Theory and Statistical Applications of Stochastic Processes

Yuliya Mishura, National University of Kyiv, Ukraine  Georgiy Shevchenko, National University of Kyiv, Ukraine

• Preface xiIntroduction xiiiPart 1 Theory of Stochastic Processes 1Chapter 1 Stochastic Processes General Properties. Trajectories, Finite-dimensional Distributions 31.1 Definition of a stochastic process 31.2 Trajectories of a stochastic process Some examples of stochastic processes 51.2.1 Definition of trajectory and some examples 51.2.2 Trajectory of a stochastic process as a random element.81.3 Finite-dimensional distributions of stochastic processes: consistency conditions.101.3.1 Definition and properties of finite-dimensional distributions 101.3.2 Consistency conditions.111.3.3 Cylinder sets and generated σ-algebra 131.3.4 Kolmogorov theorem on the construction of a stochastic process by the family of probability distributions 151.4 Properties of σ-algebra generated by cylinder sets. The notion of σ-algebra generated by a stochastic process 19Chapter 2 Stochastic Processes with Independent Increments 212.1 Existence of processes with independent increments in terms of incremental characteristic functions 212.2 Wiener process 242.2.1 One-dimensional Wiener process 242.2.2 Independent stochastic processes Multidimensional Wiener process 242.3 Poisson process 272.3.1 Poisson process defined via the existence theorem 272.3.2 Poisson process defined via the distributions of the increments 282.3.3 Poisson process as a renewal process 302.4 Compound Poisson process 332.5 Lévy processes 342.5.1 Wiener process with a drift 362.5.2 Compound Poisson process as a Lévy process 362.5.3 Sum of a Wiener process with a drift and a Poisson process 362.5.4 Gamma process 372.5.5 Stable Lévy motion372.5.6 Stable Lévy subordinator with stability parameter α ∈ (0, 1) 38Chapter 3 Gaussian Processes Integration with Respect to Gaussian Processes 393.1 Gaussian vectors 393.2 Theorem of Gaussian representation (theorem on normal correlation) 423.3 Gaussian processes. 443.4 Examples of Gaussian processes 463.4.1 Wiener process as an example of a Gaussian process 463.4.2 Fractional Brownian motion.483.4.3 Sub-fractional and bi-fractional Brownian motion 503.4.4 Brownian bridge 503.4.5 Ornstein–Uhlenbeck process 513.5 Integration of non-random functions with respect to Gaussian processes 523.5.1 General approach 523.5.2 Integration of non-random functions with respect to the Wiener process 543.5.3 Integration w.r.t the fractional Brownian motion 573.6 Two-sided Wiener process and fractional Brownian motion: Mandelbrot–van Ness representation of fractional Brownian motion 603.7 Representation of fractional Brownian motion as the Wiener integral on the compact integral 63Chapter 4 Construction, Properties and Some Functionals of the Wiener Process and Fractional Brownian Motion 674.1 Construction of a Wiener process on the interval [0, 1] 674.2 Construction of a Wiener process on R+ 724.3 Nowhere differentiability of the trajectories of a Wiener process 744.4 Power variation of the Wiener process and of the fractional Brownian motion774.4.1 Ergodic theorem for power variations 774.5 Self-similar stochastic processes 794.5.1 Definition of self-similarity and some examples 794.5.2 Power variations of self-similar processes on finite intervals.80Chapter 5 Martingales and Related Processes 855.1 Notion of stochastic basis with filtration 855.2 Notion of (sub-, super-) martingale: elementary properties 865.3 Examples of (sub-, super-) martingales 875.4 Markov moments and stopping times 905.5 Martingales and related processes with discrete time 965.5.1 Upcrossings of the interval and existence of the limit of submartingale 965.5.2 Examples of martingales having a limit and of uniformly and non-uniformly integrable martingales 1025.5.3 Lévy convergence theorem 1045.5.4 Optional stopping 1055.5.5 Maximal inequalities for (sub-, super-) martingales 1085.5.6 Doob decomposition for the integrable processes with discrete time 1115.5.7 Quadratic variation and quadratic characteristics: Burkholder–Davis–Gundy inequalities 1135.5.8 Change of probability measure and Girsanov theorem for discrete-time processes 1165.5.9 Strong law of large numbers for martingales with discrete time 1205.6 Lévy martingale stopped 1265.7 Martingales with