Applied Diffusion Processes from Engineering to Finance
, by Janssen, Jacques; Manca, Oronzio; Manca, Raimondo
- ISBN: 9781848212497 | 1848212496
- Cover: Hardcover
- Copyright: 3/11/2013
This book promotes the interaction among Engineering, Finance and Insurance for solving real life problems in these three topics. An introduction to the diffusion phenomena with a description of its origin are given in Chapter 1. Moreover, in this chapter some problems in engineering, such as the mass diffusion and heat conduction, and in finance and insurance such as the derivative pricing and risk theory are provide and a detailed description of their mathematical models are given in Chapter 1. The mathematical models of diffusion are derived and explained in Chapter 2. A microscopic description is employed to achieve the probabilistic model of diffusion processes starting from random walks or Brownian motion. The determination of transport equation in macroscopic description is carried out starting from a logical form of an extensive quantity balance. This general equation is given in an integral form and considering as generic quantity the total energy of a thermodynamic system by means of the Gauss theorem a differential form is provided. Heat conduction equation is discussed together with initial and boundary conditions. Classical techniques to solve the partial differential equations (PDE) are provided in the Chapter 4. Fourier technique to obtain analytical solutions of PDE in different coordinate systems is developed and some examples are carried out. The use of Laplace transform is shown giving its definition and properties together with some indication to obtain the inversion of Laplace transform. Applications of Laplace transform in the solution of time dependent problems are achieved. The Green's function method is illustrated and the determination of Green functions is developed. Some applications are given for different cases. In the next three chapters, technical, numerical and Monte Carlo methods are explained in view to apply them to get the solution of the different problems presented in Chapters 5, 6 and 7. The discretization techniques are presented to explain the need to have powerful ways to carried out numerical solutions in complex problems of engineering, finance and insurance. Finite elements method is described and some simple examples are showed. Finite difference and volume methods are presented and their application is provided. Chapters 8, 9, 10 and 11 give more advanced topics such as non linear problems, Lévy processes, copula approach and semi-Markov models in interaction with diffusion models. The last chapter presents as a conclusion, actual and future Interactions among Engineering, Finance and Insurance as a fructuous source of developments of new models more adapted to approach the complexity of our three basic fields ,showing so the great originality of this book. The audience of this book is large both for professional, research and academic needs including engineers, mathematicians, physicians, actuaries and finance researchers.
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