Hyperfinite stochastic integration for Lévy processes with finite-variation jump part
In classical stochastic integration, one defines the stochastic integral of a bouded, adapted path-continuous real-valued process with respect to a Lévy process in terms of an Itô integral and two Riemann-Stieltjes integrals. In other words, by decomposing the Lévy process into its diffuse part (a Wiener process) and its jump part, classical stochastic integration with respect to Lévy processes can be divided into integration with respect to Wiener processes and integration with respect to pure-jump Lévy processes; each of these two kinds of integrals are treated by different methods. The aim of this paper is to use methods of nonstandard analysis to calculate such stochastic integrals as the standard parts of hyperfinite stochastic integrals, which can be defined using an internal Riemann-Stieltjes sum, thus providing a pathwise method for calculating stochastic integrals. Indeed, given any Lévy process \(z\) with finite-variation jump part, the paper shows how to find a hyperfinite Lévy process \(Z\), called a \textit{pure Lindstrøm lifting} of \(z\), whose standard part is \(z\) and which can be decomposed in terms of an internal random walk and the difference of two increasing hyperfinite Lévy processes. Then, stochastic integration of an adapted process \(y\) with respect to such a Levy process \(z\) can be defined as the standard part of the hyperfinite Riemann-Stieltjes sum of \(y^*\) with respect to \(Z\). The paper then uses this approach to stochastic integration to give a simple, nonstandard proof of the generalized Itô formula for Lévy jump-diffusions whose jumps are bounded below in norm.
scientific article; zbMATH DE number 5724583
scientific article