continuous time 127Chapter 6 Regularity of Trajectories of Stochastic Processes 1316.1 Continuity in probability and in L2(Ω,F, P) 1316.2 Modification of stochastic processes: stochastically equivalent and indistinguishable processes 1336.3 Separable stochastic processes: existence of separable modification 1356.4 Conditions of D-regularity and absence of the discontinuities of the second kind for stochastic processes 1386.4.1 Skorokhod conditions of D-regularity in terms of three-dimensional distributions 1386.4.2 Conditions of absence of the discontinuities of the second kind formulated in terms of conditional probabilities of large increments 1446.5 Conditions of continuity of trajectories of stochastic processes 1486.5.1 Kolmogorov conditions of continuity in terms of two-dimensional distributions 1486.5.2 Hölder continuity of stochastic processes: a sufficient condition 1526.5.3 Conditions of continuity in terms of conditional probabilities 154Chapter 7 Markov and Diffusion Processes 1577.1 Markov property 1577.2 Examples of Markov processes 1637.2.1 Discrete-time Markov chain 1637.2.2 Continuous-time Markov chain 1657.2.3 Process with independent increments 1687.3 Semigroup resolvent operator and generator related to the homogeneous Markov process 1687.3.1 Semigroup related to Markov process 1687.3.2 Resolvent operator and resolvent equation 1697.3.3 Generator of a semigroup.1717.4 Definition and basic properties of diffusion process 1757.5 Homogeneous diffusion process Wiener process as a diffusion process 1797.6 Kolmogorov equations for diffusions 181Chapter 8 Stochastic Integration 1878.1 Motivation..1878.2 Definition of Itô integral 1898.2.1 Itô integral of Wiener process 1958.3 Continuity of Itô integral 1978.4 Extended Itô integral 1998.5 Itô processes and Itô formula 2038.6 Multivariate stochastic calculus 2128.7 Maximal inequalities for Itô martingales 2158.7.1 Strong law of large numbers for Itô local martingales 2188.8 Lévy martingale characterization of Wiener process 2208.9 Girsanov theorem 2238.10 Itô representation 228Chapter 9 Stochastic Differential Equations.2339.1 Definition, solvability conditions, examples 2339.1.1 Existence and uniqueness of solution 2349.1.2 Some special stochastic differential equations 2389.2 Properties of solutions to stochastic differential equations 2419.3 Continuous dependence of solutions on coefficients 2459.4 Weak solutions to stochastic differential equations. 2479.5 Solutions to SDEs as diffusion processe 2499.6 Viability, comparison and positivity of solutions to stochastic differential equations 2529.6.1 Comparison theorem for one-dimensional projections of stochastic differential equations 2579.6.2 Non-negativity of solutions to stochastic differential equations 2589.7 Feynman–Kac formula 2589.8 Diffusion model of financial markets 2609.8.1 Admissible portfolios, arbitrage and equivalent martingale measure 2639.8.2 Contingent claims, pricing and hedging 266Part 2 Statistics of Stochastic Processes 271Chapter 10 Parameter Estimation 27310.1 Drift and diffusion parameter estimation in the linear regression model with discrete time 27310.1.1 Drift estimation in the linear regression model with discrete time in the case when the initial value is known 27410.1.2 Drift estimation in the case when the initial value is unknown 27710.2 Estimation of the diffusion coefficient in a linear regression model with discrete time 27710.3 Drift and diffusion parameter estimation in the linear model with continuous time and the Wiener noise 27810.3.1 Drift parameter estimation 27910.3.2 Diffusion parameter estimation 28010.4 Parameter estimation in linear models with fractional Brownian motion 28110.4.1 Estimation of Hurst index 28110.4.2 Estimation of the diffusion parameter 28310.5 Drift parameter estimation 28410.6 Drift parameter estimation in the simplest autoregressive model 28510.7 Drift parameters estimation in the homogeneous diffusion model 289Chapter 11 Filtering Problem Kalman-Bucy Filter 29311.1 General setting 29311.2 Auxiliary properties of the non-observable process 29411.3 What is an optimal filter 29511.4 Representation of an optimal filter via an integral equation with respect to an observable process 29611.5 Integral Wiener-Hopf equation 299Appendices 311Appendix 1 313Appendix 2 329Bibliography 363Index 369