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

We consider difference equations in balanced, i.i.d. environments which are not necessary elliptic. In this setting we prove a parabolic Harnack inequality (PHI) for non-negative solutions to the discrete heat equation satisfying a (rather mild) growth condition, and we identify the optimal Harnack constant for the PHI. We show by way of an example that a growth condition is necessary and that our growth condition is sharp. Along the way we also prove a parabolic oscillation inequality and a (weak) quantitative homogenization result, which we believe to be of independent interest. 

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A Parabolic Harnack Principle for Balanced Difference Equations in Random Environments

As a consequence of the financial crises, risk management became more important and real-world dynamics of interest-rate models moved into the focus of interest. Since risk-neutral dynamics are classically important to compute prices of financial derivatives, it is interesting when real-world dynamics can be related to risk-neutral dynamics via an equivalent change of measures. In this article we give deterministic conditions in a general Heath-Jarrow-Morton framework driven by a Hilbert space valued Brownian motion and a Poisson random measure. Our conditions are of Lipschitz type and therefore easy to verify.

A Note on Real-World and Risk-Neutral Dynamics for Heath-Jarrow-Morton Frameworks

. In this paper we study a family of nonlinear (conditional) expectations that can be understood as a stochastic process with uncertain parameters. We develop a general framework which can be seen as a version of the martingale problem method of Stroock and Varadhan with parameter uncertainty. In particular, we show that our methodology includes the important class of nonlinear L´evy processes as introduced by Neufeld and Nutz, and we introduce the new classes of stochastic partial diﬀerential equations and spin systems with parameter uncertainty. Moreover, we study properties of the nonlinear expectations. We prove the dynamic programming principle, i.e., the tower property, and we establish conditions for the (strong) USC b –Feller property and a strong Markov selection principle. Finally, we investigate ergodicity for spin systems with uncertain parameters such as the nonlinear contact process.

Stochastic Processes under Parameter Uncertainty

We use the abstract method of (local) martingale problems in order to give criteria for convergence of stochastic processes. Extending previous notions, the formulation we use is neither restricted to Markov processes (or semimartingales), nor to continuous or cadlag paths. We illustrate our findings both, by finding generalizations of known results, and proving new results. For the latter, we work on processes with fixed times of discontinuity.

The Martingale Problem Method Revisited

Abstract It is well known that the law of a one-dimensional diffusion on natural scale is fully characterized by its speed measure. Stone proved a continuous dependence of such diffusions on their speed measures. In this paper we establish the converse direction, i.e., we prove a continuous dependence of the speed measures on their diffusions. Furthermore, we take a topological point of view on the relation. More precisely, for suitable topologies, we establish a homeomorphic relation between the set of regular diffusions on natural scale without absorbing boundaries and the set of locally finite speed measures. 

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David Criens

Papers

A Parabolic Harnack Principle for Balanced Difference Equations in Random Environments

A Note on Real-World and Risk-Neutral Dynamics for Heath-Jarrow-Morton Frameworks

Stochastic Processes under Parameter Uncertainty

The Martingale Problem Method Revisited

On the Relation of One-Dimensional Diffusions on Natural Scale and Their Speed Measures