Theory of Stochastic Processes I

Biography of I.I. Gikhman Iosif Ilyich Gikhman was born on the 26th of May 1918 in the city of Uman, Ukraine. He studied in Kiev, graduating in 1939, then remained there to teach and do research under the supervision of N. Bogolyubov, defending a "candidate" thesis on the influence of random processes on dynamical systems in 1942 and a doctoral dissertation on Markov processes and mathematical statistics in 1955. I.I. Gikhman is one of the founders of the theory of stochastic differential equations and also contributed significantly to mathematical statistics, limit theorems, multidimensional martingales, and stochastic control. He died in 1985, in Donetsk. Biography of A.V. Skorokhod Anatoli Vladimirovich Skorokhod was born on September 10th, 1930 in the city Nikopol, Ukraine. He graduated from Kiev University in 1953, after which his graduate studies at Moscow University, were directed by E.B. Dynkin. From 1956 to 1964 Anatoli Skorokhod was a professor of Kiev university. Threafter he worked at the Institute of Mathematics of the Ukrainian Academy of Science, but he has also, since 1993, been professor of Statistics and Probability at Michigan State University. Skorokhod was elected to the Ukrainian Academy of Sciences in 1985 and became a Fellow of American Academy of Arts and Sciences in 2000. His mathematical research interests are the theory of stochastic processes, stochastic differential equations, Markov processes, randomly perturbed dynamical systems.

• I. Basic Notions of Probability Theory.- § 1. Axioms and Definitions.- § 2. Independence.- § 3. Conditional Probabilities and Conditional Expectations.- § 4. Random Functions and Random Mappings.- II. Random Sequences.- § 1. Preliminary Remarks.- § 2. Semi-Martingales and Martingales.- § 3. Series.- § 4. Markov Chains.- § 5. Markov Chains with a Countable Number of States.- § 6. Random Walks on a Lattice.- § 7. Local Limit Theorems for Lattice Walks.- § 8. Ergodic Theorems.- III. Random Functions.- § 1. Some Classes of Random Functions.- § 2. Separable Random Functions.- § 3. Measurable Random Functions.- § 4. A Criterion for the Absence of Discontinuities of the Second Kind.- § 5. Continuous Processes.- IV. Linear Theory of Random Processes.- § 1. Correlation Functions.- § 2. Spectral Representations of Correlation Functions.- § 3. A Basic Analysis of Hilbert Random Functions.- § 4. Stochastic Measures and Integrals.- § 5. Integral Representation of Random Functions.- § 6. Linear Transformations.- § 7. Physically Realizable Filters.- § 8. Forecasting and Filtering of Stationary Processes.- § 9. General Theorems on Forecasting Stationary Processes.- V. Probability Measures on Functional Spaces.- § 1. Measures Associated with Random Processes.- § 2. Measures in Metric Spaces.- § 3. Measures on Linear Spaces. Characteristic Functionals.- § 4. Measures in ?p Spaces.- § 5. Measures in Hilbert Spaces.- § 6. Gaussian Measures in a Hilbert Space.- VI. Limit Theorems for Random Processes.- § 1. Weak Convergences of Measures in Metric Spaces.- § 2. Conditions for Weak Convergence of Measures in Hilbert Spaces.- § 3. Sums of Independent Random Variables with Values in a Hilbert Space.- § 4. Limit Theorems for Continuous Random Processes.- §5. Limit Theorems for Processes without Discontinuities of the Second Kind.- VII. Absolute Continuity of Measures Associated with Random Processes.- § 1. General Theorems on Absolute Continuity.- § 2. Admissible Shifts in Hilbert Spaces.- § 3. Absolute Continuity of Measures under Mappings of Spaces.- § 4. Absolute Continuity of Gaussian Measures in a Hilbert Space.- § 5. Equivalence and Orthogonality of Measures Associated with Stationary Gaussian Processes.- § 6. General Properties of Densities of Measures Associated with Markov Processes.- VIII. Measurable Functions on Hilbert Spaces.- § 1. Measurable Linear Functionals and Operators on Hilbert Spaces.- § 2. Measurable Polynomial Functions. Orthogonal Polynomials.- § 3. Measurable Mappings.- § 4. Calculation of Certain Characteristics of Transformed Measures.- Historical and Bibliographical Remarks.- Corrections.