Basic Stochastic Processes

Pierre Devolder, Université catholique de Louvain, Belgium.Jacques Janssen, Solvay Business School, Brussels, Belgium.Raimondo Manca, University "La Sapienza", Rome, Italy.

• INTRODUCTION  xiCHAPTER 1. BASIC PROBABILISTIC TOOLS FOR STOCHASTIC MODELING 11.1. Probability space and random variables 11.2. Expectation and independence 41.3. Main distribution probabilities 71.3.1. Binomial distribution 71.3.2. Negative exponential distribution 81.3.3. Normal (or Laplace–Gauss) distribution 81.3.4. Poisson distribution 111.3.5. Lognormal distribution 111.3.6. Gamma distribution 121.3.7. Pareto distribution 131.3.8. Uniform distribution 161.3.9. Gumbel distribution 161.3.10. Weibull distribution 161.3.11. Multi-dimensional normal distribution 171.3.12. Extreme value distribution 191.4. The normal power (NP) approximation 281.5. Conditioning 311.6. Stochastic processes 391.7. Martingales 43CHAPTER 2. HOMOGENEOUS AND NON-HOMOGENEOUS RENEWAL MODELS 472.1. Introduction 472.2. Continuous time non-homogeneous convolutions 492.2.1. Non-homogeneous convolution product 492.3. Homogeneous and non-homogeneous renewal processes 532.4. Counting processes and renewal functions 562.5. Asymptotical results in the homogeneous case 612.6. Recurrence times in the homogeneous case 632.7. Particular case: the Poisson process 662.7.1. Homogeneous case 662.7.2. Non-homogeneous case 682.8. Homogeneous alternating renewal processes 692.9. Solution of non-homogeneous discrete timevevolution equation 712.9.1. General method 712.9.2. Some particular formulas 732.9.3. Relations between discrete time and continuous time renewal equations 74CHAPTER 3. MARKOV CHAINS 773.1. Definitions 773.2. Homogeneous case 783.2.1. Basic definitions 783.2.2. Markov chain state classification 813.2.3. Computation of absorption probabilities 873.2.4. Asymptotic behavior 883.2.5. Example: a management problem in an insurance company 933.3. Non-homogeneous Markov chains 953.3.1. Definitions 953.3.2. Asymptotical results 983.4. Markov reward processes 993.4.1. Classification and notation 993.5. Discrete time Markov reward processes (DTMRWPs) 1023.5.1. Undiscounted case 1023.5.2. Discounted case 1053.6. General algorithms for the DTMRWP 1113.6.1. Homogeneous MRWP 1123.6.2. Non-homogeneous MRWP 112CHAPTER 4. HOMOGENEOUS AND NON-HOMOGENEOUS SEMI-MARKOV MODELS 1134.1. Continuous time semi-Markov processes 1134.2. The embedded Markov chain 1174.3. The counting processes and the associated semi-Markov process 1184.4. Initial backward recurrence times 1204.5. Particular cases of MRP 1224.5.1. Renewal processes and Markov chains 1224.5.2. MRP of zero-order (PYKE (1962)) 1224.5.3. Continuous Markov processes 1244.6. Examples 1244.7. Discrete time homogeneous and non-homogeneous semi-Markov processes 1274.8. Semi-Markov backward processes in discrete time 1294.8.1. Definition in the homogeneous case 1294.8.2. Semi-Markov backward processes in discrete time for the non-homogeneous case 1304.8.3. DTSMP numerical solutions 1334.9. Discrete time reward processes 1374.9.1. Undiscounted SMRWP 1374.9.2. Discounted SMRWP 1414.9.3. General algorithms for DTSMRWP 1444.10. Markov renewal functions in the homogeneous case 1464.10.1. Entrance times 1464.10.2. The Markov renewal equation 1504.10.3. Asymptotic behavior of an MRP 1514.10.4. Asymptotic behavior of SMP 1534.11. Markov renewal equations for the non-homogeneous case 1584.11.1. Entrance time 1584.11.2. The Markov renewal equation 162CHAPTER 5. STOCHASTIC CALCULUS  1655.1. Brownian motion 1655.2. General definition of the stochastic integral 1675.2.1. Problem of stochastic integration 1675.2.2. Stochastic integration of simple predictable processes and semi-martingales 1685.2.3. General definition of the stochastic integral 1705.3. Itô’s formula 1775.3.1. Quadratic variation of a