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

In this article we derive Lipschitz- and linear growth conditions for the martingale property of a stochastic exponential driven by an infinite-dimensional stochastic differential equation. At the heart of the proof lies a local change of measure and a Yamada-Watanabe-type argument. We illustrate two applications. First, we obtain sufficient conditions for the equivalence of laws of solutions to infinite-dimensional stochastic differential equations. Second, we give sufficient conditions for the existence of a candidate equivalent local martingale measure in an infinite-dimensional Heath-Jarrow-Morton framework.

The Martingale Property in Terms of Infinite-Dimensional SDEs

. In this paper we study a family of non-linear (conditional) expectations that can be understood as a continuous semimartingale with uncertain local characteristics. Here, the diﬀerential characteristics are prescribed by a set-valued function that depends on time and path in a non-Markovian way. We provide a dynamic programming principle for the non-linear expectation and we link the corresponding value function to a variational form of a non-linear path-dependent partial diﬀerential equation. Furthermore, we prove that the non-linear expectation solves a non-linear martingale problem, which conﬁrms our interpretation as a non-linear semimartingale.

Non-linear continuous semimartingales

We give a collection of explicit sufficient conditions for the true martingale property of a wide class of exponentials of semimartingales. We express the conditions in terms of semimartingale characteristics. This turns out to be very convenient in financial modelling in general. Especially it allows us to carefully discuss the question of well-definedness of semimartingale Libor models, whose construction crucially relies on a sequence of measure changes.

Martingale property of exponential semimartingales: a note on explicit conditions and applications to asset price and Libor models

The separating time for two probability measures on a filtered space is an extended stopping time which captures the phase transition between equivalence and singularity. More specifically, two probability measures are equivalent before their separating time and singular afterwards. In this paper, we investigate the separating time for two laws of general one-dimensional regular continuous strong Markov processes, so-called general diffusions, which are parameterized via scale functions and speed measures. Our main result is a representation of the corresponding separating time as (loosely speaking) a hitting time of a deterministic set which is characterized via speed and scale. As hitting times are fairly easy to understand, our result gives access to explicit and easy-to-check sufficient and necessary conditions for two laws of general diffusions to be (locally) absolutely continuous and/or singular. Most of the related literature treats the case of stochastic differential equations. In our setting we encounter several novel features, which are due to general speed and scale on the one hand, and to the fact that we do not exclude (instantaneous or sticky) reflection on the other hand. These new features are discussed in a variety of examples. As an application of our main theorem, we investigate the no arbitrage concept no free lunch with vanishing risk (NFLVR) for a single asset financial market whose (discounted) asset is modeled as a general diffusion which is bounded from below (e.g., non-negative). More precisely, we derive deterministic criteria for NFLVR and we identify the (unique) equivalent local martingale measure as the law of a certain general diffusion on natural scale.

Separating Times for One-Dimensional Diffusions

. In this note we consider a family of nonlinear (conditional) expectations that can be un-derstood as a multidimensional diﬀusion with uncertain drift and certain volatility. Here, the drift is prescribed by a set-valued function that depends on time and path in a Markovian way. We establish the Feller property for the associated sublinear Markovian semigroup and we observe a smoothing eﬀect as our framework carries enough randomness. Furthermore, we link the corresponding value function to a semilinear Kolmogorov equation.

David Criens

Papers

The Martingale Property in Terms of Infinite-Dimensional SDEs

Non-linear continuous semimartingales

Martingale property of exponential semimartingales: a note on explicit conditions and applications to asset price and Libor models

Separating Times for One-Dimensional Diffusions

A class of multidimensional nonlinear diffusions with the Feller property