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Theory and Statistical Applications of Stochastic Processes.

By: Contributor(s): Material type: TextTextPublisher: Newark : John Wiley & Sons, Incorporated, 2018Copyright date: ©2017Edition: 1st edDescription: 1 online resource (405 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781119476634
Subject(s): Genre/Form: Additional physical formats: Print version:: Theory and Statistical Applications of Stochastic ProcessesDDC classification:
  • 519.23
LOC classification:
  • QA274 .M574 2017
Online resources:
Contents:
Cover -- Half-Title Page -- Title Page -- Copyright Page -- Contents -- Preface -- Introduction -- PART 1. Theory of Stochastic Processes -- 1. Stochastic Processes. General Properties. Trajectories, Finite-dimensional Distributions -- 1.1. Definition of a stochastic process -- 1.2. Trajectories of a stochastic process. Some examples of stochastic processes -- 1.2.1. Definition of trajectory and some examples -- 1.2.2. Trajectory of a stochastic process as a random element -- 1.3. Finite-dimensional distributions of stochastic processes: consistency conditions -- 1.3.1. Definition and properties of finite-dimensional distributions -- 1.3.2. Consistency conditions -- 1.3.3. Cylinder sets and generated σ-algebra -- 1.3.4. Kolmogorov theorem on the construction of a stochastic process by the family of probability distributions -- 1.4. Properties of σ-algebra generated by cylinder sets. The notion of σ-algera generated by a stochastic process -- 2. Stochastic Processes with Independent Increments -- 2.1. Existence of processes with independent increments in terms of incremental characteristic functions -- 2.2. Wiener process -- 2.2.1. One-dimensional Wiener process -- 2.2.2. Independent stochastic processes. Multidimensional Wiener process -- 2.3. Poisson process -- 2.3.1. Poisson process defined via the existence theorem -- 2.3.2. Poisson process defined via the distributions of the increments -- 2.3.3. Poisson process as a renewal process -- 2.4. Compound Poisson process -- 2.5. Lévy processes -- 2.5.1. Wiener process with a drift -- 2.5.2. Compound Poisson process as a Lévy process -- 2.5.3. Sum of a Wiener process with a drift and a Poisson process -- 2.5.4. Gamma process -- 2.5.5. Stable Lévy motion -- 2.5.6. Stable Lévy subordinator with stability parameter α Є (0, 1) -- 3. Gaussian Processes. Integration with Respect to Gaussian Processes.
3.1. Gaussian vectors -- 3.2. Theorem of Gaussian representation (theorem on normal correlation) -- 3.3. Gaussian processes -- 3.4. Examples of Gaussian processes -- 3.4.1. Wiener process as an example of a Gaussian process -- 3.4.2. Fractional Brownian motion -- 3.4.3. Sub-fractional and bi-fractional Brownian motion -- 3.4.4. Brownian bridge -- 3.4.5. Ornstein-Uhlenbeck process -- 3.5. Integration of non-random functions with respect to Gaussian processes -- 3.5.1. General approach -- 3.5.2. Integration of non-random functions with respect to the Wiener process -- 3.5.3. Integration w.r.t. the fractional Brownian motion -- 3.6. Two-sided Wiener process and fractional Brownian motion: Mandelbrot-van Ness representation of fractional Brownian motion -- 3.7. Representation of fractional Brownian motion as the Wiener integral on the compact integral -- 4. Construction, Properties and Some Functionals of the Wiener Process and Fractional Brownian Motion -- 4.1. Construction of a Wiener process on the interval [0, 1] -- 4.2. Construction of a Wiener process on R+ -- 4.3. Nowhere differentiability of the trajectories of a Wiener process -- 4.4. Power variation of the Wiener process and of the fractional Brownian motion -- 4.4.1. Ergodic theorem for power variations -- 4.5. Self-similar stochastic processes -- 4.5.1. Definition of self-similarity and some examples -- 4.5.2. Power variations of self-similar processes on finite intervals -- 5. Martingales and Related Processes -- 5.1. Notion of stochastic basis with filtration -- 5.2. Notion of (sub-, super-) martingale: elementary properties -- 5.3. Examples of (sub-, super-) martingales -- 5.4. Markov moments and stopping times -- 5.5. Martingales and related processes with discrete time -- 5.5.1. Upcrossings of the interval and existence of the limit of submartingale.
