Cover
Title page
Preface
Part I Foundations
1 Why equations with Lévy noise?
1.1 Discrete-time dynamical systems
1.2 Deterministic continuous-time systems
1.3 Stochastic continuous-time systems
1.4 Courrège's theorem
1.5 Itô's approach
1.6 Infinite-dimensional case
2 Analytic preliminaries
2.1 Notation
2.2 Sobolev and Hölder spaces
2.3 L^p and C_ρ-spaces
2.4 Lipschitz functions and composition operators
2.5 Differential operators
3 Probabilistic preliminaries
3.1 Basic definitions
3.2 Kolmogorov existence theorem
3.3 Random elements in Banach spaces
3.4 Stochastic processes in Banach spaces
3.5 Gaussian measures on Hilbert spaces
3.6 Gaussian measures on topological spaces
3.7 Submartingales
3.8 Semimartingales
3.9 Burkholder-Davies-Gundy inequalities
4 Lévy processes
4.1 Basic properties
4.2 Two building blocks - Poisson and Wiener processes
4.3 Compound Poisson processes in a Hilbert space
4.4 Wiener processes in a Hilbert space
4.5 Lévy-Khinchin decomposition
4.6 Lévy-Khinchin formula
4.7 Laplace transforms of convolution semigroups
4.8 Expansion with respect to an orthonormal basis
4.9 Square integrable Lévy processes
4.10 Lévy processes on Banach spaces
5 Lévy semigroups
5.1 Basic properties
5.2 Generators
6 Poisson random measures
6.1 Introduction
6.2 Stochastic integral of deterministic fields
6.3 Application to construction of Lévy processes
6.4 Moment estimates in Banach spaces
7 Cylindrical processes and reproducing kernels
7.1 Reproducing kernel Hilbert space
7.2 Cylindrical Poisson processes
7.3 Compensated Poisson measure as a martingale
8 Stochastic integration
8.1 Operator-valued angle bracket process
8.2 Construction of the stochastic integral
8.3 Space of integrands
8.4 Local properties of stochastic integrals
8.5 Stochastic Fubini theorem
8.6 Stochastic integral with respect to a Lévy process
8.7 Integration with respect to a Poisson random measure
8.8 L^p-theory for vector-valued integrands
Part II Existence and Regularity
9 General existence and uniqueness results
9.1 Deterministic linear equations
9.2 Mild solutions
9.3 Equivalence of weak and mild solutions
9.4 Linear equations
9.5 Existence of weak solutions
9.6 Markov property
9.7 Equations with general Lévy processes
9.8 Generators and a martingale problem
10 Equations with non-Lipschitz coefficients
10.1 Dissipative mappings
10.2 Existence theorem
10.3 Reaction-diffusion equation
11 Factorization and regularity
11.1 Finite-dimensional case
11.2 Infinite-dimensiona1 case
11.3 Applications to time continuity
11.4 The case of an arbitrary martingale
12 Stochastic parabolic problems
12.1 Introduction
12.2 Space-time continuity in the Wiener case
12.3 The jump case
12.4 Stochastic heat equation
12.5 Equations with fractional Laplacian and stable noise
13 Wave and delay equations
13.1 Stochastic wave equation on [0,1]
13.2 Stochastic wave equation on R^d driven by impulsive noise
13.3 Stochastic delay equations
14 Equations driven by a spatially homogeneous noise
14.1 Tempered distributions
14.2 Lévy processes in S'(R^d)
14.3 RKHS of a square integrable Lévy process in S'(R^d)
14.4 Spatially homogeneous Lévy processes
14.5 Examples
14.6 RKHS of a homogeneous noise
14.7 Stochastic equations on R^d
14.8 Stochastic heat equation
14.9 Space-time regularity in the Wiener case
14.10 Stochastic wave equation
15 Equations with noise on the boundary
15.1 Introduction
15.2 Weak and mild solutions
15.3 Analytical preliminaries
15.4 L² case
15.5 Poisson perturbation
Part III Applications
16 Invariant measures
16.1 Basic definitions
16.2 Existence results
16.3 Invariant measures for the reaction-diffusion equation
17 Lattice systems
17.1 Introduction
17.2 Global interactions
17.3 Regular case
17.4 Non-Lipschitz case
17.5 Kolmogorov's formula
17.6 Gibbs measures
18 Stocbastic Burgers equation
18.1 Burgers system
18.2 Uniqueness and local existence of solutions
18.3 Stochastic Burgers equation with additive noise
19 Environmental pollution model
19.1 Model
20 Bond market models
20.1 Forward curves and the HJM postulate
20.2 HJM condition
20.3 HJMM equation
20.4 Linear volatility
20.5 BGM equation
20.6 Consistency problem
Appendix A Operators on Hilbert spaces
Appendix B Co-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