Maximal upper bounds for the moments of stochastic integrals and solutions of stochastic differential equations with respect to fractional Brownian motion with Hurst index $H

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We study stochastic differential equations with Wiener integral considered with respect to fractional Brownian motion with Hurst index $H<1/2$. We prove the existence and uniqueness of a strong solution of the equations and find maximal upper bounds for moments of a solution and its increments. We obtain estimates for the distribution of the supremum of a solution on an arbitrary interval. The modulus of continuity of solutions is found and estimates for the distributions of the norms of solutions are obtained in some Lipschitz spaces.