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.

• Introduction xiiiChapter 1 Diffusion Phenomena and Models 11.1 General presentation of diffusion process 11.2 General balance equations 61.3 Heat conduction equation 101.4 Initial and boundary conditions 12Chapter 2 Probabilistic Models of Diffusion Processes 172.1 Stochastic differentiation 172.2 Itô’s formula 192.3 Stochastic differential equations (SDE) 242.4 Itô and diffusion processes 282.5 Some particular cases of diffusion processes 322.6 Multidimensional diffusion processes 362.7 The Stroock–Varadhan martingale characterization of diffusions (Karlin and Taylor) 412.8 The Feynman–Kac formula (Platen and Heath) 42Chapter 3 Solving Partial Differential Equations of Second Order 473.1 Basic definitions on PDE of second order 473.2 Solving the heat equation 513.3 Solution by the method of Laplace transform 653.4 Green’s functions 75Chapter 4 Problems in Finance 854.1 Basic stochastic models for stock prices 854.2 The bond investments 904.3 Dynamic deterministic continuous time model for instantaneous interest rate 934.4 Stochastic continuous time dynamic model for instantaneous interest rate 984.5 Multidimensional Black and Scholes model 110Chapter 5 Basic PDE in Finance 1115.1 Introduction to option theory 1115.2 Pricing the plain vanilla call with the Black–Scholes–Samuelson model 1155.3 Pricing no plain vanilla calls with the Black-Scholes-Samuelson model 1205.4 Zero-coupon pricing under the assumption of no arbitrage 127Chapter 6 Exotic and American Options Pricing Theory 1456.1 Introduction 1456.2 The Garman–Kohlhagen formula 1466.3 Binary or digital options 1496.4 “Asset or nothing” options 1506.5 Numerical examples 1526.6 Path-dependent options 1536.7 Multi-asset options 1576.8 American options 165Chapter 7 Hitting Times for Diffusion Processes and Stochastic Models in Insurance 1777.1 Hitting or first passage times for some diffusion processes 1777.2 Merton’s model for default risk 1937.3 Risk diffusion models for insurance 201Chapter 8 Numerical Methods 2198.1 Introduction 2198.2 Discretization and numerical differentiation 2208.3 Finite difference methods 2229.1 Nonlinear model in heat conduction 232Chapter 9 Advanced Topics in Engineering: Nonlinear Models 2319.2 Integral method applied to diffusive problems 2339.3 Integral method applied to nonlinear problems 2399.4 Use of transformations in nonlinear problems 243Chapter 10 Lévy Processes 25510.1 Motivation 25510.2 Notion of characteristic functions 25710.3 Lévy processes 25710.4 Lévy–Khintchine formula 25910.5 Examples of Lévy processes 26110.6 Variance gamma (VG) process 26410.7 The Brownian–Poisson model with jumps 26610.8 Risk neutral measures for Lévy models in finance 27510.9 Conclusion 276Chapter 11 Advanced Topics in Insurance: Copula Models and VaR Techniques 277               11.1 Introduction 27711.2 Sklar theorem (1959) 27911.3 Particular cases and Fréchet bounds 28011.4 Dependence 28811.5 Applications in finance: pricing of the bivariate digital put option 29311.6 VaR application in insurance 296Chapter 12 Advanced Topics in Finance: Semi-Markov Models 30712.1 Introduction 30712.2 Homogeneous semi-Markov process 30812.3 Semi-Markov option model 32812.4 Semi-Markov VaR models 33212.5 Conclusion 339Chapter 13 Monte Carlo Semi-Markov Simulation Methods 34113.1 Presentation of our simulation model 34113.2 The semi-Markov Monte Carlo model in a homogeneous environment 34513.3 A credit risk example 35013.4 Semi-Markov Monte Carlo with initial recurrence backward time in homogeneous case 36213.5 The SMMC applied to claim reserving problem 36313.6 An example of claim reserving calculation 366Conclusion 379Bibliography 381Index 393

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.

