## Bernoulli

- Bernoulli
- Volume 14, Number 2 (2008), 499-518.

### Stochastic calculus for convoluted Lévy processes

Christian Bender and Tina Marquardt

#### Abstract

We develop a stochastic calculus for processes which are built by convoluting a pure jump, zero expectation Lévy process with a Volterra-type kernel. This class of processes contains, for example, fractional Lévy 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 Itô formula which separates the different contributions from the memory due to the convolution and from the jumps.

#### Article information

**Source**

Bernoulli, Volume 14, Number 2 (2008), 499-518.

**Dates**

First available in Project Euclid: 22 April 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1208872115

**Digital Object Identifier**

doi:10.3150/07-BEJ115

**Mathematical Reviews number (MathSciNet)**

MR2544099

**Zentralblatt MATH identifier**

1173.60017

**Keywords**

convoluted Lévy process fractional Lévy process Itô formula Skorokhod integration

#### Citation

Bender, Christian; Marquardt, Tina. Stochastic calculus for convoluted Lévy processes. Bernoulli 14 (2008), no. 2, 499--518. doi:10.3150/07-BEJ115. https://projecteuclid.org/euclid.bj/1208872115