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 differential equations and diffusion processes

(1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

https://link.springer.com/content/pdf/10.1057%2Fjors.1987.184.pdf

Applied Probability and Queues

A Primer on Feller Semigroups and Feller Processes- Feller Generators and Symbols- Construction of Feller Processes- Transformations of Feller Processes- Sample Path Properties- Global Properties- Approximation- Open Problems- References- Index

Lévy-type processes : construction, approximation and sample path properties

Deep inside an old town clock, trouble is brewing. A little girl is the witness to a long-simmering spousal feud between two wooden figurines on the town clock. When the old woman realizes she is going to be replaced by a young blonde model, she takes action. But the old man outwits her and brings an unexpected change.

Time for Change

We study the existence of a unique stationary distribution and ergodicity for a 2-dimensional affine process. The first coordinate is supposed to be a so-called alpha-root process with \alpha\in(1,2]. The existence of a unique stationary distribution for the affine process is proved in case of \alpha\in(1,2]; further, in case of \alpha=2, the ergodicity is also shown.

Stationarity and ergodicity for an affine two factor model

We prove Chung-type laws of the iterated logarithm for general Levy processes at zero. In particular, we provide tools to translate small deviation estimates directly into laws of the iterated logarithm. This reveals laws of the iterated logarithm for Levy processes at small times in many concrete examples. In some cases, exotic norming functions are derived.

Small time Chung-type LIL for L\'{e}vy processes

First we provide a simple set of sufficient conditions for the weak convergence of scaled affine processes with state space $R_+ \times R^d$. We specialize our result to one-dimensional continuous state branching processes with immigration. As an application, we study the asymptotic behavior of least squares estimators of some parameters of a two-dimensional critical affine diffusion process.

On parameter estimation for critical affine processes

We ask for necessary and sufficient conditions for almost sure finiteness of the perpetual integrals of a Levy process. Zero-one laws are already known for Brownian motion with drift and spectrally one-sided Levy processes. Under the assumption that local times exist, we use fluctuation theory and Jeulin's lemma to prove the zero-one law.

Perpetual Integrals for Levy Processes

Conditioning Markov processes to avoid a domain is a classical problem that has been studied in many settings. Ingredients for standard arguments involve the leading order tail asymptotics of the distribution of the first hitting time of the domain of interest and its relation to an underlying harmonic function. 
In the present article we condition stable processes to avoid intervals. The required tail asymptotics in the stable setting for $\alpha\geq 1$ go back to classical work of Blumenthal et al. and Port from the 1960s. For $\alpha<1$, we appeal to recent results centred around the so-called deep factorisation of the stable process to compute hitting probabilities and, moreover, to identify the associated harmonic functions for all $\alpha\in (0,2)$. With these in hand, we thus prove that conditioning to avoid an interval is possible in the classical sense and that the resulting process is a Doob $h$-transform of the stable process killed on entering the aforesaid interval. Appealing to the representation of the conditioned process as a Doob $h$-transform, we verify that the conditioned process is transient.

Leif Doering

Papers

Stationarity and ergodicity for an affine two factor model

Small time Chung-type LIL for L\'{e}vy processes

On parameter estimation for critical affine processes

Perpetual Integrals for Levy Processes

Stable processes conditioned to avoid an interval