The Theory of Stochastic Processes III (Grundlehren Der Mathematischen Wissenschaften) （Reprint）

Kotz, S. (TRN)/ Gihman, I. I./ Skorohod, A. V.

Springer Verlag（2012/11発売）

• 製本 Paperback:紙装版/ペーパーバック版
• 言語 ENG
• 商品コード 9781461580676
• DDC分類 519

Full Description

It was originally planned that the Theory of Stochastic Processes would consist of two volumes: the first to be devoted to general problems and the second to specific cJasses of random processes. It became apparent, however, that the amount of material related to specific problems of the theory could not possibly be incJuded in one volume. This is how the present third volume came into being. This voJume contains the theory of martingales, stochastic integrals, stochastic differential equations, diffusion, and continuous Markov processes. The theory of stochastic processes is an actively developing branch of mathe- matics, and it would be an unreasonable and impossible task to attempt to encompass it in a single treatise (even a multivolume one). Therefore, the authors, guided by their own considerations concerning the relative importance of various results, naturally had to be selective in their choice of material.The authors are fully aware that such a selective process is not perfecL Even a number of topics that are, in the authors' opinion, of great importance could not be incJuded, for example, limit theorems for particular cJasses of random processes, the theory of random fields, conditional Markov processes, and information and statistics of random processes. With the publication of this last volume, we recall with gratitude oUf associates who assisted us in this endeavor, and express our sincere thanks to G. N. Sytaya, L. V. Lobanova, P. V. Boiko, N. F. Ryabova, N. A. Skorohod, V. V. Skorohod, N. I. Portenko, and L. I. Gab.

Contents

I. Martingales and Stochastic Integrals.- 1. Martingales and Their Generalizations.- 2. Stochastic Integrals.- 3. Ito's Formula.- II. Stochastic Differential Equations.- 1. General Problems of the Theory of Stochastic Differential Equations.- 2. Stochastic Differential Equations without an After-Effect.- 3. Limit Theorems for Sequences of Random Variables and Stochastic Differential Equations.- III. Stochastic Differential Equations for Continuous Processes and Continuous Markov Processes in Rm.- 1. Ito Processes.- 2. Stochastic Differential Equations for Processes of Diffusion Type.- 3. Diffusion Processes in Rm.- 4. Continuous Homogeneous Markov Processes in Rm.- Remarks.- Appendix: Corrections to Volumes I and II.