Title: Stability for gains from large investors' strategies in M1/J1 topologies,
Abstract: We prove continuity of a controlled SDE solution in Skorokhod's $M_1$ and
$J_1$ topologies and also uniformly, in probability, as a non-linear functional
of the control strategy. The functional comes from a finance problem to model
price impact of a large investor in an illiquid market. We show that
$M_1$-continuity is the key to ensure that proceeds and wealth processes from
(self-financing) càdlàg trading strategies are determined as the
continuous extensions for those from continuous strategies. We demonstrate by
examples how continuity properties are useful to solve different stochastic
control problems on optimal liquidation and to identify asymptotically
realizable proceeds.