We prove continuity of a controlled SDE solution in Skorokhod's $J_1$ and $M_1$ 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 demonstrate by examples how continuity properties are useful to solve different stochastic control problems on optimal liquidation, and to identify asymptotically realizable proceeds and wealth processes from (self-financing) c\`{a}dl\`{a}g trading strategies.
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