Initially the theory of convergence in law of stochastic processes was developed quite independently from the theory of martingales, semimartingales and stochastic integrals. Apart from a few exceptions essentially concerning diffusion processes, it is only recently that the relation between the two theories has been thoroughly studied. The authors of this Grundlehren volume, two of the international leaders in the field, propose a systematic exposition of convergence in law for stochastic processes, from the point of view of semimartingale theory, with emphasis on results that are useful for mathematical theory and mathematical statistics. This leads them to develop in detail some particularly useful parts of the general theory of stochastic processes, such as martingale problems, and absolute continuity or contiguity results. The book contains an elementary introduction to the main topics: theory of martingales and stochastic integrales, Skorokhod topology, etc., as well as a large number of results which have never appeared in book form, and some entirely new results. It should be useful to the professional probabilist or mathematical statistician, and of interest also to graduate students
CONTENT
I. The General Theory of Stochastic Processes, Semimartingales and Stochastic Integrals -- II. Characteristics of Semimartingales and Processes with Independent Increments -- III. Martingale Problems and Changes of Measures -- IV. Hellinger Processes, Absolute Continuity and Singularity of Measures -- V. Contiguity, Entire Separation, Convergence in Variation -- VI. Skorokhod Topology and Convergence of Processes -- VII. Convergence of Processes with Independent Increments -- VIII. Convergence to a Process with Independent Increments -- IX. Convergence to a Semimartingale -- X Limit Theorems, Density Processes and Contiguity -- Bibliographical Comments -- References -- Index of Symbols -- Index of Terminology -- Index of Topics -- Index of Conditions for Limit Theorems
