Mathematics and Financial Economics

, Volume 13, Issue 3, pp 429–458 | Cite as

Bubbles in assets with finite life

  • Henri Berestycki
  • Cameron Bruggeman
  • Regis Monneau
  • José A. ScheinkmanEmail author


We study the speculative value of a finitely lived asset when investors disagree and short sales are limited. In this case, investors are willing to pay a speculative value for the resale option they obtain when they acquire the asset. Using martingale arguments, we characterize the equilibrium speculative value as a solution to a fixed point problem for a monotone operator \(\mathbb F\). A Dynamic Programming Principle applies and is used to show that the minimal solution to the fixed-point problem is a viscosity solution of a naturally associated (non-local) obstacle problem. Combining the monotonicity of the operator \({\mathbb {F}}\) and a comparison principle for viscosity solutions to the obstacle problem we obtain several comparison of solution results. We also use a characterization of the exercise boundary of the obstacle problem to study the effect of an increase in the costs of transactions on the value of the bubble and on the volume of trade, and in particular to quantify the effect of a small transaction (Tobin) tax.


Asset-price bubble Finitely-lived financial asset Heterogeneous beliefs Non-local free boundary problem Dynamic programming Stochastic control 

JEL Classifications

G1 G12 G14 G18 



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Henri Berestycki
    • 1
  • Cameron Bruggeman
    • 2
  • Regis Monneau
    • 3
  • José A. Scheinkman
    • 4
    • 5
    • 6
    Email author
  1. 1.École des hautes études en sciences sociales, CNRS, Centre d’analyse et mathématique socialesPSL UniversityParisFrance
  2. 2.Lyft Inc.San FranciscoUSA
  3. 3.CERMICS, Ecole des Ponts ParisTechUniversité Paris-EstMarne-la-ValléeFrance
  4. 4.Columbia UniversityNew York CityUSA
  5. 5.Princeton UniversityPrincetonUSA
  6. 6.NBERCambridgeUSA

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