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This fascinating book begins with fundamental definitions and notations of urn models before moving on to stochastic processes and applications of urn models in the field of finance. The Polya urn model is simple but has rich content and diverse applications because it includes correlations. Applications of Polya models such as phase transitions in nonlinear Polya models are studied here, and the relation between temporal correlation and phase transition is also discussed.

In a continuous limit, the self-exciting negative binomial distribution model and Hawkes model, which has Poisson noise, can be obtained. In these models, it is possible to observe phase transition as a branching process, which is one of the absorption phase transitions. If connected urns are considered, the process can be extended to represent correlations between several urns, corresponding to complex networks among the urns and leading to consideration of how the network affects the urn processes.

In this book, the method is applied to default portfolios, including correlations. In finance, correlation is an important issue in the clustering of a default, and several topics involving applications of urn models to risk assessment for default portfolios in finance are explained. Especially in default portfolios, some sectors affect many other sectors while other sectors do not; thus the origin of default contagion, a phenomenon to which urn models with networks are applied here.