financial Topics
      1. Investment of Surplus
      2. Diffusion Processes
      3. Equity Linked Life Insurance
2. Probability Distributions
   1. Random Variables and Their Characteristics
      1. Distributions of Random Variables
      2. Basic Characteristics
      3. Independence and Conditioning
      4. Convolution
      5. Transforms
   2. Parameterized Families of Distributions
      1. Discrete Distributions
      2. Absolutely Continuous Distributions
      3. Parameterized Distributions with Heavy Tail
      4. Operations on Distributions
      5. Some Special Functions
   3. Associated Distributions
   4. Distributions with Monotone Hazard Rates
      1. Discrete Distributions
      2. Absolutely Continuous Distributions
   5. Heavy-Tailed Distributions
      1. Definition and Basic Properties
      2. Subexponential Distributions
      3. Criteria for Subexponentiality and the Class S*
      4. Pareto Mixtures of Exponentials
   6. Detection of Heavy-Tailed Distributions
      1. Large Claims
      2. Quantile Plots
      3. Mean Residual Hazard Function
      4. Extreme Value Statistics
3. Premiums and Ordering of Risks
   1. Premium Calculation Principles
      1. Desired Properties of "Good" Premiums
      2. Basic Premium Principles
      3. Quantile Function: Two More Premium Principles
   2. Ordering of Distributions
      1. Concepts of Utility Theory
      2. Stochastic Order
      3. Stop-Loss Order
      4. The Zero Utility Principle
   3. Some Aspects of Reinsurance
4. Distributions of Aggregate Claim Amount
   1. Individual and Collective Model
   2. Compound Distributions
      1. Definition and Elementary Properties
      2. Three Special Cases
      3. Some Actuarial Applications
      4. Ordering of Compounds
      5. The Larger Claims in the Portfolio
   3. Claim Number Distributions
      1. Classical Examples; Panjer's Recurrence Relation
      2. Discrete Compound Poisson Distributions
      3. Mixed Poisson Distributions
   4. Recursive Computation Methods
      1. The Individual Model: De Pril's Algorithm
      2. The Collective Model: Panjer's Algorithm
      3. A Continuous Version of Panjer's Algorithm
   5. Lundberg Bounds
      1. Geometric Compounds
      2. More General Compound Distributions
      3. Estimation of the Adjustment Coefficient
   6. Approximation by Compound Distributions
      1. The Total Variation Distance
      2. The Compound Poisson Approximation
      3. Homogeneous Portfolio
      4. Higher-Order Approximations
   7. Inverting the Fourier Transform
5. Risk Processes
   1. Time-Dependent Risk Models
      1. The Ruin Problem
      2. Computation of the Ruin Function
      3. A Dual Queueing Model
      4. A Risk Model in Continuous Time
   2. Poisson Arrival Processes
      1. Homogeneous Poisson Processes
      2. Compound Poisson Processes
   3. Ruin Probabilities: The Compound Poisson Model
      1. An Integro-Differential Equation
      2. An Integral Equation
      3. Laplace Transforms, Pollaczek-Khinchin Formula
      4. Severity of Ruin
   4. Bounds, Asymptotics and Approximations
      1. Lundberg Bounds
      2. The Cramér-Lundberg Approximation
      3. Subexponential Claim Sizes
      4. Approximation by Moment Fitting
      5. Ordering of Ruin Functions
   5. Numerical Evaluation of Ruin Functions
   6. Finite-Horizon Ruin Probabilities
      1. Deterministic Claim Sizes
      2. Seal's Formulae
      3. Exponential Claim Sizes
6. Renewal Processes and Random Walks
   1. Renewal Processes
      1. Definition and Elementary Properties
      2. The Renewal Function; Delayed Renewal Processes
      3. Renewal Equations and Lorden's Inequality
      4. Key Renewal Theorem
      5. Another Look at the Aggregate Claim Amount
   2. Extensions and Actuarial Applications
      1. Weighted Renewal Functions
      2. A Blackwell-Type Renewal Theorem
      3. Approximation to the Aggregate Claim Amount
      4. Lundberg-Type Bounds
   3. Random Walks
      1. Ladder Epochs
      2. Random Walks with and without Drift
      3. Ladder Heights; Negative Drift
   4. The Wiener-Hopf Factorization
      1. General Representation Formulae
      2. An Analytical Factorization; Examples
      3. Ladder Height Distributions
   5. Ruin Probabilities: Sparre Andersen Model
      1. Formulae of Pollaczek-Khinchin Type
      2. Lundberg Bounds
      3. The Cramér-Lundberg Approximation
      4. Compound Poisson Model with Aggregate Claims
      5. Subexponential Claim Sizes
7. Markov Chains
   1. Definition and Basic Properties
      1. Initial Distribution and Transition Probabilities
      2. Computation of the n-Step Transition Matrix
      3. Recursive Stochastic Equations
      4. Bonus-Malus Systems
   2. Stationary Markov Chains
      1. Long-Run Behaviour
      2. Application of the Perron-Frobenius Theorem
      3. Irreducibility and Aperiodicity
      4. Stationary Initial Distributions
   3. Markov Chains with Rewards
      1. Interest and Discounting
      2. Discounted and Undiscounted Rewards
      3. Efficiency of Bonus-Malus Systems
   4. Monotonicity and Stochastic Ordering
      1. Monotone Transition Matrices
      2. Comparison of Markov Chains
      3. Application to Bonus-Malus Systems
   5. An Actuarial Application of Branching Processes
8. Continuous-Time Markov Models
