• Bernoulli
  • Volume 25, Number 2 (2019), 1105-1140.

Stability for gains from large investors’ strategies in $M_{1}$/$J_{1}$ topologies

Dirk Becherer, Todor Bilarev, and Peter Frentrup

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We prove continuity of a controlled SDE solution in Skorokhod’s $M_{1}$ and $J_{1}$ topologies and also uniformly, in probability, as a nonlinear 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.

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Bernoulli, Volume 25, Number 2 (2019), 1105-1140.

Received: December 2016
Revised: July 2017
First available in Project Euclid: 6 March 2019

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continuity of proceeds illiquid markets no-arbitrage optimal liquidation Skorokhod space Skorokhod topologies stability stochastic differential equation transient price impact


Becherer, Dirk; Bilarev, Todor; Frentrup, Peter. Stability for gains from large investors’ strategies in $M_{1}$/$J_{1}$ topologies. Bernoulli 25 (2019), no. 2, 1105--1140. doi:10.3150/17-BEJ1014.

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