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 derive equivalent conditions for the (local) absolute continuity of two laws of semimartingales on random sets. Our result generalizes previous results for classical semimartingales by replacing a strong uniqueness assumption by a weaker uniqueness assumption. The main tool is a generalized Girsanov’s theorem, which relates laws of two possibly explosive semimartingales to a candidate density process. Its proof is based on an extension theorem for consistent families of probability measures. Moreover, we show that in a one-dimensional Ito-diffusion setting our result reproduces the known deterministic characterizations for (local) absolute continuity. Finally, we give a Khasminskii-type test for the absolute continuity of multidimensional Ito-diffusions and derive linear growth conditions for the martingale property of stochastic exponentials.

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Absolute continuity of semimartingales

We derive deterministic criteria for the existence and nonexistence of equivalent (local) martingale measures for financial markets driven by multi-dimensional time-inhomogeneous diffusions. Our conditions can be used to construct financial markets in which the no unbounded profit with bounded risk condition holds, while the classical no free lunch with vanishing risk condition fails.

Deterministic criteria for the absence and existence of arbitrage in multi-dimensional diffusion markets

We prove limit theorems for cylindrical martingale problems associated with Levy generators. Furthermore, we give sufficient and necessary conditions for the Feller property of well-posed problems with continuous coefficients. We discuss two applications. First, we derive continuity and linear growth conditions for the existence of weak solutions to infinite-dimensional stochastic differential equations driven by Levy noise. Second, we derive continuity, local boundedness and linear growth conditions for limit theorems and the Feller property of weak solutions to stochastic partial differential equations driven by Wiener noise.

Limit Theorems for Cylindrical Martingale Problems Associated with Lévy Generators

We derive integral tests for the existence and absence of arbitrage in a financial market with one risky asset which is either modeled as stochastic exponential of an Ito process or a positive diffusion with Markov switching. In particular, we derive conditions for the existence of the minimal martingale measure. We also show that for Markov switching models the minimal martingale measure preserves the independence of the noise and we study how the minimal martingale measure can be modified to change the structure of the switching mechanism. Our main mathematical tools are new criteria for the martingale and strict local martingale property of certain stochastic exponentials.

No Arbitrage in Continuous Financial Markets

We study monotone and convex stochastic orders for processes with independent increments. Our contributions are twofold: First, we relate stochastic orders of the Poisson component to orders of their (generalized) Levy measures. The relation is proven using an interpolation formula for infinitely divisible laws. Second, we derive explicit conditions on the characteristics of the processes. In this case, we prove the conditions via constructions of couplings.

David Criens

Papers

Absolute continuity of semimartingales

Deterministic criteria for the absence and existence of arbitrage in multi-dimensional diffusion markets

Limit Theorems for Cylindrical Martingale Problems Associated with Lévy Generators

No Arbitrage in Continuous Financial Markets

Monotone and Convex Stochastic Orders for Processes with Independent Increments