  We consider a central trading desk which aggregates the inflow of clients'
orders with unobserved toxicity, i.e. persistent adverse directionality. The
desk chooses either to internalise the inflow or externalise it to the market
in a cost effective manner. In this model, externalising the order flow creates
both price impact costs and an additional market feedback reaction for the
inflow of trades. The desk's objective is to maximise the daily trading P&L
subject to end of the day inventory penalization. We formulate this setting as
a partially observable stochastic control problem and solve it in two steps.
First, we derive the filtered dynamics of the inventory and toxicity, projected
to the observed filtration, which turns the stochastic control problem into a
fully observed problem. Then we use a variational approach in order to derive
the unique optimal trading strategy. We illustrate our results for various
scenarios in which the desk is facing momentum and mean-reverting toxicity. Our
implementation shows that the P&L performance gap between the partially
observable problem and the full information case are of order $0.01\%$ in all
tested scenarios.
