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Affine diffusions and related processes: simulation, theory and applications. (English) Zbl 1387.60002

Bocconi & Springer Series 6. Milano: Bocconi University Press; Cham: Springer (ISBN 978-3-319-05220-5/hbk; 978-3-319-05221-2/ebook). xiii, 252 p. (2015).
Motivated by applications in the mathematics of finance, the author provides an up-to-date treatment of simulations of affine diffusions and some related processes, from theory down to explicit algorithms. For readers familiar with stochastic analysis of diffusions, the book is self-contained. It starts with one-dimensional affine diffusions which essentially are of either Ornstein-Uhlenbeck or Cox-Ingersoll-Ross (CIR) type. They are applied to interest rate modelling. This is followed by chapters on simulation schemes for stochastic differential equations, in particular the Euler-Maruyama scheme, strong and weak approximations, and simulation of the CIR process, including exact simulation methods, discretization schemes, weak order schemes, and some numerical results.
The next chapter starts with setting the general framework for higher-dimensional affine diffusions and continues with the Heston model [S. Heston, Rev. Financ. Stud. 6, No. 2, 327–343 (1993; Zbl 1384.35131)], a two-dimensional affine diffusion, modelling the price and volatility of a single stock. Several pricing models for European option and simulation schemes are given. The chapter ends with a section on affine term structure short rate models. In Chapter 5 affine diffusions on positive semi-definite matrices, especially Wishart processes [M.-F. Bru, J. Theor. Probab. 4, No. 4, 725–751 (1991; Zbl 0737.60067)] are treated. Originally introduced to model perturbation of experimental data in biology, Wishart processes, which may be viewed as generalization of the CIR process, have found interest in connection with modelling short-term interest rates, the volatility of a single stock, and in particular, the instantaneous covariance of a basket of assets, extending Heston’s model. After some stochastic analysis for diffusions on positive semidefinite matrices, exact simulation schemes, high-order discretization schemes and some numerical results are provided.
The last chapter concerns processes of Wright-Fisher type. These are not affine diffusions, but together with the latter they belong to the class of polynomial processes [C. Cuchiero et al., Finance Stoch. 16, No. 4, 711–740 (2012; Zbl 1270.60079)]. After the classical Wright-Fisher process, known from biology as modelling gene frequency, a mean-reverting process on correlation matrices of Wright-Fischer type, (MRC process) introduced by A. Ahdida and the author [Stochastic Processes Appl. 123, No. 4, 1472–1520 (2013; Zbl 1271.65014)] is considered. The idea is to model the dependence dynamics between different stochastic differential equations through their driving Brownian motion, by assuming that their instantaneous quadratic covariation is described by the correlation process. That could serve as model for a basket of financial assets, consistent with the model for each single asset. After basic existence and uniqueness results for MRC processes, second-order discretization schemes are presented.
Clearly, the book will a useful for anyone interested in current perspectives of modelling the price of financial assets, in theory as well as practical application.

MSC:

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60H30 Applications of stochastic analysis (to PDEs, etc.)
91G70 Statistical methods; risk measures
62P05 Applications of statistics to actuarial sciences and financial mathematics
65C30 Numerical solutions to stochastic differential and integral equations
91B25 Asset pricing models (MSC2010)
91B70 Stochastic models in economics
91G60 Numerical methods (including Monte Carlo methods)
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