  We consider an optimal liquidation problem with instantaneous price impact
and stochastic resilience for small instantaneous impact factors. Within our
modelling framework, the optimal portfolio process converges to the solution of
an optimal liquidation problem with with general semi-martingale controls when
the instantaneous impact factor converges to zero. Our results provide a
unified framework within which to embed the two most commonly used modelling
frameworks in the liquidation literature and show how liquidation problems with
portfolio processes of unbounded variation can be obtained as limiting cases in
models with small instantaneous impact as well as a microscopic foundation for
the use of semi-martingale liquidation strategies. Our convergence results are
based on novel convergence results for BSDEs with singular terminal conditions
and novel representation results of BSDEs in terms of uniformly continuous
functions of forward processes.