   1. Homogeneous Markov Processes
      1. Matrix Transition Function
      2. Kolmogorov Differential Equations
      3. An Algorithmic Approach
      4. Monotonicity of Markov Processes
      5. Stationary Initial Distributions
   2. Phase-Type Distributions
      1. Some Matrix Algebra and Calculus
      2. Absorption Time
      3. Operations on Phase-Type Distributions
   3. Risk Processes with Phase-Type Distributions
      1. The Compound Poisson Model
      2. Numerical Issues
   4. Nonhomogeneous Markov Processes
      1. Definition and Basic Properties
      2. Construction of Nonhomogeneous Markov Processes
      3. Application to Life and Pension Insurance
   5. Mixed Poisson Processes
      1. Definition and Elementary Properties
      2. Markov Processes with Infinite State Space
      3. Mixed Poisson Processes as Pure Birth Processes
      4. The Claim Arrival Epochs
      5. The Inter-Occurrence Times
      6. Examples
9. Martingale Techniques I
   1. Discrete-Time Martingales
      1. Fair Games
      2. Filtrations and Stopping Times
      3. Martingales, Sub- and Supermartingales
      4. Life-Insurance Model with Multiple Decrements
      5. Convergence Results
      6. Optional Sampling Theorems
      7. Doob's Inequality
      8. The Doob-Meyer Decomposition
   2. Change of the Probability Measure
      1. The Likelihood Ratio Martingale
      2. Kolmogorov's Extension Theorem
      3. Exponential Martingales for Random Walks
      4. Finite-Horizon Ruin Probabilities
      5. Simulation of Ruin Probabilities
10. Martingale Techniques II
    1. Continuous-Time Martingales
       1. Stochastic Processes and Filtrations
       2. Stopping Times
       3. Martingales, Sub- and Supermartingales
       4. Brownian Motion and Related Processes
       5. Uniform Integrability
    2. Some Fundamental Results
       1. Doob's Inequality
       2. Convergence Results
       3. Optional Sampling Theorems
       4. The Doob-Meyer Decomposition
       5. Kolmogorov's Extension Theorem
       6. Change of the Probability Measure
    3. Ruin Probabilities and Martingales
       1. Ruin Probabilities for Additive Processes
       2. Finite-Horizon Ruin Probabilities
       3. Law of Large Numbers for Additive Processes
       4. An Identity for Finite-Horizon Ruin Probabilities
11. Piecewise Deterministic Markov Processes
    1. Markov Processes with Continuous State Space
       1. Transition Kernels
       2. The Infinitesimal Generator
       3. Dynkin's Formula
       4. The Full Generator
    2. Construction and Properties of PDMP
       1. Behaviour between Jumps
       2. The Jump Mechanism
       3. The Generator of a PDMP
       4. An Application to Health Insurance
    3. The Compound Poisson Model Revisited
       1. Exponential Martingales via PDMP
       2. Change of the Probability Measure
       3. Cramér-Lundberg Approximation
       4. A Stopped Risk Reserve Process
       5. Characteristics of the Ruin Time
    4. Compound Poisson Model in an Economic Environment
       1. Interest and Discounting
       2. A Discounted Risk Reserve Process
       3. The Adjustment Coefficient
       4. Decreasing Economic Factor
    5. Exponential Martingales: the Sparre Andersen Model
       1. An Integral Equation
       2. Backward Markovization Technique
       3. Forward Markovization Technique
12. Point Processes
    1. Stationary Point Processes
       1. Definition and Elementary Properties
       2. Palm Distributions and Campbell's Formula
       3. Ergodic Theorems
       4. Marked Point Processes
       5. Ruin Probabilities in the Time-Stationary Model
    2. Mixtures and Compounds of Point Processes
       1. Nonhomogeneous Poisson Processes
       2. Cox Processes
       3. Compounds of Point Processes
       4. Comparison of Ruin Probabilities
    3. The Markov-Modulated Risk Model via PDMP
       1. A System of Integro-Differential Equations
       2. Law of Large Numbers
       3. The Generator and Exponential Martingales
       4. Lundberg Bounds
       5. Cramér-Lundberg Approximation
       6. Finite-Horizon Ruin Probabilities
    4. Periodic Risk Model
    5. The Björk-Grandell Model via PDMP
       1. Law of Large Numbers
       2. The Generator and Exponential Martingales
       3. Lundberg Bounds
       4. Cramér-Lundberg Approximation
       5. Finite-Horizon Ruin Probabilities
    6. Subexponential Claim Sizes
       1. General Results
       2. Poisson Cluster Arrival Processes
       3. Superposition of Renewal Processes
       4. The Markov-Modulated Risk Model
       5. The Björk-Grandell Risk Model
13. Diffusion Models
    1. Stochastic Differential Equations
       1. Stochastic Integrals and Itô's Formula
       2. Diffusion Processes
       3. Lévy's Characterization Theorem
    2. Perturbed Risk Processes
       1. Lundberg Bounds
       2. Modified Ladder Heights
       3. Cramér-Lundberg Approximation
       4. Subexponential Claim Sizes
    3. Other Applications to Insurance and Finance
       1. The Black-Scholes Model
       2. Equity Linked Life Insurance
       3. Stochastic Interest Rates in Life Insurance
    4. Simple Interest Rate Models
       1. Zero-Coupon Bonds
       2. The Vasicek Model
       3. The Cox-Ingersoll-Ross Model

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