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.
The aim of this book is to promote interaction between Engineering, Finance and Insurance, as there are many models and solution methods in common for solving real-life problems in these three topics.
The authors point out the strict inter-relations that exist among the diffusion models used in Engineering, Finance and Insurance.
In each of the three fields the basic diffusion models are presented and their strong similarities are discussed. Analytical, numerical and Monte Carlo simulation methods are explained with a view to applying them to get the solutions of the different problems presented in the book. Advanced topics such as non-linear problems, Lévy processes and semi-Markov models in interactions with the diffusion models are discussed, as well as possible future interactions among Engineering, Finance and Insurance.
Introduction xiiiChapter 1 Diffusion Phenomena and Models 1
1.1 General presentation of diffusion process 1
1.2 General balance equations 6
1.3 Heat conduction equation 10
1.4 Initial and boundary conditions 12
Chapter 2 Probabilistic Models of Diffusion Processes 17
2.1 Stochastic differentiation 17
2.2 Itô’s formula 19
2.3 Stochastic differential equations (SDE) 24
2.4 Itô and diffusion processes 28
2.5 Some particular cases of diffusion processes 32
2.6 Multidimensional diffusion processes 36
2.7 The Stroock–Varadhan martingale characterization of diffusions (Karlin and Taylor) 41
2.8 The Feynman–Kac formula (Platen and Heath) 42
Chapter 3 Solving Partial Differential Equations of Second Order 47
3.1 Basic definitions on PDE of second order 47
3.2 Solving the heat equation 51
3.3 Solution by the method of Laplace transform 65
3.4 Green’s functions 75
Chapter 4 Problems in Finance 85
4.1 Basic stochastic models for stock prices 85
4.2 The bond investments 90
4.3 Dynamic deterministic continuous time model for instantaneous interest rate 93
4.4 Stochastic continuous time dynamic model for instantaneous interest rate 98
4.5 Multidimensional Black and Scholes model 110
Chapter 5 Basic PDE in Finance 111
5.1 Introduction to option theory 111
5.2 Pricing the plain vanilla call with the Black–Scholes–Samuelson model 115
5.3 Pricing no plain vanilla calls with the Black-Scholes-Samuelson model 120
5.4 Zero-coupon pricing under the assumption of no arbitrage 127
Chapter 6 Exotic and American Options Pricing Theory 145
6.1 Introduction 145
6.2 The Garman–Kohlhagen formula 146
6.3 Binary or digital options 149
6.4 “Asset or nothing” options 150
6.5 Numerical examples 152
6.6 Path-dependent options 153
6.7 Multi-asset options 157
6.8 American options 165
Chapter 7 Hitting Times for Diffusion Processes and Stochastic Models in Insurance 177
7.1 Hitting or first passage times for some diffusion processes 177
7.2 Merton’s model for default risk 193
7.3 Risk diffusion models for insurance 201
Chapter 8 Numerical Methods 219
8.1 Introduction 219
8.2 Discretization and numerical differentiation 220
8.3 Finite difference methods 222
9.1 Nonlinear model in heat conduction 232
Chapter 9 Advanced Topics in Engineering: Nonlinear Models 231
9.2 Integral method applied to diffusive problems 233
9.3 Integral method applied to nonlinear problems 239
9.4 Use of transformations in nonlinear problems 243
Chapter 10 Lévy Processes 255
10.1 Motivation 255
10.2 Notion of characteristic functions 257
10.3 Lévy processes 257
10.4 Lévy–Khintchine formula 259
10.5 Examples of Lévy processes 261
10.6 Variance gamma (VG) process 264
10.7 The Brownian–Poisson model with jumps 266
10.8 Risk neutral measures for Lévy models in finance 275
10.9 Conclusion 276
Chapter 11 Advanced Topics in Insurance: Copula Models and VaR Techniques 277
11.1 Introduction 277
11.2 Sklar theorem (1959) 279
11.3 Particular cases and Fréchet bounds 280
11.4 Dependence 288
11.5 Applications in finance: pricing of the bivariate digital put option 293
11.6 VaR application in insurance 296
Chapter 12 Advanced Topics in Finance: Semi-Markov Models 307
12.1 Introduction 307
12.2 Homogeneous semi-Markov process 308
12.3 Semi-Markov option model 328
12.4 Semi-Markov VaR models 332
12.5 Conclusion 339
Chapter 13 Monte Carlo Semi-Markov Simulation Methods 341
13.1 Presentation of our simulation model 341
13.2 The semi-Markov Monte Carlo model in a homogeneous environment 345
13.3 A credit risk example 350
13.4 Semi-Markov Monte Carlo with initial recurrence backward time in homogeneous case 362
13.5 The SMMC applied to claim reserving problem 363
13.6 An example of claim reserving calculation 366
Conclusion 379
Bibliography 381
Index 393