## About this book

### Introduction

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.

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.

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.

### Keywords

Noncausal Stochastic Calculus random variable stochastic derivative principle of causality

### Bibliographic information

- DOI https://doi.org/10.1007/978-4-431-56576-5
- Copyright Information Springer Japan KK 2017
- Publisher Name Springer, Tokyo
- eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
- Print ISBN 978-4-431-56574-1
- Online ISBN 978-4-431-56576-5
- Buy this book on publisher's site

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