Bernoulli

  • Bernoulli
  • Volume 22, Number 3 (2016), 1383-1430.

Integration theory for infinite dimensional volatility modulated Volterra processes

Fred Espen Benth and André Süss

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Abstract

We treat a stochastic integration theory for a class of Hilbert-valued, volatility-modulated, conditionally Gaussian Volterra processes. We apply techniques from Malliavin calculus to define this stochastic integration as a sum of a Skorohod integral, where the integrand is obtained by applying an operator to the original integrand, and a correction term involving the Malliavin derivative of the same altered integrand, integrated against the Lebesgue measure. The resulting integral satisfies many of the expected properties of a stochastic integral, including an Itô formula. Moreover, we derive an alternative definition using a random-field approach and relate both concepts. We present examples related to fundamental solutions to partial differential equations.

Article information

Source
Bernoulli, Volume 22, Number 3 (2016), 1383-1430.

Dates
Received: October 2013
Revised: July 2014
First available in Project Euclid: 16 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.bj/1458132986

Digital Object Identifier
doi:10.3150/15-BEJ696

Mathematical Reviews number (MathSciNet)
MR3474820

Zentralblatt MATH identifier
1341.60047

Keywords
Gaussian random fields Malliavin calculus stochastic integration Volterra processes

Citation

Benth, Fred Espen; Süss, André. Integration theory for infinite dimensional volatility modulated Volterra processes. Bernoulli 22 (2016), no. 3, 1383--1430. doi:10.3150/15-BEJ696. https://projecteuclid.org/euclid.bj/1458132986


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