  This paper deals with a nonlinear filtering problem in which a
multi-dimensional signal process is additively affected by a process $\nu$
whose components have paths of bounded variation. The presence of the process
$\nu$ prevents from directly applying classical results and novel estimates
need to be derived. By making use of the so-called reference probability
measure approach, we derive the Zakai equation satisfied by the unnormalized
filtering process, and then we deduce the corresponding Kushner-Stratonovich
equation. Under the condition that the jump times of the process $\nu$ do not
accumulate over the considered time horizon, we show that the unnormalized
filtering process is the unique solution to the Zakai equation, in the class of
measure-valued processes having a square-integrable density. Our analysis paves
the way to the study of stochastic control problems where a decision maker can
exert singular controls in order to adjust the dynamics of an unobservable
It\^o-process.
