The Symbol Associated with the Solution of a Stochastic Differential Equation

TL;DR

This paper investigates the symbol of solutions to Levy-driven stochastic differential equations, deriving explicit formulas and introducing indices that generalize known concepts to analyze the process's fine properties.

Contribution

It provides an explicit probabilistic formula for the symbol of Levy-driven SDE solutions and introduces generalized indices for detailed process analysis.

Findings

Explicit formula for the symbol of the solution process

Introduction of generalized Blumenthal-Getoor indices

Application of indices to analyze fine properties of the process

Abstract

We consider a stochastic differential equations which is driven by a Levy process. It turns out that the solution process is a Feller process if the coefficient of the SDE is bounded. Using a probabilistic formula we calculate the symbol, which appears in the Fourier representation of the generator, explicitely. Using the symbol we introduce indices which are generalizations of the well known Blumenthal-Getoor index. These indices are then used to obtain some fine properties of the solution process.