Statistical inference on the drift parameter in symmetric stable Lévy process with a deterministic drift

ABSTRACT

In statistical inference on the drift parameter in the symmetric stable Lévy process with the parameter of stability with a deterministic drift , there are some simple options how to do it. We may, for example, base this inference on the properties of the stable distribution. Although this method is very simple, it turns out that it is more appropriate to use inverse methods. For the hypotheses testing about the drift parameter , it is more proper to standardize the observed process and to use inverse methods based on the first exit time of the observed process of a prespecified interval until some given time. These procedures are illustrated, and their times of decision are compared with the direct approach. Other generalizations are possible when the random part is a stochastic integral of a known, deterministic function with respect to the symmetric stable Lévy process or a symmetric stochastic integral of a random (but observable) process with respect to the symmetric stable Lévy process.

KEYWORDS:

• Symmetric stable Lévy process
• symmetric random part
• inverse methods
• stochastic integral
• hypotheses testing

AMS SUBJECT CLASSIFICATIONS:

• 60G18
• 60G51
• 60G52
• 62F03
• 62F05