The linear space of generalized Brownian motions with applications

In this paper, we define, motivated by recent works of Chang and Skoug, stochastic integrals for a generalized Brownian motion ( ${\textrm{gBm}}$) $X$ and then use it to study the representation problem on the linear space $H(X)$ spanned by $X$. We next establish a translation theorem for $L^p$-functionals of $X$, $p \geq 1$, and then use this translation to establish an integration by parts formula for $L^p$-functionals of $X$.