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