Stochastic calculus for convoluted Lévy processes

Details Stochastic calculus for convoluted Lévy processes Stochastic calculus for convoluted Lévy processes

2008-05-14

arxiv.org/abs/0805.2084v1

Bernoulli 2008, Vol. 14, No. 2, 499-518

Mathematics

Probability

Published in at http://dx.doi.org/10.3150/07-BEJ115 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/07-BEJ115

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.

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