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Chapter 1. Introduction

The modeling of systems by differential equations usually requires that the parameters involved be completely known. Such models often originate from problems in physics or economics where we have insufficient information on parameter values. For example, the values can vary in time or space due to unknown conditions of the surroundings or of the medium. In some cases the parameter values may depend in a complicated way on the microscopic properties of the medium. In addition, the parameter values may fluctuate due to some external or internal “noise”, which is random – or at least appears so to us.

Helge Holden, Bernt Øksendal, Jan Ubøe, Tusheng Zhang

Chapter 2. Framework

In this chapter we develop the general framework to be used in this book. The starting point for the discussion will be the standard white noise structures and how constructions of this kind can be given a rigorous treatment. White noise analysis can be addressed in several different ways. The presentation here is to a large extent influenced by ideas and methods used by the authors. In particular, we emphasize the use of multidimensional structures, i.e., the white noise we are about to consider will in general take on values in a multidimensional space and will also be indexed by amultidimensional parameter set.

Helge Holden, Bernt Øksendal, Jan Ubøe, Tusheng Zhang

Chapter 3. Applications to Stochastic Ordinary Differential Equations

As mentioned in the introduction, the framework that we developed in Chapter 2 for the main purpose of solving stochastic partial differential equations, can also be used to obtain new results – as well as new proofs of old results – for stochastic ordinary differential equations. In this chapter we will illustrate this by discussing some important examples.

Helge Holden, Bernt Øksendal, Jan Ubøe, Tusheng Zhang

Chapter 4. Stochastic Partial Differential Equations Driven by Brownian White Noise

In this chapter we will apply the general theory developed in Chapter 2 to solve various stochastic partial differential equations (SPDEs) driven by Brownian white noise. In fact, as pointed out in Chapter 1, our main motivation for setting up this machinery was to enable us to solve some of the basic SPDEs that appear frequently in applications.

Helge Holden, Bernt Øksendal, Jan Ubøe, Tusheng Zhang

Chapter 5. Stochastic Partial Differential Equations Driven by Lévy Processes

In the last decades there has been an increased interest in stochastic models based on other processes than the Brownian motion B(



Helge Holden, Bernt Øksendal, Jan Ubøe, Tusheng Zhang


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