Reducing Obizhaeva–Wang-type trade execution problems to LQ stochastic control problems

Julia Ackermann (), Thomas Kruse () and Mikhail Urusov ()
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Julia Ackermann: University of Wuppertal
Thomas Kruse: University of Wuppertal
Mikhail Urusov: University of Duisburg-Essen

Finance and Stochastics, 2024, vol. 28, issue 3, No 5, 813-863

Abstract: Abstract We start with a stochastic control problem where the control process is of finite variation (possibly with jumps) and acts as integrator both in the state dynamics and in the target functional. Problems of such type arise in the stream of literature on optimal trade execution pioneered by Obizhaeva and Wang (J. Financ. Mark. 16:1–32, 2013) (models with finite resilience). We consider a general framework where the price impact and the resilience are stochastic processes. Both are allowed to have diffusive components. First we continuously extend the problem from processes of finite variation to progressively measurable processes. Then we reduce the extended problem to a linear–quadratic (LQ) stochastic control problem. Using the well-developed theory on LQ problems, we describe the solution to the obtained LQ one and translate it back to the solution for the (extended) initial trade execution problem. Finally, we illustrate our results by several examples. Among other things, the examples discuss the Obizhaeva–Wang model with random (terminal and moving) targets, the necessity to extend the initial trade execution problem to a reasonably large class of progressively measurable processes (even going beyond semimartingales), and the effects of diffusive components in the price impact process and/or the resilience process.

Keywords: Optimal trade execution; Stochastic price impact; Stochastic resilience; Finite-variation stochastic control; Continuous extension of cost functional; Progressively measurable execution strategy; Linear–quadratic stochastic control; Backward stochastic differential equation; 91G10; 93E20; 60H10; 60G99 (search for similar items in EconPapers)
JEL-codes: C02 G10 G11 (search for similar items in EconPapers)
Date: 2024
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s00780-024-00537-1

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