This paper deals with a nonlinear ltering problem in which a multi-dimensional signal process is additively a ected by a process whose components have paths of bounded variation. The presence of the process 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 satis ed by the unnormalized ltering process, and then we deduce the corresponding Kushner-Stratonovich equation. Under the condition that the jump times of the process do not accumulate over the considered time horizon, we show that the unnormalized ltering 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 Itô-process.
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