Anastasia Ruzmaikina
Stieljes integrals of H\"older continuous functions with applications
to fractional Brownian motion
(602K, Postscript)
ABSTRACT. 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.