Table of Contents -- Title -- Copyright -- Introduction -- 1: Basic Probabilistic Tools for Stochastic Modeling -- 1.1. Probability space and random variables -- 1.2. Expectation and independence -- 1.3. Main distribution probabilities -- 1.4. The normal power (NP) approximation -- 1.5. Conditioning -- 1.6. Stochastic processes -- 1.7. Martingales -- 2: Homogeneous and Non-homogeneous Renewal Models -- 2.1. Introduction -- 2.2. Continuous time non-homogeneous convolutions -- 2.3. Homogeneous and non-homogeneous renewal processes -- 2.4. Counting processes and renewal functions -- 2.5. Asymptotical results in the homogeneous case -- 2.6. Recurrence times in the homogeneous case -- 2.7. Particular case: the Poisson process -- 2.8. Homogeneous alternating renewal processes -- 2.9. Solution of non-homogeneous discrete time evolution equation -- 3: Markov Chains -- 3.1. Definitions -- 3.2. Homogeneous case -- 3.3. Non-homogeneous Markov chains -- 3.4. Markov reward processes -- 3.5. Discrete time Markov reward processes (DTMRWPs) -- 3.6. General algorithms for the DTMRWP -- 4: Homogeneous and Non-homogeneous Semi-Markov Models -- 4.1. Continuous time semi-Markov processes -- 4.2. The embedded Markov chain -- 4.3. The counting processes and the associated semi-Markov process -- 4.4. Initial backward recurrence times -- 4.5. Particular cases of MRP -- 4.6. Examples -- 4.7. Discrete time homogeneous and non-homogeneous semi-Markov processes -- 4.8. Semi-Markov backward processes in discrete time -- 4.9. Discrete time reward processes -- 4.10. Markov renewal functions in the homogeneous case -- 4.11. Markov renewal equations for the non-homogeneous case -- 5: Stochastic Calculus -- 5.1. Brownian motion -- 5.2. General definition of the stochastic integral -- 5.3. Itô's formula.

5.4. Stochastic integral with standard Brownian motion as an integrator process -- 5.5. Stochastic differentiation -- 5.6. Stochastic differential equations -- 5.7. Multidimensional diffusion processes -- 5.8. Relation between the resolution of PDE and SDE problems. The Feynman-Kac formula [PLA 06] -- 5.9. Application to option theory -- 6: Lévy Processes -- 6.1. Notion of characteristic functions -- 6.2. Lévy processes -- 6.3. Lévy-Khintchine formula -- 6.4. Subordinators -- 6.5. Poisson measure for jumps -- 6.6. Markov and martingale properties of Lévy processes -- 6.7. Examples of Lévy processes -- 6.8. Variance gamma (VG) process -- 6.9. Hyperbolic Lévy processes -- 6.10. The Esscher transformation -- 6.11. The Brownian-Poisson model with jumps -- 6.12. Complete and incomplete markets -- 6.13. Conclusion -- 7: Actuarial Evaluation, VaR and Stochastic Interest Rate Models -- 7.1. VaR technique -- 7.2. Conditional VaR value -- 7.3. Solvency II -- 7.4. Fair value -- 7.5. Dynamic stochastic time continuous time model for instantaneous interest rate -- 7.6. Zero-coupon pricing under the assumption of no arbitrage -- 7.7. Market evaluation of financial flows -- Bibliography -- Index -- End User License Agreement.