Skip to main content
SpringerLink
Log in

Asymptotics for a Symmetric Equation in Price Formation

  • Published:
Applied Mathematics and Optimization Submit manuscript

Abstract

We study the existence and asymptotics for large time of the solutions to a one dimensional evolution equation with non-standard right-hand side. The right-hand side involves the derivative of the solution computed at a given point. Existence is proven through a fixed point argument. When the problem is considered in a bounded interval, it is shown that the solution decays exponentially to the stationary state. This problem is a particular case of a mean-field free boundary model proposed by Lasry and Lions on price formation and dynamic equilibria.

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

Similar content being viewed by others

References

  1. Caffarelli, L., Salsa, S.: A geometric approach to free boundary problems. In: Graduate Studies in Mathematics, vol. 68. Am. Math. Soc., Providence (2005)

    Google Scholar 

  2. Carrillo, J.A., Toscani, G.: Exponential convergence toward equilibrium for homogeneous Fokker–Planck-type equations. Math. Methods Appl. Sci. 21(13), 1269–1286 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Carrillo, J.A., Toscani, G.: Asymptotic L 1-decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J. 49(1), 113–142 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Frank, T.D.: Nonlinear Fokker–Planck Equations: Fundamentals and Applications. Springer, Berlin/New York (2005)

    MATH  Google Scholar 

  5. González, M.d.M., Gualdani, M.P.: Asymptotics for a free-boundary model in price formation. In preparation

  6. Lasry, J.-M., Lions, P.-L.: Towards a self-consistent theory of volatility. J. Math. Pures Appl. (9) 86(6), 541–551 (2006)

    MATH  MathSciNet  Google Scholar 

  7. Lasry, J.-M., Lions, P.-L.: Large investor trading impacts on volatility. Ann. Inst. H. Poincaré Anal. Non Linéaire 24(2), 311–323 (2007)

    Article  MathSciNet  Google Scholar 

  8. Lasry, J.-M., Lions, P.-L.: Instantaneous self-fulfilling of long-term prophecies on the probabilistic distribution of financial asset values. Ann. Inst. H. Poincaré Anal. Non Linéaire 24(3), 361–368 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Risken, H.: The Fokker–Planck Equation: Methods of Solutions and Applications, 2nd edn. Springer Series in Synergetics. Springer, Berlin (1989)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ETSEIB, Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Av. Diagonal 647, 08028, Barcelona, Spain

    María del Mar González

  2. Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX, 78712, USA

    Maria Pia Gualdani

Authors
  1. María del Mar González
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maria Pia Gualdani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to María del Mar González.

Additional information

Maria P. Gualdani is supported by the NSF Grant DMS-0807636.

Rights and permissions

Reprints and permissions

About this article

Cite this article

González, M.d.M., Gualdani, M.P. Asymptotics for a Symmetric Equation in Price Formation. Appl Math Optim 59, 233–246 (2009). https://doi.org/10.1007/s00245-008-9052-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-008-9052-y

Keywords

Search

Navigation