Stochastic calculus for convoluted L\'{e}vy processes
Abstract
We develop a stochastic calculus for processes which are built by convoluting a pure jump, zero expectation L\'{e}vy process with a Volterra-type kernel. This class of processes contains, for example, fractional L\'{e}vy processes as studied by Marquardt [Bernoulli 12 (2006) 1090--1126.] The integral which we introduce is a Skorokhod integral. Nonetheless, we avoid the technicalities from Malliavin calculus and white noise analysis and give an elementary definition based on expectations under change of measure. As a main result, we derive an It\^{o} formula which separates the different contributions from the memory due to the convolution and from the jumps.
Cite
@article{arxiv.0805.2084,
title = {Stochastic calculus for convoluted L\'{e}vy processes},
author = {Christian Bender and Tina Marquardt},
journal= {arXiv preprint arXiv:0805.2084},
year = {2008}
}
Comments
Published in at http://dx.doi.org/10.3150/07-BEJ115 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)