5.5.2. Examples of martingales having a limit and of uniformly and non-uniformly integrable martingales -- 5.5.3. Lévy convergence theorem -- 5.5.4. Optional stopping -- 5.5.5. Maximal inequalities for (sub-, super-) martingales -- 5.5.6. Doob decomposition for the integrable processes with discrete time -- 5.5.7. Quadratic variation and quadratic characteristics: Burkholder-Davis-Gundy inequalities -- 5.5.8. Change of probability measure and Girsanov theorem for discrete-time processes -- 5.5.9. Strong law of large numbers for martingales with discrete time -- 5.6. Lévy martingale stopped -- 5.7. Martingales with continuous time -- 6. Regularity of Trajectories of Stochastic Processes -- 6.1. Continuity in probability and in L2(Ω,F, P) -- 6.2. Modification of stochastic processes: stochastically equivalent and indistinguishable processes -- 6.3. Separable stochastic processes: existence of separable modification -- 6.4. Conditions of D-regularity and absence of the discontinuities of the second kind for stochastic processes -- 6.4.1. Skorokhod conditions of D-regularity in terms of three-dimensional distributions -- 6.4.2. Conditions of absence of the discontinuities of the second kind formulated in terms of conditional probabilities of large increments -- 6.5. Conditions of continuity of trajectories of stochastic processes -- 6.5.1. Kolmogorov conditions of continuity in terms of two-dimensional distributions -- 6.5.2. Hölder continuity of stochastic processes: a sufficient condition -- 6.5.3. Conditions of continuity in terms of conditional probabilities -- 7. Markov and Diffusion Processes -- 7.1. Markov property -- 7.2. Examples of Markov processes -- 7.2.1. Discrete-time Markov chain -- 7.2.2. Continuous-time Markov chain -- 7.2.3. Process with independent increments.
7.3. Semigroup resolvent operator and generator related to the homogeneous Markov process -- 7.3.1. Semigroup related to Markov process -- 7.3.2. Resolvent operator and resolvent equation -- 7.3.3. Generator of a semigroup -- 7.4. Definition and basic properties of diffusion process -- 7.5. Homogeneous diffusion process. Wiener process as a diffusion process -- 7.6. Kolmogorov equations for diffusions -- 8. Stochastic Integration -- 8.1. Motivation -- 8.2. Definition of Itô integral -- 8.2.1. Itô integral of Wiener process -- 8.3. Continuity of Itô integral -- 8.4. Extended Itô integral -- 8.5. Itô processes and Itô formula -- 8.6. Multivariate stochastic calculus -- 8.7. Maximal inequalities for Itô martingales -- 8.7.1. Strong law of large numbers for Itô local martingales -- 8.8. Lévy martingale characterization of Wiener process -- 8.9. Girsanov theorem -- 8.10. Itô representation -- 9. Stochastic Differential Equations -- 9.1. Definition, solvability conditions, examples -- 9.1.1. Existence and uniqueness of solution -- 9.1.2. Some special stochastic differential equations -- 9.2. Properties of solutions to stochastic differential equations -- 9.3. Continuous dependence of solutions on coefficients -- 9.4. Weak solutions to stochastic differential equations -- 9.5. Solutions to SDEs as diffusion processes -- 9.6. Viability, comparison and positivity of solutions to stochastic differential equations -- 9.6.1. Comparison theorem for one-dimensional projections of stochastic differential equations -- 9.6.2. Non-negativity of solutions to stochastic differential equations -- 9.7. Feynman-Kac formula -- 9.8. Diffusion model of financial markets -- 9.8.1. Admissible portfolios, arbitrage and equivalent martingale measure -- 9.8.2. Contingent claims, pricing and hedging -- PART 2. Statistics of Stochastic Processes -- 10. Parameter Estimation.
10.1. Drift and diffusion parameter estimation in the linear regression model with discrete time -- 10.1.1. Drift estimation in the linear regression model with discrete time in the case when the initial value is known -- 10.1.2. Drift estimation in the case when the initial value is unknown -- 10.2. Estimation of the diffusion coefficient in a linear regression model with discrete time -- 10.3. Drift and diffusion parameter estimation in the linear model with continuous time and the Wiener noise -- 10.3.1. Drift parameter estimation -- 10.3.2. Diffusion parameter estimation -- 10.4. Parameter estimation in linear models with fractional Brownian motion -- 10.4.1. Estimation of Hurst index -- 10.4.2. Estimation of the diffusion parameter -- 10.5. Drift parameter estimation -- 10.6. Drift parameter estimation in the simplest autoregressive model -- 10.7. Drift parameters estimation in the homogeneous diffusion model -- 11. Filtering Problem. Kalman-Bucy Filter -- 11.1. General setting -- 11.2. Auxiliary properties of the non-observable process -- 11.3. What is an optimal filter -- 11.4. Representation of an optimal filter via an integral equation with respect to an observable process -- 11.5. Integral Wiener-Hopf equation -- Appendices -- Appendix 1: Selected Facts from Calculus, Measure Theory and the Theory of Operators -- Appendix 2: Selected Facts from Probability Theory and Auxiliary Computations for Stochastic Processes -- Bibliography -- Index -- Other titles from iSTE in Mathematics and Statistics -- EULA.
