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In this paper, stochastic differential equations in a Hilbert space with a standard, cylindrical fractional Brownian motion with the Hurst parameter in the interval (1/2,1) are investigated. Existence and uniqueness of mild solutions, continuity of the sample paths and state space regularity of the solutions, and the existence of limiting measures are verified. The equivalence of the probability laws for the solution evaluated at different times and different initial conditions and the convergence of these probability laws to the limiting probability are verified. These results are applied to specific stochastic parabolic and hyperbolic differential equations. The solution of a specific parabolic equation with the fractional Brownian motion only in the boundary condition is shown to have many results that are analogues of the results for a fractional Brownian motion in the domain.

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Abstract-is observable, where Q Q Q 1 determines the quadratic form for the state in the integrand of the cost functional. A weighted least squares algorithm is modified by using a random regularization to ensure that the family of estimated models is uniformly controllable and observable. A diminishing excitation is used with the adaptive control to ensure that the family of estimates is strongly consistent. A lagged certainty equivalence control using this family of estimates is shown to be self-optimizing for an ergodic, quadratic cost functional.

Abstract-A Kiefer-Wolfowitz or simultaneous perturbation algorithm that uses either one-sided or two-sided randomized differences and truncations at randomly varying bounds is given in this paper. At each iteration of the algorithm only two observations are required in contrast to 2`observations, where`is the dimension, in the classical algorithm. The algorithm given here is shown to be convergent under only some mild conditions. A rate of convergence and an asymptotic normality of the algorithm are also established.Index Terms-Kiefer-Wolfowitz algorithm, perturbation algorithm, simultaneous stochastic approximation, stochastic approximation with randomized differences.