Optimal trade execution under small market impact and portfolio liquidation with semimartingale strategies
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 general semimartingale 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 semimartingale 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.