• Introduction xiiiChapter 1 Diffusion Phenomena and Models 11.1 General presentation of diffusion process 11.2 General balance equations 61.3 Heat conduction equation 101.4 Initial and boundary conditions 12Chapter 2 Probabilistic Models of Diffusion Processes 172.1 Stochastic differentiation 172.2 Itô’s formula 192.3 Stochastic differential equations (SDE) 242.4 Itô and diffusion processes 282.5 Some particular cases of diffusion processes 322.6 Multidimensional diffusion processes 362.7 The Stroock–Varadhan martingale characterization of diffusions (Karlin and Taylor) 412.8 The Feynman–Kac formula (Platen and Heath) 42Chapter 3 Solving Partial Differential Equations of Second Order 473.1 Basic definitions on PDE of second order 473.2 Solving the heat equation 513.3 Solution by the method of Laplace transform 653.4 Green’s functions 75Chapter 4 Problems in Finance 854.1 Basic stochastic models for stock prices 854.2 The bond investments 904.3 Dynamic deterministic continuous time model for instantaneous interest rate 934.4 Stochastic continuous time dynamic model for instantaneous interest rate 984.5 Multidimensional Black and Scholes model 110Chapter 5 Basic PDE in Finance 1115.1 Introduction to option theory 1115.2 Pricing the plain vanilla call with the Black–Scholes–Samuelson model 1155.3 Pricing no plain vanilla calls with the Black-Scholes-Samuelson model 1205.4 Zero-coupon pricing under the assumption of no arbitrage 127Chapter 6 Exotic and American Options Pricing Theory 1456.1 Introduction 1456.2 The Garman–Kohlhagen formula 1466.3 Binary or digital options 1496.4 “Asset or nothing” options 1506.5 Numerical examples 1526.6 Path-dependent options 1536.7 Multi-asset options 1576.8 American options 165Chapter 7 Hitting Times for Diffusion Processes and Stochastic Models in Insurance 1777.1 Hitting or first passage times for some diffusion processes 1777.2 Merton’s model for default risk 1937.3 Risk diffusion models for insurance 201Chapter 8 Numerical Methods 2198.1 Introduction 2198.2 Discretization and numerical differentiation 2208.3 Finite difference methods 2229.1 Nonlinear model in heat conduction 232Chapter 9 Advanced Topics in Engineering: Nonlinear Models 2319.2 Integral method applied to diffusive problems 2339.3 Integral method applied to nonlinear problems 2399.4 Use of transformations in nonlinear problems 243Chapter 10 Lévy Processes 25510.1 Motivation 25510.2 Notion of characteristic functions 25710.3 Lévy processes 25710.4 Lévy–Khintchine formula 25910.5 Examples of Lévy processes 26110.6 Variance gamma (VG) process 26410.7 The Brownian–Poisson model with jumps 26610.8 Risk neutral measures for Lévy models in finance 27510.9 Conclusion 276Chapter 11 Advanced Topics in Insurance: Copula Models and VaR Techniques 277               11.1 Introduction 27711.2 Sklar theorem (1959) 27911.3 Particular cases and Fréchet bounds 28011.4 Dependence 28811.5 Applications in finance: pricing of the bivariate digital put option 29311.6 VaR application in insurance 296Chapter 12 Advanced Topics in Finance: Semi-Markov Models 30712.1 Introduction 30712.2 Homogeneous semi-Markov process 30812.3 Semi-Markov option model 32812.4 Semi-Markov VaR models 33212.5 Conclusion 339Chapter 13 Monte Carlo Semi-Markov Simulation Methods 34113.1 Presentation of our simulation model 34113.2 The semi-Markov Monte Carlo model in a homogeneous environment 34513.3 A credit risk example 35013.4 Semi-Markov Monte Carlo with initial recurrence backward time in homogeneous case 36213.5 The SMMC applied to claim reserving problem 36313.6 An example of claim reserving calculation 366Conclusion 379Bibliography 381Index 393

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.

