Contents -- Title Page -- Copyright Page -- Introduction -- Chapter 1: Diffusion Phenomena and Models -- 1.1. General presentation of diffusion process -- 1.2. General balance equations -- 1.3. Heat conduction equation -- 1.4. Initial and boundary conditions -- Chapter 2: Probabilistic Models of Diffusion Processes -- 2.1. Stochastic differentiation -- 2.2. Itô's formula -- 2.3. Stochastic differential equations (SDE) -- 2.4. Itô and diffusion processes -- 2.5. Some particular cases of diffusion processes -- 2.6. Multidimensional diffusion processes -- 2.7. The Stroock-Varadhan martingale characterization of diffusions (Karlin and Taylor [KAR 81]) -- 2.8. The Feynman-Kac formula (Platen and Heath [PLA 06]) -- Chapter 3: Solving Partial Differential Equations of Second Order -- 3.1. Basic definitions on PDE of second order -- 3.2. Solving the heat equation -- 3.3. Solution by the method of Laplace transform -- 3.4. Green's functions -- Chapter 4: Problems in Finance -- 4.1. Basic stochastic models for stock prices -- 4.2. The bond investments -- 4.3. Dynamic deterministic continuous time model for instantaneous interest rate -- 4.4. Stochastic continuous time dynamic model for instantaneous interest rate -- 4.5. Multidimensional Black and Scholes model -- Chapter 5: Basic PDE in Finance -- 5.1. Introduction to option theory -- 5.2. Pricing the plain vanilla call with the Black-Scholes-Samuelson model -- 5.3. Pricing no plain vanilla calls with the Black-Scholes-Samuelson model -- 5.4. Zero-coupon pricing under the assumption of no arbitrage -- Chapter 6: Exotic and American Options Pricing Theory -- 6.1. Introduction -- 6.2. The Garman-Kohlhagen formula -- 6.3. Binary or digital options -- 6.4. "Asset or nothing" options -- 6.5. Numerical examples -- 6.6. Path-dependent options -- 6.7. Multi-asset options -- 6.8. American options.

Chapter 7: Hitting Times for Diffusion Processes and Stochastic Models in Insurance -- 7.1. Hitting or first passage times for some diffusion processes -- 7.2. Merton's model for default risk -- 7.3. Risk diffusion models for insurance -- Chapter 8: Numerical Methods -- 8.1. Introduction -- 8.2. Discretization and numerical differentiation -- 8.3. Finite difference methods -- Chapter 9: Advanced Topics in Engineering: Nonlinear Models -- 9.1. Nonlinear model in heat conduction -- 9.2. Integral method applied to diffusive problems -- 9.3. Integral method applied to nonlinear problems -- 9.4. Use of transformations in nonlinear problems -- Chapter 10: Lévy Processes -- 10.1. Motivation -- 10.2. Notion of characteristic functions -- 10.3. Lévy processes -- 10.4. Lévy-Khintchine formula -- 10.5. Examples of Lévy processes -- 10.6. Variance gamma (VG) process -- 10.7. The Brownian-Poisson model with jumps -- 10.8. Risk neutral measures for Lévy models in finance -- 10.9. Conclusion -- Chapter 11: Advanced Topics in Insurance: Copula Models and VaR Techniques -- 11.1. Introduction -- 11.2. Sklar theorem (1959) -- 11.3. Particular cases and Fréchet bounds -- 11.4. Dependence -- 11.5. Applications in finance: pricing of the bivariate digital put option [CHE 04] -- 11.6. VaR application in insurance -- Chapter 12: Advanced Topics in Finance: Semi-Markov Models -- 12.1. Introduction -- 12.2. Homogeneous semi-Markov process -- 12.3. Semi-Markov option model -- 12.4. Semi-Markov VaR models -- 12.5. Conclusion -- Chapter 13: Monte Carlo Semi-Markov Simulation Methods -- 13.1. Presentation of our simulation model -- 13.2. The semi-Markov Monte Carlo model in a homogeneous environment -- 13.3. A credit risk example -- 13.4. Semi-Markov Monte Carlo with initial recurrence backward time in homogeneous case -- 13.5. The SMMC applied to claim reserving problem.

13.6. An example of claim reserving calculation -- Conclusion -- Bibliography -- Index.