Skip to main content
SpringerLink
Log in

On the Minimizers of Energy Forms with Completely Monotone Kernel

  • Published:
Applied Mathematics & Optimization Aims and scope Submit manuscript

Abstract

Motivated by the problem of optimal portfolio liquidation under transient price impact, we study the minimization of energy functionals with completely monotone displacement kernel under an integral constraint. The corresponding minimizers can be characterized by Fredholm integral equations of the second type with constant free term. Our main result states that minimizers are analytic and have a power series development in terms of even powers of the distance to the midpoint of the domain of definition and with nonnegative coefficients. We show moreover that our minimization problem is equivalent to the minimization of the energy functional under a nonnegativity constraint.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. This fact relies on the well-known result that \(G(|\cdot |)\) is positive definite in the sense of Bochner for every bounded, convex, and nonincreasing function \(G:[0,\infty )\rightarrow [0,\infty )\). This latter result is often attributed to Pólya [15], although it is also an easy consequence of Young [21].

References

  1. Alfonsi, A., Schied, A.: Capacitary measures for completely monotone kernels via singular control. SIAM J. Control Optim. 51(2), 1758–1780 (2013)

    Article  MathSciNet  Google Scholar 

  2. Alfonsi, A., Schied, A., Slynko, A.: Order book resilience, price manipulation, and the positive portfolio problem. SIAM J. Financ. Math. 3, 511–533 (2012)

    Article  MathSciNet  Google Scholar 

  3. Almgren, R.: Optimal execution with nonlinear impact functions and trading-enhanced risk. Appl. Math. Financ. 10, 1–18 (2003)

    Article  Google Scholar 

  4. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Bangkok (1994)

    Book  Google Scholar 

  5. Bernstein, S.: Sur la définition et les propriétś des fonctions analytiques d’une variable réelle. Math. Ann. 75, 449–468 (1914)

    Article  MathSciNet  Google Scholar 

  6. Chen, Z., Haykin, S.: On different facets of regularization theory. Neural Comput. 14(12), 2791–2846 (2002)

    Article  Google Scholar 

  7. Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1954)

    Article  MathSciNet  Google Scholar 

  8. Gatheral, J.: No-dynamic-arbitrage and market impact. Quant. Financ. 10(7), 749–759 (2010)

    Article  MathSciNet  Google Scholar 

  9. Gatheral, J., Schied, A., Slynko, A.: Transient linear price impact and Fredholm integral equations. Math. Financ. 22(3), 445–474 (2012)

    Article  MathSciNet  Google Scholar 

  10. Hilbert, D.: Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. (Zweite Mitteilung.). Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1904, 213–259 (1904)

    MATH  Google Scholar 

  11. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2013)

    MATH  Google Scholar 

  12. Huberman, G., Stanzl, W.: Price manipulation and quasi-arbitrage. Econometrica 72(4), 1247–1275 (2004)

    Article  MathSciNet  Google Scholar 

  13. Milne-Thomson, L.M.: The Calculus of Finite Differences. American Mathematical Society, Providence (2000)

    MATH  Google Scholar 

  14. Petrov, F.: Inequality connected to Lagrange’s interpolation formula. MathOverflow contribution (http://mathoverflow.net/q/263277, Version: 02/28/2017) (2017)

  15. Pólya, G.: Remarks on characteristic functions. In: Neyman, J., (ed.), Proceedings of the Berkeley Symposium of Mathematical Statistics and Probability, pp. 115–123. University of California Press (1949)

  16. Riesz, F.: Sur une inégalité intégrale. J. Lond. Math. Soc. 1(3), 162–168 (1930)

    Article  Google Scholar 

  17. Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton (1970)

    Google Scholar 

  18. Schechter, S.: On the inversion of certain matrices. Math. Tables Other Aids Comput. 13(66), 73–77 (1959)

    Article  MathSciNet  Google Scholar 

  19. Terrell, B.: How to show that this polynomial has \(n\) positive roots? Math StackExchange contribution (http://math.stackexchange.com/q/2106024, Version: 01/20/2017) (2017)

  20. Williamson, R.E.: Multiply monotone functions and their Laplace transforms. Duke Math. J. 23, 189–207 (1956)

    Article  MathSciNet  Google Scholar 

  21. Young, W.H.: On the Fourier series of bounded functions. Proc. Lond. Math. Soc. 12, 41–70 (1913)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Canada

    Alexander Schied

  2. Department of Mathematics, University of Mannheim, Mannheim, Germany

    Elias Strehle

Authors
  1. Alexander Schied
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elias Strehle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Schied.

Additional information

This paper is dedicated to Jim Gatheral on the occasion of his 60th birthday.

The authors gratefully acknowledge financial support by Deutsche Forschungsgemeinschaft through Research Grant SCHI 500/3-2. A.S. also acknowledges partial support from the Natural Sciences and Engineering Research Council of Canada through Grant RGPIN-2017-04054.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schied, A., Strehle, E. On the Minimizers of Energy Forms with Completely Monotone Kernel. Appl Math Optim 83, 177–205 (2021). https://doi.org/10.1007/s00245-018-9516-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-018-9516-7

Keywords

Mathematics Subject Classification

Search

Navigation