Schulte, Daniel

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Schulte, Daniel

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    PublicationAccess status: Open Access ,
    Über operator-stable-like Prozesse und ihre Eigenschaften
    (2018)
    Schulte, Daniel
    In this thesis, we will introduce a new class of stochastic processes, so-called operator-stable-like processes. Roughly speaking, they behave locally like operator stable processes, but they need not to be homogenous in space. They are a subclass of Feller processes. A special case of operator-stable-like processes are stable-like processes. We modify the Levy measure of a operator stable law without normal component in such a way that the exponent E (d times d matrix) is no longer constant but depends on the position x (x is a element of RR^d) in space. If the exponent E(x) satisfies certain conditions, we call the resulting family of Levy measures phi(x,.) operator-stable-like Levy measures. According to the Levy measure, we construct an associated stochastic differential equation (SDE) and prove that the solution of this SDE is a Feller process. We show that for symmetric operator-stable-like Levy measures the symbol of the constructed Feller process can be represented with these Levy measures. We call the associated process operator-stable-like process. In the second part we investigate the properties of the symbol. We show a scaling property. The scaling property is helpful to get lower and upper bounds for the symbol. Then we use these estimations to study the properties of the processes. We will focus on maximal estimations, the existence of moments, the short- and long-time behaviour of the sample paths and the p-variation.
    Source Type:Doctoral Thesis
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