Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Stochastic equations in infinite dimensions

Stochastic differential equations and diffusion processes

Convergence Of Probability Measures

Contents: Transition Functions and Resolvents.- Existence and Uniqueness of Q-Functions.- Examples of Continuous Time Markov Chains.- More on the Uniqueness Problem.- Classification of States and Invariant Measures.- Strong and Exponential Ergodicity.- Reversibility, Monotonictity, and Other Properties.- Birth and Death Processes.- Population Processes.- Bibliography.- Symbol Index.- Author Index.- Subject Index.

Book Reviews - Continuous Time Markov Chains: An Applications-Oriented Approach

A note on the monotone stochastic order for processes with independent increments

We derive limit theorems for cylindrical martingale problems associated to L\'evy generators. Furthermore, we give sufficient and necessary conditions for the Feller property of well-posed problems with continuous coefficients and limit theorems for solution measures to stochastic (partial) differential equations.

Limit Theorems for Cylindrical Martingale Problems associated with L\'evy Generators

In this article we relate the set of structure preserving equivalent martingale measures $(\mathcal{M})$ for financial models driven by semimartingales with conditionally independent increments to a set of measurable and integrable functions $(\mathscr{Y})$. More precisely, we prove that $(\mathcal{M}\not = \emptyset)$ if, and only if, $(\mathscr{Y}\not = \emptyset)$, and connect the sets $(\mathcal{M})$ and $(\mathscr{Y})$ to the semimartingale characteristics of the driving process. As examples we consider integrated Levy models with independent stochastic factors and time-changed Levy models and derive mild conditions for $(\mathcal{M} \not = \emptyset)$.

Structure Preserving Equivalent Martingale Measures for $\mathscr{H}$-SII Models

We prove a dual Yamada--Watanabe theorem for one-dimensional stochastic differential equations driven by quasi-left continuous semimartingales with independent increments. In particular, our result covers stochastic differential equations driven by (time-inhomogeneous) Levy processes.

A Dual Yamada--Watanabe Theorem for L\'evy driven stochastic differential equations

Consider two laws $P$ and $Q$ of multidimensional possibly explosive diffusions with common diffusion coefficient $\mathfrak {a}$ and drift coefficients $\mathfrak {b}$ and $\mathfrak {b}+ \mathfrak {a}\mathfrak {c}$, respectively, and the law $P^{\circ }$ of an auxiliary diffusion with diffusion coefficient $\langle \mathfrak {c}, \mathfrak {a}\mathfrak {c}\rangle ^{-1}\mathfrak {a}$ and drift coefficient $\langle \mathfrak {c}, \mathfrak {a}\mathfrak {c}\rangle ^{-1}\mathfrak {b}$. We show that $P \ll Q$ if and only if the auxiliary diffusion $P^{\circ }$ explodes almost surely and that $P\perp Q$ if and only if the auxiliary diffusion $P^{\circ }$ almost surely does not explode. As applications we derive a Khasminskii-type integral test for absolute continuity and singularity, an integral test for explosion of time-changed Brownian motion, and we discuss applications to mathematical finance.

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David Criens

Papers

A note on the monotone stochastic order for processes with independent increments

Limit Theorems for Cylindrical Martingale Problems associated with L\'evy Generators

Structure Preserving Equivalent Martingale Measures for $\mathscr{H}$-SII Models

A Dual Yamada--Watanabe Theorem for L\'evy driven stochastic differential equations

On absolute continuity and singularity of multidimensional diffusions