1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

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Convex bodies : the Brunn-Minkowski theory

Concepts from Insurance and Finance. Probability Distributions. Premiums and Ordering of Risks. Distributions of Aggregate Claim Amount. Risk Processes. Renewal Processes and Random Walks. Markov Chains. Continuous-Time Markov Models. Martingale Techniques I. Martingale Techniques II. Piecewise Deterministic Markov Processes. Point Processes. Diffusion Models. Distribution Tables. References. Index.

Stochastic Processes for Insurance and Finance

This paper considers a wide range of stochastic reserving models for use in general insurance, beginning with stochastic models which reproduce the traditional chain-ladder reserve estimates. The models are extended to consider parametric curves and smoothing models for the shape of the development run-off, which allow extrapolation for the estimation of tail factors. The Bornhuetter-Ferguson technique is also considered, within a Bayesian framework, which allows expert opinion to be used to provide prior estimates of ultimate claims. The primary advantage of stochastic reserving models is the availability of measures of precision of reserve estimates, and in this respect, attention is focused on the root mean squared error of prediction (prediction error). Of greater interest is a full predictive distribution of possible reserve outcomes, and different methods of obtaining that distribution are described. The techniques are illustrated with examples throughout, and the wider issues discussed, in particular, the concept of a ‘best estimate’; reporting the variability of claims reserves; and use in dynamic financial analysis models.

Stochastic Claims Reserving in General Insurance

Preface. Acknowledgement. 1 Introduction and Notation. 1.1 Claims Process. 1.2 Structural Framework to the Claims-Reserving Problem. 1.3 Outstanding Loss Liabilities, Classical Notation. 1.4 General Remarks. 2 Basic Methods. 2.1 Chain-Ladder Method (Distribution-Free). 2.2 Bornhuetter-Ferguson Method. 2.3 Number of IBNyR Claims, Poisson Model. 2.4 Poisson Derivation of the CL Algorithm. 3 Chain-Ladder Models. 3.1 Mean Square Error of Prediction. 3.2 Chain-Ladder Method. 3.3 Bounds in the Unconditional Approach. 3.4 Analysis of Error Terms in the CL Method. 4 Bayesian Models. 4.1 Benktander-Hovinen Method and Cape-Cod Model. 4.2 Credible Claims Reserving Methods. 4.3 Exact Bayesian Models. 4.4 Markov Chain Monte Carlo Methods. 4.5 Buhlmann-Straub Credibility Model. 4.6 Multidimensional Credibility Models. 4.7 Kalman Filter. 5 Distributional Models. 5.1 Log-Normal Model for Cumulative Claims. 5.2 Incremental Claims. 6 Generalized Linear Models. 6.1 Maximum Likelihood Estimators. 6.2 Generalized Linear Models Framework. 6.3 Exponential Dispersion Family. 6.4 Parameter Estimation in the EDF. 6.5 Other GLM Models. 6.6 Bornhuetter-Ferguson Method, Revisited. 7 Bootstrap Methods. 7.1 Introduction. 7.2 Log-Normal Model for Cumulative Sizes. 7.3 Generalized Linear Models. 7.4 Chain-Ladder Method. 7.5 Mathematical Thoughts about Bootstrapping Methods. 7.6 Synchronous Bootstrapping of Seemingly Unrelated Regressions. 8 Multivariate Reserving Methods. 8.1 General Multivariate Framework. 8.2 Multivariate Chain-Ladder Method. 8.3 Multivariate Additive Loss Reserving Method. 8.4 Combined Multivariate CL and ALR Method. 9 Selected Topics I: Chain-Ladder Methods. 9.1 Munich Chain-Ladder. 9.2 CL Reserving: A Bayesian Inference Model. 10 Selected Topics II: Individual Claims Development Processes. 10.1 Modelling Claims Development Processes for Individual Claims. 10.2 Separating IBNeR and IBNyR Claims. 11 Statistical Diagnostics. 11.1 Testing Age-to-Age Factors. 11.2 Non-Parametric Smoothing. Appendix A: Distributions. A.1 Discrete Distributions. A.2 Continuous Distributions. Bibliography. Index.

Stochastic claims reserving methods in insurance

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

Sundt and Jewell have shown that a nondegenerate claim number distribution Q = {q,}, ~ No satisfies the recursion qn+l=( a b + -h--~)q, for all n _> 0 if and only if Q is a binomial, Poisson or negativebinomial distribution. This recursion is of interest since it yields a recursion for the aggregate claims distribution in the collective model of risk theory when the claim size distribution is integer-valued as well. A similar characterization of claim number distributions satisfying the above recursion for all n >_ 1 has been obtained by Willmot. In the present paper we extend these results and the subsequent recursion for the aggregate claims distribution to the case where the recursion holds for all n >_ k with arbitrary k. Our results are of interest in catastrophe excess-of-loss reinsurance.

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An extension of panjer's recursion

The Claim Arrival Process - The Claim Number Process - The Claim Number Process as a Markov Process - The Mixed Claim Number Process - The Aggregate Claims Process - The Risk Process in Reinsurance - The Reserve Process and the Ruin Problem Appendix: Special Distributions

Lectures on Risk Theory

The present paper introduces a group of transformations on the collection of all bivariate copulas. This group contains an involution which is particularly useful since it provides (1) a criterion under which a given symmetric copula can be transformed into an asymmetric one and (2) a condition under which for a given copula the value of every measure of concordance is equal to zero. The group also contains a subgroup which is of particular interest since its four elements preserve symmetry, the order between two copulas and the value of every measure of concordance.

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Bivariate copulas: Transformations, asymmetry and measures of concordance

In the present paper we propose a multivariate version of the additive model of loss reserving. The multivariate additive model is a linear model with a particular design matrix and a particular variance structure and is suitable for certain portfolios consisting of several correlated subportfolios. Under the assumptions of the multivariate additive model, we derive a formula for the Gauss–Markov predictor for a non-observable incremental claim. We also show that the Gauss–Markov predictors for the reserve of a particular accident year and for the total reserve are obtained by summation over the Gauss–Markov predictors for the corresponding non-observable incremental claims, and that this is also true for the Gauss–Markov predictors for the corresponding quantities of the aggregate portfolio.

Multivariate loss prediction in the multivariate additive model

In the present paper we review and extend two stochastic models for loss resenting and study their impact on extensions of the additive method and of the chain-ladder method. The first of these models is a particular linear model while the second one is a sequential model which is composed of a finite number of conditional linear models. These models lead to multivariate extensions of the additive method and of the chain-ladder method, respectively, which turn out to resoh-e the problem of additiviq'.

Klaus D. Schmidt

Papers

An extension of panjer's recursion

Lectures on Risk Theory

Bivariate copulas: Transformations, asymmetry and measures of concordance

Multivariate loss prediction in the multivariate additive model

Optimal and Additive Loss Reserving for Dependent Lines of Business