Vector-valued stochastic processes. II. A Radon-Nikodým theorem for vector-valued processes with finite variation

Abstract:

Given a real-valued process $A$ with finite variation $\left | A \right |$ and a vector-valued process $B$ with finite variation $\left | B \right |$ such that for each $\omega$, the Stieltjes measure $dB{\mathbf {.}}(\omega )$ is absolutely continuous with respect to $dA.(\omega )$, there exists a vector-valued process $H$ which, under certain separability conditions, satisfies ${B_t} = \int _{[0,t]} {{H_s}d{A_s}}$ and ${\left | B \right |_t} = \int _{[0,t]} {\left \| {{H_s}} \right \|d{{\left | A \right |}_s}}$ for every $t \geq 0$. If, moreover, $A$ and $B$ are optional or predictable, then so is $H$.