KOBV-Portal

UID:

almafu_9959327037102883

Umfang: 1 online resource : , illustrations

ISBN: 9781119476634 , 1119476631 , 9781119441601 , 1119441609 , 9781119476597 , 1119476593

Inhalt: This book is concerned with the theory of stochastic processes and the theoretical aspects of statistics for stochastic processes. It combines classic topics such as construction of stochastic processes, associated filtrations, processes with independent increments, Gaussian processes, martingales, Markov properties, continuity and related properties of trajectories with contemporary subjects: integration with respect to Gaussian processes, It' integration, stochastic analysis, stochastic differential equations, fractional Brownian motion and parameter estimation in diffusion models.

Anmerkung: 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. , 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.

Weitere Ausg.: Print version: Mishura, I︠U︡lii︠a︡ S. Theory and statistical applications of stochastic processes. London : ISTE ; Hoboken, NJ : Wiley, 2017 ISBN 1786300508

Weitere Ausg.: ISBN 9781786300508

Sprache: Englisch

Schlagwort(e): Electronic books. ; Electronic books. ; Electronic books.

URL: https://onlinelibrary.wiley.com/doi/book/10.1002/9781119441601

Kooperativer Bibliotheksverbund

Berlin Brandenburg