Stability for gains from large investors' strategies in M1/J1 topologies

TL;DR

This paper establishes the continuity of solutions to controlled stochastic differential equations in specific topologies, facilitating the analysis of large investors' strategies in illiquid markets and ensuring the robustness of proceeds and wealth processes.

Contribution

It proves the continuity of controlled SDE solutions in Skorokhod's M1 and J1 topologies, crucial for modeling large investor strategies in illiquid markets.

Findings

M1-continuity ensures proceeds are well-defined for trading strategies.

Continuity properties aid in solving optimal liquidation problems.

Results help identify asymptotically realizable proceeds.

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) c\`{a}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.