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

• Published in: Bulletin des sciences mathématiques Vol. 134; no. 4; pp. 423 - 445
• Main Author: Herzberg, Frederik S.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier SAS 01.06.2010; Elsevier
• Subjects: Exact sciences and technology; General mathematics; General, history and biography; Itô formula; Lévy processes; Markov processes; Mathematical analysis; Mathematics; Measure and integration; Nonstandard analysis; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Stochastic integration; Stochastic processes; secondary; Nonstandard analysis; Itô formula; Lévy processes; primary; Stochastic integration; Integration; Lévy process; primary 03H05, 28E05, 60G51; Stochastic process; secondary 60H05; Stochastic integral; Jump diffusion; Stochastic theory; Smooth function; Jump process
• ISSN: 0007-4497; 1952-4773
• DOI: 10.1016/j.bulsci.2010.02.004

This article links the hyperfinite theory of stochastic integration with respect to certain hyperfinite Lévy processes with the elementary theory of pathwise stochastic integration with respect to pure-jump Lévy processes with finite-variation jump part. Since the hyperfinite Itô integral is also defined pathwise, these results show that hyperfinite stochastic integration provides a pathwise definition of the stochastic integral with respect to Lévy jump-diffusions with finite-variation jump part. As an application, we provide a short and direct nonstandard proof of the generalized Itô formula for stochastic differentials of smooth functions of Lévy jump-diffusions whose jumps are bounded from below in norm.