Bernoulli

  • Bernoulli
  • Volume 16, Number 2 (2010), 389-417.

Approximating a geometric fractional Brownian motion and related processes via discrete Wick calculus

Christian Bender and Peter Parczewski

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Abstract

We approximate the solution of some linear systems of SDEs driven by a fractional Brownian motion BH with Hurst parameter H∈(½, 1) in the Wick–Itô sense, including a geometric fractional Brownian motion. To this end, we apply a Donsker-type approximation of the fractional Brownian motion by disturbed binary random walks due to Sottinen. Moreover, we replace the rather complicated Wick products by their discrete counterpart, acting on the binary variables, in the corresponding systems of Wick difference equations. As the solutions of the SDEs admit series representations in terms of Wick powers, a key to the proof of our Euler scheme is an approximation of the Hermite recursion formula for the Wick powers of BH.

Article information

Source
Bernoulli, Volume 16, Number 2 (2010), 389-417.

Dates
First available in Project Euclid: 25 May 2010

Permanent link to this document
https://projecteuclid.org/euclid.bj/1274821076

Digital Object Identifier
doi:10.3150/09-BEJ223

Mathematical Reviews number (MathSciNet)
MR2668907

Zentralblatt MATH identifier
1248.60044

Keywords
discrete Wick calculus fractional Brownian motion weak convergence Wick–Itô integral

Citation

Bender, Christian; Parczewski, Peter. Approximating a geometric fractional Brownian motion and related processes via discrete Wick calculus. Bernoulli 16 (2010), no. 2, 389--417. doi:10.3150/09-BEJ223. https://projecteuclid.org/euclid.bj/1274821076


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