Capacitary measures for completely monotone kernels via singular control

Aurélien Alfonsi () and Alexander Schied
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Aurélien Alfonsi: CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique - ENPC - École nationale des ponts et chaussées, MATHRISK - Mathematical Risk handling - Inria Paris-Rocquencourt - Inria - Institut National de Recherche en Informatique et en Automatique - UPEM - Université Paris-Est Marne-la-Vallée - ENPC - École nationale des ponts et chaussées
Alexander Schied: Department of Mathematics - University of Mannheim = Universität Mannheim

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Abstract: We give a singular control approach to the problem of minimizing an energy functional for measures with given total mass on a compact real interval, when energy is defined in terms of a completely monotone kernel. This problem occurs both in potential theory and when looking for optimal financial order execution strategies under transient price impact. In our setup, measures or order execution strategies are interpreted as singular controls, and the capacitary measure is the unique optimal control. The minimal energy, or equivalently the capacity of the underlying interval, is characterized by means of a nonstandard infinite-dimensional Riccati differential equation, which is analyzed in some detail. We then show that the capacitary measure has two Dirac components at the endpoints of the interval and a continuous Lebesgue density in between. This density can be obtained as the solution of a certain Volterra integral equation of the second kind.

Date: 2013
Note: View the original document on HAL open archive server: https://enpc.hal.science/hal-00659421v2
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Published in SIAM Journal on Control and Optimization, 2013, 51 (2), pp.1758-1780. ⟨10.1137/120862223⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-00659421

DOI: 10.1137/120862223

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