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Cover -- Half-Title Page -- Title Page -- Copyright Page -- Contents -- Preface -- Introduction -- PART 1. Theory of Stochastic Processes -- 1. Stochastic Processes. General Properties. Trajectories, Finite-dimensional Distributions -- 1.1. Definition of a stochastic process -- 1.2. Trajectories of a stochastic process. Some examples of stochastic processes -- 1.2.1. Definition of trajectory and some examples -- 1.2.2. Trajectory of a stochastic process as a random element -- 1.3. Finite-dimensional distributions of stochastic processes: consistency conditions -- 1.3.1. Definition and properties of finite-dimensional distributions -- 1.3.2. Consistency conditions -- 1.3.3. Cylinder sets and generated σ-algebra -- 1.3.4. Kolmogorov theorem on the construction of a stochastic process by the family of probability distributions -- 1.4. Properties of σ-algebra generated by cylinder sets. The notion of σ-algera generated by a stochastic process -- 2. Stochastic Processes with Independent Increments -- 2.1. Existence of processes with independent increments in terms of incremental characteristic functions -- 2.2. Wiener process -- 2.2.1. One-dimensional Wiener process -- 2.2.2. Independent stochastic processes. Multidimensional Wiener process -- 2.3. Poisson process -- 2.3.1. Poisson process defined via the existence theorem -- 2.3.2. Poisson process defined via the distributions of the increments -- 2.3.3. Poisson process as a renewal process -- 2.4. Compound Poisson process -- 2.5. Lévy processes -- 2.5.1. Wiener process with a drift -- 2.5.2. Compound Poisson process as a Lévy process -- 2.5.3. Sum of a Wiener process with a drift and a Poisson process -- 2.5.4. Gamma process -- 2.5.5. Stable Lévy motion -- 2.5.6. Stable Lévy subordinator with stability parameter α Є (0, 1) -- 3. Gaussian Processes. Integration with Respect to Gaussian Processes.

3.1. Gaussian vectors -- 3.2. Theorem of Gaussian representation (theorem on normal correlation) -- 3.3. Gaussian processes -- 3.4. Examples of Gaussian processes -- 3.4.1. Wiener process as an example of a Gaussian process -- 3.4.2. Fractional Brownian motion -- 3.4.3. Sub-fractional and bi-fractional Brownian motion -- 3.4.4. Brownian bridge -- 3.4.5. Ornstein-Uhlenbeck process -- 3.5. Integration of non-random functions with respect to Gaussian processes -- 3.5.1. General approach -- 3.5.2. Integration of non-random functions with respect to the Wiener process -- 3.5.3. Integration w.r.t. the fractional Brownian motion -- 3.6. Two-sided Wiener process and fractional Brownian motion: Mandelbrot-van Ness representation of fractional Brownian motion -- 3.7. Representation of fractional Brownian motion as the Wiener integral on the compact integral -- 4. Construction, Properties and Some Functionals of the Wiener Process and Fractional Brownian Motion -- 4.1. Construction of a Wiener process on the interval [0, 1] -- 4.2. Construction of a Wiener process on R+ -- 4.3. Nowhere differentiability of the trajectories of a Wiener process -- 4.4. Power variation of the Wiener process and of the fractional Brownian motion -- 4.4.1. Ergodic theorem for power variations -- 4.5. Self-similar stochastic processes -- 4.5.1. Definition of self-similarity and some examples -- 4.5.2. Power variations of self-similar processes on finite intervals -- 5. Martingales and Related Processes -- 5.1. Notion of stochastic basis with filtration -- 5.2. Notion of (sub-, super-) martingale: elementary properties -- 5.3. Examples of (sub-, super-) martingales -- 5.4. Markov moments and stopping times -- 5.5. Martingales and related processes with discrete time -- 5.5.1. Upcrossings of the interval and existence of the limit of submartingale.

