Franziska Kühn

Existence and estimates of moments for Lévy-type processes

In this paper, we establish the existence of moments and moment estimates for Lévy-type processes. We discuss whether the existence of moments is a time dependent distributional property, give sufficient conditions for the existence of moments and prove estimates of fractional moments. Our results apply in particular to SDEs and stable-like processes.

Solutions of Lévy-driven SDEs with unbounded coefficients as Feller processes

Let (Lt)t≥0 be a k-dimensional Lévy process and σ ∶ R → R a continuous function such that the Lévy-driven stochastic differential equation (SDE) dXt = σ(Xt−)dLt, X0 ∼ μ has a unique weak solution. We show that the solution is a Feller process whose domain of the generator contains the smooth functions with compact support if, and only if, the Lévy measure ν of the driving Lévy process (Lt)t≥0 satisfies ν({y ∈ R; ∣σ(x)y + x∣ < r}) ∣x∣→∞ ÐÐÐ→ 0. This generalizes a result by Schilling & Schnurr [14] which states that the solution to the SDE has this property if σ is bounded.

On martingale problems and Feller processes

Let A be a pseudo-differential operator with negative definite symbol q. In this paper we establish a sufficient condition such that the well-posedness of the (A,C c (R))-martingale problem implies that the unique solution to the martingale problem is a Feller process. This provides a proof of a former claim by van Casteren. As an application we prove new existence and uniqueness results for Lévy-driven stochastic differential equations and stable-like processes with unbounded coefficients.

Transition probabilities of Lévy-type processes: Parametrix construction

We present an existence result for Lévy-type processes which requires only weak regularity assumptions on the symbol q(x, ξ) with respect to the space variable x. Applications range from existence and uniqueness results for Lévy-driven SDEs with Hölder continuous coefficients to existence results for stable-like processes and Lévy-type processes with symbols of variable order. Moreover, we obtain heat kernel estimates for a class of Lévy and Lévy-type processes. The paper includes an extensive list of Lévy(-type) processes satisfying the assumptions of our results.

Moments of Lévy-Type Processes

This chapter is concerned with generalized moments \(\mathbb{E}^{x}f(X_{t})\) of a Feller process (X t ) t ≥ 0. We present a sufficient condition for the existence of generalized moments in terms of the triplet (Theorem 2.4) and show that generalized moments exist backward in time (Theorem 2.1). Furthermore, we will derive estimates for fractional moments of Feller processes. In the second part, Sect. 2.3, the absolute continuity of a class of Levy-type processes with Holder continuous symbols is proved by combining the moment estimates with an idea by Fournier and Printems (Bernoulli, 16:343–360, 2010).

Moderate Deviations and Strassen’s Law for Additive Processes

We establish a moderate deviation principle for processes with independent increments under certain growth conditions for the characteristics of the process. Using this moderate deviation principle, we give a new proof of Strassen’s functional law of the iterated logarithm. In particular, we show that any square-integrable Lévy process satisfies Strassen’s law.