Bernoulli

  • 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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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.bj/1551862845

Digital Object Identifier
doi:10.3150/17-BEJ1014

Mathematical Reviews number (MathSciNet)
MR3920367

Zentralblatt MATH identifier
07049401

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

Citation

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. https://projecteuclid.org/euclid.bj/1551862845


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