Abstract
We show that an anticipating stochastic forward integral introduced in [9] by means of fractional calculus is an extension of other forward integrals known from the literature. The latter provide important classes of integrable processes. In particular, we investigate the deterministic case for integrands and integrators from optimal Besov spaces. Here the forward integral agrees with the continuous extension of the Lebesgue-Stieltjes integral to these function spaces.
Generalized quadratic variation processes are defined in a similar manner. A survey on applications to anticipating stochastic differential equations with driving processes Z0, Z 1,…, Z m is given, where Z0 has a generalized bracket [Z0] and [Z0], Z 1,…, Zm are smooth of fractional order greater than 1/2.
It is a pleasure to thank John Appleby, Connor Griffin, Gabriella Iovino, Brian O’Kelly, Anatoly Patrick and Paul Quigley (all members of the DCU ‘Default Club’ of 1998-99) for stimulating discussions.
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© 2002 Springer Basel AG
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Zähle, M. (2002). Forward Integrals and Stochastic Differential Equations. In: Dalang, R.C., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications III. Progress in Probability, vol 52. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8209-5_20
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DOI: https://doi.org/10.1007/978-3-0348-8209-5_20
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9474-6
Online ISBN: 978-3-0348-8209-5
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