Dependence Modelling
Course Description
Advanced course on modelling dependencies between random variables with applications to risk management.
Lecture Notes
1. Bivariate Risks
Joint distributions of two random variables, covariance, correlation, and limitations of linear dependence measures.
2. Tractable Dependence Models
Comonotonicity, countermonotonicity, and simple dependence structures with closed-form properties.
3. Multivariate Gaussian and Elliptical Risks
Multivariate normal distribution, elliptical distributions, and their dependence properties.
4. Discrete Mixtures
Mixture models for generating dependence, conditionally independent structures, and latent factor models.
5. Copulas and Marginals
Sklar's theorem, copula definition, separation of marginals and dependence structure.
6. Max-stable Distributions and Copulas
Max-stable distributions, extreme value copulas, Gumbel family, tail dependence in extremes.
7. Max Domains of Attractions
Convergence to extreme value distributions, Fisher-Tippett theorem, domain of attraction conditions.
8. Archimedean Copulas
Generator functions, Clayton, Frank, and Gumbel copulas, Archimedean copulas as frailty models, simulation methods.
10. Non-parametric Statistics
Empirical copulas, rank-based estimation, and distribution-free inference methods.
11. Parametric Statistics of Dependent Risks
Maximum likelihood estimation, pseudo-MLE, goodness-of-fit tests.
12. Risk Measures
Value-at-Risk, Expected Shortfall, coherent risk measures, and their properties.
13. Case Study: Mixture Copula Model
Practical application of mixture copulas to real insurance data, model selection and validation.
14. Risk Aggregation
Distribution of sums of dependent risks, bounds under dependence uncertainty, diversification benefit.