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May 2019 Stability for gains from large investors’ strategies in $M_{1}$/$J_{1}$ topologies
Dirk Becherer, Todor Bilarev, Peter Frentrup
Bernoulli 25(2): 1105-1140 (May 2019). DOI: 10.3150/17-BEJ1014

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 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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Dirk Becherer. Todor Bilarev. Peter Frentrup. "Stability for gains from large investors’ strategies in $M_{1}$/$J_{1}$ topologies." Bernoulli 25 (2) 1105 - 1140, May 2019. https://doi.org/10.3150/17-BEJ1014

Information

Received: 1 December 2016; Revised: 1 July 2017; Published: May 2019
First available in Project Euclid: 6 March 2019

zbMATH: 07049401
MathSciNet: MR3920367
Digital Object Identifier: 10.3150/17-BEJ1014

Keywords: continuity of proceeds , illiquid markets , no-arbitrage , optimal liquidation , Skorokhod space , Skorokhod topologies , stability , Stochastic differential equation , transient price impact

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

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Vol.25 • No. 2 • May 2019
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