Stieltjes integrals of Hölder continuous functions with applications to fractional Brownian motion
Abstract
We give a new estimate on Stieltjes integrals of Hölder continuous functions and use it to prove an existenceuniqueness theorem for solutions of ordinary differential equations with Hölder continuous forcing. We construct stochastic integrals with respect to fractional Brownian motion, and establish sufficient conditions for its existence. We prove that stochastic differential equations with fractional Brownian motion have a unique solution with probability 1 in certain classes of Höldercontinuous functions. We give tail estimates of the maximum of stochastic integrals from tail estimates of the Hölder coefficient of fractional Brownian motion. In addition we apply the techniques used for ordinary Brownian motion to construct stochastic integrals of deterministic functions with respect to fractional Brownian motion and give tail estimates of its maximum.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2000
 arXiv:
 arXiv:math/0005147
 Bibcode:
 2000math......5147R
 Keywords:

 Mathematics  Probability;
 Mathematics  Functional Analysis