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 dl\`{a}g 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 and wealth processes from (self-financing) c\`{adl\`{a}g trading strategies} .
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) c\`{adl\`{a}g 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 dl\`{a}g trading strategies} .
R
J_1 and M
M_1 and J
A
show that M_1-continuity is the key to ensure that proceeds and wealth processes from (self-financing) c\`{a
D
,
D
and wealth processes from (self-financing) c\`{a