图书简介

1. Preliminaries. 1.1. Markov processes. 1.2. Random evolutions. 1.3. Determinant theorem. 1.4. Kurtz’s diffusion approximation theorem. 1.5. Special functions. 1.6. Hypergeometric functions. 1.7. Generalized functions. 1.8. Integral transforms. 1.9. Auxiliary lemmas. 2. Telegraph Processes. 2.1. Definition of the process and structure of distribution. 2.2. Kolmogorov equation. 2.3. Telegraph equation. 2.4. Characteristic function. 2.5. Transition density. 2.6. Probability distribution function. 2.7. Convergence to the Wiener process. 2.8. Laplace transform of transition density. 2.9. Moment analysis. 2.11. Telegraph-type processes with several velocities. 2.12. Euclidean distance between two telegraph processes. 2.13. Sum of two telegraph processes. 2.14. Linear combinations of telegraph processes. 3. Planar Random Motion with a Finite Number of Directions. 3.1. Description of the model and the main result. 3.2. Proof of the Main Theorem. 3.3. Diffusion area. 3.4. Polynomial representations of the generator. 3.5. Limiting differential operator. 3.6. Weak convergence to the Wiener process. 4. Integral Transforms of the Distributions of Markov Random Flights. 4.1. Description of process and structure of distribution. 4.2. Recurrent integral relations. 4.3. Laplace transforms of conditional characteristic functions. 4.4. Conditional characteristic functions. 4.5. Integral equation for characteristic function. 4.6. Laplace transform of characteristic function. 4.7. Initial conditions. 4.8. Limit theorem. 4.9. Random flight with rare switching. 4.10. Hyper-parabolic operators. 4.11. Random flight with arbitrary dissipation function. 4.12. Integral equation for transition density. 5. Markov Random Flight in the Plane R2. 5.1. Conditional densities. 5.2 Distribution of the process. 5.3. Characteristic function. 5.4 Telegraph equation. 5.5. Limit theorem. 5.6. Alternative derivation of transition density. 5.7. Moments. 5.8. Random flight with Gaussian starting point. 5.9. Euclidean distance between two random flights. 6. Markov Random Flight in the Space R3. 6.1. Characteristic function. 6.2. Discontinuous term of distribution. 6.3. Limit theorem. 6.4. Asymptotic relation for the transition density. 6.5. Fundamental solution to Kolmogorov equation. 7. Markov Random Flight in the Space R4. 7.1. Conditional densities. 7.2. Distribution of the process. 7.3. Characteristic function. 7.4. Limit theorem. 7.5. Moments. 8. Markov Random Flight in the Space R6. 8.1. Conditional densities. 8.2. Distribution of the process. 9. Applied Models. 9.1. Slow diffusion. 9.2. Fluctuations of water level in reservoir. 9.3. Pollution model. 9.4. Physical applications. 9.5 Option pricing.

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