Book contents
- Frontmatter
- Contents
- Preface
- Part I Foundations
- 1 Why equations with Lévy noise?
- 2 Analytic preliminaries
- 3 Probabilistic preliminaries
- 4 Lévy processes
- 5 Lévy semigroups
- 6 Poisson random measures
- 7 Cylindrical processes and reproducing kernels
- 8 Stochastic integration
- Part II Existence and Regularity
- Part III Applications
- Appendix A Operators on Hilbert spaces
- Appendix B C0-semigroups
- Appendix C Regularization of Markov processes
- Appendix D Itô formulae
- Appendix E Lévy–Khinchin formula on [0,+∞)
- Appendix F Proof of Lemma 4.24
- List of symbols
- References
- Index
1 - Why equations with Lévy noise?
from Part I - Foundations
Published online by Cambridge University Press: 10 November 2010
- Frontmatter
- Contents
- Preface
- Part I Foundations
- 1 Why equations with Lévy noise?
- 2 Analytic preliminaries
- 3 Probabilistic preliminaries
- 4 Lévy processes
- 5 Lévy semigroups
- 6 Poisson random measures
- 7 Cylindrical processes and reproducing kernels
- 8 Stochastic integration
- Part II Existence and Regularity
- Part III Applications
- Appendix A Operators on Hilbert spaces
- Appendix B C0-semigroups
- Appendix C Regularization of Markov processes
- Appendix D Itô formulae
- Appendix E Lévy–Khinchin formula on [0,+∞)
- Appendix F Proof of Lemma 4.24
- List of symbols
- References
- Index
Summary
The book is devoted to stochastic evolution equations with Lévy noise. Such equations are important because, roughly speaking, stochastic dynamical systems, or equivalently Markov processes, can be represented as solutions to such equations. In this introductory chapter, it is shown how that is the case. To motivate better the construction of the associated stochastic equations, the chapter starts with discrete-time systems.
Discrete-time dynamical systems
A deterministic discrete-time dynamical system consists of a set E, usually equipped with a σ-field ε of subsets of itself, and a mapping F, usually measurable, acting from E into E. If the position of the system at time t = 0, 1, …, is denoted by X(t) then by definition X(t + 1) = F(X(t)), t = 0, 1, … The sequences (X(t), t = 0, 1, …) are the so-called trajectories or paths of the dynamical system, and their asymptotic properties are of prime interest in the theory. The set E is called the state space and the transformation F determines the dynamics of the system.
If the present state x determines only the probability P(x, Γ) that at the next moment the system will be in the set Γ then one says that the system is stochastic. Thus a stochastic dynamical system consists of the state space E, a σ-field ε and a function P = P(x, Γ), x ∈ E, Γ ∈ ε, such that, for each Γ ∈ ε, P(·, Γ) is a measurable function and, for each x ∈ E, P(x, ·) is a probability measure. We call P the transition function or transition probability.
- Type
- Chapter
- Information
- Stochastic Partial Differential Equations with Lévy NoiseAn Evolution Equation Approach, pp. 3 - 12Publisher: Cambridge University PressPrint publication year: 2007