• Introduction xiiiChapter 1 Diffusion Phenomena and Models 11.1 General presentation of diffusion process 11.2 General balance equations 61.3 Heat conduction equation 101.4 Initial and boundary conditions 12Chapter 2 Probabilistic Models of Diffusion Processes 172.1 Stochastic differentiation 172.2 Itô’s formula 192.3 Stochastic differential equations (SDE) 242.4 Itô and diffusion processes 282.5 Some particular cases of diffusion processes 322.6 Multidimensional diffusion processes 362.7 The Stroock–Varadhan martingale characterization of diffusions (Karlin and Taylor) 412.8 The Feynman–Kac formula (Platen and Heath) 42Chapter 3 Solving Partial Differential Equations of Second Order 473.1 Basic definitions on PDE of second order 473.2 Solving the heat equation 513.3 Solution by the method of Laplace transform 653.4 Green’s functions 75Chapter 4 Problems in Finance 854.1 Basic stochastic models for stock prices 854.2 The bond investments 904.3 Dynamic deterministic continuous time model for instantaneous interest rate 934.4 Stochastic continuous time dynamic model for instantaneous interest rate 984.5 Multidimensional Black and Scholes model 110Chapter 5 Basic PDE in Finance 1115.1 Introduction to option theory 1115.2 Pricing the plain vanilla call with the Black–Scholes–Samuelson model 1155.3 Pricing no plain vanilla calls with the Black-Scholes-Samuelson model 1205.4 Zero-coupon pricing under the assumption of no arbitrage 127Chapter 6 Exotic and American Options Pricing Theory 1456.1 Introduction 1456.2 The Garman–Kohlhagen formula 1466.3 Binary or digital options 1496.4 “Asset or nothing” options 1506.5 Numerical examples 1526.6 Path-dependent options 1536.7 Multi-asset options 1576.8 American options 165Chapter 7 Hitting Times for Diffusion Processes and Stochastic Models in Insurance 1777.1 Hitting or first passage times for some diffusion processes 1777.2 Merton’s model for default risk 1937.3 Risk diffusion models for insurance 201Chapter 8 Numerical Methods 2198.1 Introduction 2198.2 Discretization and numerical differentiation 2208.3 Finite difference methods 2229.1 Nonlinear model in heat conduction 232Chapter 9 Advanced Topics in Engineering: Nonlinear Models 2319.2 Integral method applied to diffusive problems 2339.3 Integral method applied to nonlinear problems 2399.4 Use of transformations in nonlinear problems 243Chapter 10 Lévy Processes 25510.1 Motivation 25510.2 Notion of characteristic functions 25710.3 Lévy processes 25710.4 Lévy–Khintchine formula 25910.5 Examples of Lévy processes 26110.6 Variance gamma (VG) process 26410.7 The Brownian–Poisson model with jumps 26610.8 Risk neutral measures for Lévy models in finance 27510.9 Conclusion 276Chapter 11 Advanced Topics in Insurance: Copula Models and VaR Techniques 277               11.1 Introduction 27711.2 Sklar theorem (1959) 27911.3 Particular cases and Fréchet bounds 28011.4 Dependence 28811.5 Applications in finance: pricing of the bivariate digital put option 29311.6 VaR application in insurance 296Chapter 12 Advanced Topics in Finance: Semi-Markov Models 30712.1 Introduction 30712.2 Homogeneous semi-Markov process 30812.3 Semi-Markov option model 32812.4 Semi-Markov VaR models 33212.5 Conclusion 339Chapter 13 Monte Carlo Semi-Markov Simulation Methods 34113.1 Presentation of our simulation model 34113.2 The semi-Markov Monte Carlo model in a homogeneous environment 34513.3 A credit risk example 35013.4 Semi-Markov Monte Carlo with initial recurrence backward time in homogeneous case 36213.5 The SMMC applied to claim reserving problem 36313.6 An example of claim reserving calculation 366Conclusion 379Bibliography 381Index 393