Hyperfinite stochastic integration for Levy processes with finite-variation jump part

Herzberg F. Hyperfinite stochastic integration for Levy processes with finite-variation jump part. BULLETIN DES SCIENCES MATHEMATIQUES. 2010;134(4):423-445.This article links the hyperfinite theory of stochastic integration with respect to certain hyperfinite Levy processes with the elementary theory of pathwise stochastic integration with respect to pure-jump Levy processes with finite-variation jump part. Since the hyperfinite Ito integral is also defined pathwise, these results show that hyperfinite stochastic integration provides a pathwise definition of the stochastic integral with respect to Levy jump-diffusions with finite-variation jump part. As an application, we provide a short and direct nonstandard proof of the generalized Ito formula for stochastic differentials of smooth functions of Levy jump-diffusions whose jumps are bounded from below in norm. (C) 2010 Elsevier Masson SAS. All rights reserved