Construction of Lévy Processes and Their Corresponding SDEs: The Finite Variation Case
In this chapter, we will generalize the previous construction of compound Poisson processes and allow the possibility of a infinite number of jumps on a fixed interval. The stochastic process constructed in this section will satisfy that the number of jumps whose absolute size is larger than any fixed positive value is finite in any fixed interval. Therefore the fact that there are infinite number of jumps is due to the fact that most of these jumps are small in size. The conditions imposed will also imply that the generated stochastic process has paths of bounded variation and therefore Stiltjes integration can be used to give a meaning to stochastic integrals. We also introduce the associated stochastic calculus.