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Über dieses Buch

Presenting some recent results on the construction and the moments of Lévy-type processes, the focus of this volume is on a new existence theorem, which is proved using a parametrix construction. Applications range from heat kernel estimates for a class of Lévy-type processes to existence and uniqueness theorems for Lévy-driven stochastic differential equations with Hölder continuous coefficients. Moreover, necessary and sufficient conditions for the existence of moments of Lévy-type processes are studied and some estimates on moments are derived. Lévy-type processes behave locally like Lévy processes but, in contrast to Lévy processes, they are not homogeneous in space. Typical examples are processes with varying index of stability and solutions of Lévy-driven stochastic differential equations.
This is the sixth volume in a subseries of the Lecture Notes in Mathematics called Lévy Matters. Each volume describes a number of important topics in the theory or applications of Lévy processes and pays tribute to the state of the art of this rapidly evolving subject, with special emphasis on the non-Brownian world.



Chapter 1. Basics

In this chapter we collect definitions and results which we will frequently use in this book. After introducing (universal) Markov processes in Sect. 1.2, we define Lévy processes, subordinators and Feller processes and discuss their most important properties. In Sect. 1.6 we collect some facts on martingale problems. The last part of this chapter, Sect. 1.7, is devoted to the parametrix method.
Franziska Kühn

Chapter 2. 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 Lévy-type processes with Hölder continuous symbols is proved by combining the moment estimates with an idea by Fournier and Printems (Bernoulli, 16:343–360, 2010).
Franziska Kühn

Chapter 3. Parametrix Construction

In this chapter we present an existence result for Feller processes with symbols of form
$$\displaystyle{ q(x,\xi ) =\psi _{\boldsymbol{\alpha }(x)}(\xi ) }$$
where (ψ α ) αI is a family of continuous negative definite functions and \(\boldsymbol{\alpha }: \mathbb{R}^{} \rightarrow I\) a Hölder continuous mapping. We derive heat kernel estimates for the transition density and its time derivative and prove the well-posedness of the associated martingale problem.
Franziska Kühn

Chapter 4. Parametrix Construction: Proofs

This chapter is devoted to the proofs of the results presented in Chap. 3
Franziska Kühn

Chapter 5. Applications

In the first part of this chapter, Sects. 5.1 and 5.2, we investigate variable order subordination. Section 5.1 is concerned with symbols of the form
$$\displaystyle{ q(x,\xi ) = f_{\boldsymbol{\alpha }(x)}(\vert \xi \vert ^{2}) }$$
where ( f α ) αI is a family of Bernstein functions. In particular, we obtain existence results for normal tempered stable-like, relativistic stable-like and Lamperti stable-like processes. The existence of Feller processes with symbols of varying order is studied in Sect. 5.2. Section 5.3 is devoted to jump processes of mixed type. In Sect. 5.4, we prove existence and uniqueness results for solutions of Lévy-driven SDEs with Hölder continuous coefficients. Finally, in Sect. 5.5, we present transition density estimates for a class of Lévy processes.
Franziska Kühn


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