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We find a pathwise decomposition of a certain class of Brownian semistationary processes (BSS) in terms of fractional Brownian motions. To do this, we specialize in the case when the kernel of the BSS is given by ϕα(x) = L(x)xα with α ∈ (-1/2, 0) ∪ (0, 1/2) and L a continuous function slowly varying at zero. We use this decomposition to study some path properties and derive Itˆo's formula for this subclass of BSS processes. [ABSTRACT FROM AUTHOR]

A linear stochastic differential equation in an arbitrary separable Banach space is considered. To solve this equation, the corresponding linear stochastic differential equation for generalized random processes is constructed and its solution is produced as a generalized Itô process. The conditions under which the received generalized random process is the Itô process in a Banach space are found, and thus the solution of the considered linear stochastic differential equation is obtained. The heart of this approach is the conversion of the main equation in a Banach space to the equation for generalized random processes, to find the generalized solution, and then to learn the conditions under which the obtained generalized random process is the random process with values in a Banach space. [ABSTRACT FROM AUTHOR]