Abstract

We give a new estimate on Stieltjes integrals of Holder continuous functions and use it to prove an existence-uniqueness theorem for solutions of ordinary differential equations with Holder 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 Holder-continuous functions. We give tail estimates of the maximum of stochastic integrals from tail estimates of the Holder 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.

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