Summary

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.

Author Biography

Jacques Janssen is now Honorary Professor at the Solvay Business School (ULB) in Brussels, Belgium, having previously taught at EURIA (Euro-Institut d’Actuariat, University of West Brittany, Brest, France) and Télécom-Bretagne (Brest, France) as well as being a director of Jacan Insurance and Finance Services, a consultancy and training company.

Oronzio Manca is Professor of thermal sciences at Seconda Università degli Studi di Napoli in Italy. He is currently Associate Editor of ASME Journal of Heat Transfer and Journal of Porous Media and a member of the editorial advisory boards for The Open Thermodynamics Journal, Advances in Mechanical Engineering, The Open Fuels & Energy Science Journal.

Raimondo Manca is Professor of mathematical methods applied to economics, finance and actuarial science at University of Rome "La Sapienza" in Italy. He is associate editor for the journal Methodology and Computing in Applied Probability. His main research interests are multidimensional linear algebra, computational probability, application of stochastic processes to economics, finance and insurance and simulation models.

Chapter 1 Diffusion phenomena and models

1.1 The origin of Diffusion processes

1.2 Problems in engineering

1.3 Problems in Finance and Insurance

Chapter 2 Basic mathematical aspects of diffusion processes

2.1 Probabilistic model of diffusion processes

2.2Continuum models of diffusion processes

Chapter 3 pricing problems in finance and interaction with diffusion theory

Chapter 4 Technical methods for solving diffusion problems

4.1 Fourier technique

4.2 Laplace transform

4.3 Green function

4.4 Risk neutral measure

Chapter 5 Numerical methods

5.1 Discretization methods

5.2 Finite elements

5.3 Finite difference/volume methods

5.4 Methods for SDE

Chapter 6 Monte Carlo methods

6.1 Presentation of the methods

6.2 Case of deterministic models

6.3 Case of stochastic models

Chapter 7 Solution of Problems in Engineering

Chapter 8 Solution of Problems in Finance and in Insurance Chapter 9 Advanced topics in Engineering

9.1 Non linear models

Chapter 10 Advanced topics in Finance

10.1 Lévy models

10.2 Semi-Markov models

10.3 Copula methods

Chapter 11 Advanced topics in Insurance

11.1 Semi-Markov models

11.2 Copula methods

Chapter 12 Present and future Interactions among Engineering, Finance and Insurance OMRMJJ

Appendix 1 Stochastic processes

Appendix 2 Itô Calculus

Appendix 3 Partial Differential equations

Appendix 4 n-Dimensional Matrices

References

Index

Supplemental Materials

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Applied Diffusion Processes from Engineering to Finance

9781848212497

1848212496

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Summary

Author Biography

Table of Contents

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