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About this book

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.

Table of Contents

Frontmatter

Chapter I. The General Theory of Stochastic Processes, Semimartingales and Stochastic Integrals

Abstract
The “General Theory of Stochastic Processes”, in spite of its name, encompasses the rather restrictive subject of stochastic processes indexed by ℝ+. But, within this framework, it expounds deep properties related to the order structure of ℝ+, and martingales play a central rôle.
Jean Jacod, Albert N. Shiryaev

Chapter II. Characteristics of Semimartingales and Processes with Independent Increments

Abstract
We continue across our project of expounding the general theory of processes. However, here we touch upon a slightly different aspect of the theory, which at the same time is much less widely known than what was in the first chapter. This is also the aspect which will be most directly useful for limit theorems.
Jean Jacod, Albert N. Shiryaev

Chapter III. Martingale Problems and Changes of Measures

Abstract
In limit theorems, one needs to characterize the law (or distribution) of various processes, in particular of the limiting process. As is well known, the law of a process is indeed characterized by the family of its “finite-dimensional” distributions. However, one is very rarely able to explicitely compute these finite-dimensional distributions, except for PII. On the other hand, many usual processes are semimartingales; and a natural tool has emerged in Chapter II for studying them, namely their characteristics: at least, they are often easy to compute.
Jean Jacod, Albert N. Shiryaev

Chapter IV. Hellinger Processes, Absolute Continuity and Singularity of Measures

Abstract
The question of absolute continuity or singularity (ACS) of two probability measures has been investigated a long time ago, both for its theoretical interest and for its applications to mathematical statistics. S. Kakutani in 1948 [125] was the first to solve the ACS problem in the case of two measures P and P′ having a (possibly infinite) product form: P = µ 1µ 2 ⊗ ... and P′ = µ1µ2 ⊗ ..., when µ n ~ µ n (µ n and µ n are equivalent) for all n; he proved a remarquable result, known as the “Kakutani alternative”, which says that either P ~ P, or PP′ (P and P′ are mutually singular). Ten years later, Hajek [80] and Feldman [53] proved a similar alternative for Gaussian measures, and several authors gave effective criteria in terms of the covariance functions or spectral quantities, for the laws of two Gaussian processes.
Jean Jacod, Albert N. Shiryaev

Chapter V. Contiguity, Entire Separation, Convergence in Variation

Abstract
We examine here two apparently disconnected sorts of problems. The relation between them essentially comes from the fact that, in order to solve both of them, we use the same tool, namely the Hellinger processes introduced in the previous chapter.
Jean Jacod, Albert N. Shiryaev

Chapter VI. Skorokhod Topology and Convergence of Processes

Abstract
In this chapter, we lay down the last cornerstone that is needed to derive functional limit theorems for processes. Namely, we consider the space D (ℝ d ) of all càdlàg functions: ℝ+→ ℝ d we need to provide this space with a topology, such that: (1) the space is Polish (so we can apply classical limsit theorems on Polish spaces); (2) the Borel σ-field is exactly the σ-field generated by all evaluation maps (because the “law” of a process is precisely a measure on this σ-field).
Jean Jacod, Albert N. Shiryaev

Chapter VII. Convergence of Processes with Independent Increments

Abstract
With this chapter, at least, we enter the subject which has given its name to the whole book. Our final aim is to prove convergence theorems for a sequence of semimartingales toward a semimartingale. We present the material through three successive steps, corresponding to Chapters VII, VIII and IX: firstly, the prelimiting processes, as well of course as the limiting process, have independent increments; secondly, only the limiting process has independent increments; thirdly, the limiting process itself belongs to some rather broad class of semi-martingales.
Jean Jacod, Albert N. Shiryaev

Chapter VIII. Convergence to a Process with Independent Increments

Abstract
This chapter constitutes the second step on our way to general limit theorems. We consider a sequence (X n ) of semimartingales, with characteristics (B n , C n , v n ), and a limiting process X which is a PII with characteristics (B, C, v). Our main objective is to prove that the various conditions of Chapter VII still insure the (functional or finite-dimensional) convergence of (X n ) to X, although the X n ’s are no longer PII.
Jean Jacod, Albert N. Shiryaev

Chapter IX. Convergence to a Semimartingale

Abstract
Here comes the third—and last—step in our exposition of limit theorems. Not only are the pre-limiting processes X n arbitrary semimartingales, but the limit process X also is a semimartingale; not quite an arbitrary one, though: since the method is based here on convergence of martingales and on the relations between X and its characteristics, we need these characteristics to indeed characterize the distribution. ℒ(X) of X So, in most of the chapter, we will assume that ℒ(X) is the unique solution to the martingale problem associated with the characteristics of X, as introduced in Chapter III.
Jean Jacod, Albert N. Shiryaev

Chapter X. Limit Theorems, Density Processes and Contiguity

Abstract
Let us roughly describe the problems which will retain our attention in this last chapter.
Jean Jacod, Albert N. Shiryaev

Backmatter

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