Theory of Stochastic Processes II

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 Definitions and Properties of Markov Processes.- § 1. Wide-Sense Markov Processes.- § 2. Markov Random Functions.- § 3. Markov Processes.- § 4. Strong Markov Process.- § 5. Multiplicative Functional.- § 6. Properties of Sample Functions of Markov Processes.- II. Homogeneous Markov Processes.- § 1. Basic Definitions.- § 2. The Resolvent and the Generating Operator of a Weakly Measurable Markov Process.- § 3. Stochastically Continuous Processes.- § 4. Feller Processes in Locally Compact Spaces.- § 5. Strong Markov Processes in Locally Compact Spaces.- § 6. Multiplicative Additive Functionals, Excessive Functions.- III. Jump Processes.- § 1. General Definitions and Properties of Jump Processes.- § 2. Homogeneous Markov Processes with a Countable Set of States.- § 3. Semi-Markov Processes.- § 4. Markov Processes with a Discrete Component.- IV. Processes with Independent Increments.- §1. Definitions. General Properties.- § 2. Homogeneous Processes with Independent Movements. One-Dimensional Case.- § 3. Properties of Sample Functions of Homogeneous Processes with Independent Increments in ?1.- §4. Finite-Dimensional Homogeneous Processes with Independent Increments.- V. Branching Processes.- § 1. Branching Processes with Finite Number of Particles.- § 2. Branching Processes with a Continuum of States.- §3. General Markov Processes with Branching.- Historical and Bibliographical Remarks.