semi-martingale 1775.3.2. Itô’s formula 1795.4. Stochastic integral with standard Brownian motion as an integrator process 1805.4.1. Case of simple predictable processes 1815.4.2. Extension to general integrator processes 1835.5. Stochastic differentiation 1845.5.1. Stochastic differential 1845.5.2. Particular cases 1845.5.3. Other forms of Itô’s formula 1855.6. Stochastic differential equations 1915.6.1. Existence and unicity general theorem 1915.6.2. Solution of stochastic differential equations 1955.6.3. Diffusion processes 1995.7. Multidimensional diffusion processes 2025.7.1. Definition of multidimensional Itô and diffusion processes 2035.7.2. Properties of multidimensional diffusion processes 2035.7.3. Kolmogorov equations 2055.7.4. The Stroock–Varadhan martingale characterization of diffusion processes 2085.8. Relation between the resolution of PDE and SDE problems. The Feynman–Kac formula 2095.8.1. Terminal payoff 2095.8.2. Discounted payoff function 2105.8.3. Discounted payoff function and payoff rate 2105.9. Application to option theory 2135.9.1. Options 2135.9.2. Black and Scholes model 2165.9.3. The Black and Scholes partial differential equation (BSPDE) and the BS formula 2165.9.4. Girsanov theorem 2195.9.5. The risk-neutral measure and the martingale property 2215.9.6. The risk-neutral measure and the evaluation of derivative products 224CHAPTER 6. LÉVY PROCESSES 2276.1. Notion of characteristic functions 2276.2. Lévy processes 2286.3. Lévy–Khintchine formula 2306.4. Subordinators 2346.5. Poisson measure for jumps 2346.5.1. The Poisson random measure 2346.5.2. The compensated Poisson process 2356.5.3. Jump measure of a Lévy process 2366.5.4. The Itô–Lévy decomposition 2366.6. Markov and martingale properties of Lévy processes 2376.6.1. Markov property 2376.6.2. Martingale properties 2396.6.3. Itô formula 2406.7. Examples of Lévy processes 2406.7.1. The lognormal process: Black and Scholes process 2406.7.2. The Poisson process 2416.7.3. Compensated Poisson process 2426.7.4. The compound Poisson process 2426.8. Variance gamma (VG) process 2446.8.1. The gamma distribution 2446.8.2. The VG distribution 2456.8.3. The VG process 2466.8.4. The Esscher transformation 2476.8.5. The Carr–Madan formula for the European call 2496.9. Hyperbolic Lévy processes 2506.10. The Esscher transformation 2526.10.1. Definition 2526.10.2. Option theory with hyperbolic Lévy processes 2536.10.3. Value of the European option call 2556.11. The Brownian–Poisson model with jumps 2566.11.1. Mixed arithmetic Brownian–Poisson and geometric Brownian–Poisson processes 2566.11.2. Merton model with jumps 2586.11.3. Stochastic differential equation (SDE) for mixed arithmetic Brownian–Poisson and geometric Brownian–Poisson processes 2616.11.4. Value of a European call for the lognormal Merton model 2646.12. Complete and incomplete markets 2646.13. Conclusion 265CHAPTER 7. ACTUARIAL EVALUATION, VAR AND STOCHASTIC INTEREST RATE MODELS 2677.1. VaR technique 2677.2. Conditional VaR value 2717.3. Solvency II 2767.3.1. The SCR indicator 2767.3.2. Calculation of MCR 2787.3.3. ORSA approach 2797.4. Fair value 2807.4.1. Definition 2807.4.2. Market value of financial flows 2817.4.3. Yield curve 2817.4.4. Yield to maturity for a financial investment and a bond 2837.5. Dynamic stochastic time continuous time model for instantaneous interest rate 2847.5.1. Instantaneous deterministic interest rate 2847.5.2. Yield curve associated with a deterministic instantaneous interest rate 2857.5.3. Dynamic stochastic continuous time model for instantaneous interest rate 2867.5.4. The OUV stochastic model 2877.5.5. The CIR model 2897.6. Zero-coupon pricing under the assumption of no arbitrage 2927.6.1. Stochastic dynamics of zero-coupons 2927.6.2. The CIR process as rate dynamic 2957.7. Market evaluation of financial flows 298BIBLIOGRAPHY 301INDEX 309