Capacitary measures for completely monotone kernels via singular control

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.

Potential Theory on Trees, Graphs and Ahlfors Regular Metric Spaces

We investigate connections between potential theories on a Ahlfors-regular metric space X, on a graph G associated with X, and on the tree T obtained by removing the "horizontal edges" in G. Applications to the calculation of set capacity are given.Comment: 45 pages; presentation improved based on referee comment

Drift dependence of optimal trade execution strategies under transient price impact

We give a complete solution to the problem of minimizing the expected liquidity costs in presence of a general drift when the underlying market impact model has linear transient price impact with exponential resilience. It turns out that this problem is well-posed only if the drift is absolutely continuous. Optimal strategies often do not exist, and when they do, they depend strongly on the derivative of the drift. Our approach uses elements from singular stochastic control, even though the problem is essentially non-Markovian due to the transience of price impact and the lack in Markovian structure of the underlying price process. As a corollary, we give a complete solution to the minimization of a certain cost-risk criterion in our setting

Pointwise Symmetrization Inequalities for Sobolev functions and applications

We develop a technique to obtain new symmetrization inequalities that provide a unified framework to study Sobolev inequalities, concentration inequalities and sharp integrability of solutions of elliptic equationsComment: made a number of corrections, added some reference

Single- and multiplayer trade execution strategies under transient price impact

The problem of optimal execution is to trade a fixed amount of a financial asset over a fixed time horizon in a way that minimizes costs from price impact and transaction costs. Three types of price impact can be distinguished: Temporary, transient and permanent price impact. While mathematical models of optimal execution under temporary and permanent price impact can be analyzed with standard methods from the calculus of variations, models featuring transient price impact are more complex. This thesis studies optimal execution under transient price impact for a single investor and for multiple investors. Assuming that trading incurs quadratic transaction costs, existence and uniqueness of optimal execution strategies and Nash equilibria is established for a large class of transient price impact functions. Closed-form representations of Nash equilibria are derived under the assumption that price impact decays exponentially. These representations are studied in detail to arrive at an economic evaluation of order anticipation strategies and predatory trading. A second focus of this thesis is the intimate connection between problems of optimal execution and Fredholm integral equations. It is shown that, given information about certain characteristics of transient price impact, one can deduce qualitative features of optimal execution strategies, such as nonnegativity and convexity, from the corresponding Fredholm integral equations without obtaining an explicit solution