Noncausal Stochastic Calculus

Noncausal Stochastic Calculus

by Shigeyoshi Ogawa

Hardcover(1st ed. 2017)

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This book presents an elementary introduction to the theory of noncausal stochastic calculus that arises as a natural alternative to the standard theory of stochastic calculus founded in 1944 by Professor Kiyoshi Itô. As is generally known, Itô Calculus is essentially based on the "hypothesis of causality", asking random functions to be adapted to a natural filtration generated by Brownian motion or more generally by square integrable martingale.

The intention in this book is to establish a stochastic calculus that is free from this "hypothesis of causality". To be more precise, a noncausal theory of stochastic calculus is developed in this book, based on the noncausal integral introduced by the author in 1979.

After studying basic properties of the noncausal stochastic integral, various concrete problems of noncausal nature are considered, mostly concerning stochastic functional equations such as SDE, SIE, SPDE, and others, to show not only the necessity of such theory of noncausal stochastic calculus but also its growing possibility as a tool for modeling and analysis in every domain of mathematical sciences. The reader may find there many open problems as well.

Product Details

ISBN-13: 9784431565741
Publisher: Springer Japan
Publication date: 07/26/2017
Edition description: 1st ed. 2017
Pages: 210
Product dimensions: 6.10(w) x 9.25(h) x (d)

Table of Contents

1 Introduction – Why the Causality?.- 2 Preliminary – Causal calculus.- 3 Noncausal Calculus.- 4 Noncausal Integral and Wiener Chaos.- 5 Noncausal SDEs.- 6 Brownian Particle Equation.- 7 Noncausal SIE.- 8 Stochastic Fourier Transformation.- 9 Appendices to Chapter 2.- 10 Appendices 2 – Comments and Proofs.- Index.

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