5.5.2. Examples of martingales having a limit and of uniformly and non-uniformly integrable martingales -- 5.5.3. Lévy convergence theorem -- 5.5.4. Optional stopping -- 5.5.5. Maximal inequalities for (sub-, super-) martingales -- 5.5.6. Doob decomposition for the integrable processes with discrete time -- 5.5.7. Quadratic variation and quadratic characteristics: Burkholder-Davis-Gundy inequalities -- 5.5.8. Change of probability measure and Girsanov theorem for discrete-time processes -- 5.5.9. Strong law of large numbers for martingales with discrete time -- 5.6. Lévy martingale stopped -- 5.7. Martingales with continuous time -- 6. Regularity of Trajectories of Stochastic Processes -- 6.1. Continuity in probability and in L2(Ω,F, P) -- 6.2. Modification of stochastic processes: stochastically equivalent and indistinguishable processes -- 6.3. Separable stochastic processes: existence of separable modification -- 6.4. Conditions of D-regularity and absence of the discontinuities of the second kind for stochastic processes -- 6.4.1. Skorokhod conditions of D-regularity in terms of three-dimensional distributions -- 6.4.2. Conditions of absence of the discontinuities of the second kind formulated in terms of conditional probabilities of large increments -- 6.5. Conditions of continuity of trajectories of stochastic processes -- 6.5.1. Kolmogorov conditions of continuity in terms of two-dimensional distributions -- 6.5.2. Hölder continuity of stochastic processes: a sufficient condition -- 6.5.3. Conditions of continuity in terms of conditional probabilities -- 7. Markov and Diffusion Processes -- 7.1. Markov property -- 7.2. Examples of Markov processes -- 7.2.1. Discrete-time Markov chain -- 7.2.2. Continuous-time Markov chain -- 7.2.3. Process with independent increments.

7.3. Semigroup resolvent operator and generator related to the homogeneous Markov process -- 7.3.1. Semigroup related to Markov process -- 7.3.2. Resolvent operator and resolvent equation -- 7.3.3. Generator of a semigroup -- 7.4. Definition and basic properties of diffusion process -- 7.5. Homogeneous diffusion process. Wiener process as a diffusion process -- 7.6. Kolmogorov equations for diffusions -- 8. Stochastic Integration -- 8.1. Motivation -- 8.2. Definition of Itô integral -- 8.2.1. Itô integral of Wiener process -- 8.3. Continuity of Itô integral -- 8.4. Extended Itô integral -- 8.5. Itô processes and Itô formula -- 8.6. Multivariate stochastic calculus -- 8.7. Maximal inequalities for Itô martingales -- 8.7.1. Strong law of large numbers for Itô local martingales -- 8.8. Lévy martingale characterization of Wiener process -- 8.9. Girsanov theorem -- 8.10. Itô representation -- 9. Stochastic Differential Equations -- 9.1. Definition, solvability conditions, examples -- 9.1.1. Existence and uniqueness of solution -- 9.1.2. Some special stochastic differential equations -- 9.2. Properties of solutions to stochastic differential equations -- 9.3. Continuous dependence of solutions on coefficients -- 9.4. Weak solutions to stochastic differential equations -- 9.5. Solutions to SDEs as diffusion processes -- 9.6. Viability, comparison and positivity of solutions to stochastic differential equations -- 9.6.1. Comparison theorem for one-dimensional projections of stochastic differential equations -- 9.6.2. Non-negativity of solutions to stochastic differential equations -- 9.7. Feynman-Kac formula -- 9.8. Diffusion model of financial markets -- 9.8.1. Admissible portfolios, arbitrage and equivalent martingale measure -- 9.8.2. Contingent claims, pricing and hedging -- PART 2. Statistics of Stochastic Processes -- 10. Parameter Estimation.

10.1. Drift and diffusion parameter estimation in the linear regression model with discrete time -- 10.1.1. Drift estimation in the linear regression model with discrete time in the case when the initial value is known -- 10.1.2. Drift estimation in the case when the initial value is unknown -- 10.2. Estimation of the diffusion coefficient in a linear regression model with discrete time -- 10.3. Drift and diffusion parameter estimation in the linear model with continuous time and the Wiener noise -- 10.3.1. Drift parameter estimation -- 10.3.2. Diffusion parameter estimation -- 10.4. Parameter estimation in linear models with fractional Brownian motion -- 10.4.1. Estimation of Hurst index -- 10.4.2. Estimation of the diffusion parameter -- 10.5. Drift parameter estimation -- 10.6. Drift parameter estimation in the simplest autoregressive model -- 10.7. Drift parameters estimation in the homogeneous diffusion model -- 11. Filtering Problem. Kalman-Bucy Filter -- 11.1. General setting -- 11.2. Auxiliary properties of the non-observable process -- 11.3. What is an optimal filter -- 11.4. Representation of an optimal filter via an integral equation with respect to an observable process -- 11.5. Integral Wiener-Hopf equation -- Appendices -- Appendix 1: Selected Facts from Calculus, Measure Theory and the Theory of Operators -- Appendix 2: Selected Facts from Probability Theory and Auxiliary Computations for Stochastic Processes -- Bibliography -- Index -- Other titles from iSTE in Mathematics and Statistics -- EULA.

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