arXiv:2103.05957v1 [q-fin.MF] 10 Mar 2021 Small impact analysis in stochastically illiquid markets * – Convergence of optimal portfolio processes – Ulrich Horst † Evgueni Kivman ‡ March 11, 2021 Abstract We consider an optimal liquidation problem with instantaneous price impact and stochas- tic 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 com- monly 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 conver- gence results for BSDEs with singular terminal conditions and novel representation results of BSDEs in terms of uniformly continuous functions of forward processes. AMS Subject Classification: 93E20, 91B70, 60H30. Keywords: portfolio liquidation, singular BSDE, stochastic liquidity, singular control 1 Introduction The impact of limited liquidity on optimal trading strategies in financial markets has recently been widely analysed in the mathematical finance and stochastic control literature. The ma- jority of the optimal trade execution literature allows for either instantaneous (and permanent) or transient impact. The first approach, pioneered by Bertsimas and Lo [5] and Almgren and Chriss [3], divides the price impact in a purely temporary effect, which depends only on the present trading rate and does not influence future prices, and in a permanent effect, which influences the price depending only on the total volume that has been traded in the past. The temporary impact is typically assumed to be linear in the trading rate, leading to a quadratic term in the cost functional. The original modelling framework has been extended in various directions including general stochastic settings with and without model uncertainty and multi-player and mean-field-type models by many authors including Ankirchner et al. [4], Cartea and Jaimungal [8], Fu et al. [13], * Financial support through the Stiftung der Deutschen Wirtschaft (sdw) is gratefully acknowledged. † Department of Mathematics and School of Business and Economics, Humboldt-Universit¨ at zu Berlin, Unter den Linden 6, 10099 Berlin, Germany, horst@math.hu-berlin.de. ‡ Department of Mathematics, Humboldt-Universit¨ at zu Berlin, Unter den Linden 6, 10099 Berlin, Germany, kivman@math.hu-berlin.de. 1