Space-time stochastic calculus and white noise

TL;DR

This paper reviews the historical development of stochastic analysis, introduces modeling of systems with SPDEs driven by space-time white noise, and demonstrates the use of white noise calculus to solve such equations explicitly.

Contribution

It provides a comprehensive overview of space-time white noise and Hida-Malliavin calculus, and applies these tools to solve a population growth SPDE explicitly.

Findings

Explicit solution for a population growth SPDE using white noise calculus

Survey of time-space white noise and Hida-Malliavin calculus tools

Historical context of stochastic analysis development

Abstract

In the first part of this paper I give the historical background to my initial interest in stochastic analysis and to the writing of my book Stochastic Differential Equations. The first edition of this book was published by Springer in 1985, with the highly appreciated support of Catriona Byrne. In the second part I present a motivation for modelling the dynamics of a system subject to a noise by means of a stochastic partial differential equation (SPDE) driven by a time-space Brownian sheet. This is followed by a brief survey of time-space white noise and Hida-Malliavin calculus, which are useful tools for studying such equations. As an illustration I apply white noise calculus to find an explicit solution of an SPDE describing population growth in an environment subject to time-space white noise.