The optimal stopping problem revisited

  • Manuel Guerra
  • Cláudia Nunes
  • Carlos OliveiraEmail author
Regular Article


We consider an optimal stopping time problem, related with many models found in real options problems. We present analytical solutions for a broad class of gain functions, considering quite general assumptions over the model. Also, an extensive and general sensitivity analysis is provided.


Optimal Stopping Exit Option Investment Option Replacement Option 

Mathematics Subject Classification

60G40 90B50 



  1. Alvarez LH (1999) Optimal exit and valuation under demand uncertainty: a real options approach. Eur J Oper Res 114(2):320–329zbMATHGoogle Scholar
  2. Arkin V (2015) Threshold strategies in optimal stopping problem for one-dimensional diffusion processes. Theory Probab Appl 59(2):311–319MathSciNetzbMATHGoogle Scholar
  3. Belomestny D, Rüschendorf L, Urusov MA (2010) Optimal stopping of integral functionals and a “no-loss” free boundary formulation. Theory Probab Appl 54(1):14–28MathSciNetzbMATHGoogle Scholar
  4. Bronstein AL, Hughston LP, Pistorius MR, Zervos M (2006) Discretionary stopping of one-dimensional ito diffusions with a staircase reward function. J Appl Probab 43(4):984–996MathSciNetzbMATHGoogle Scholar
  5. Chevalier E, Vath VL, Roch A, Scotti S (2015) Optimal exit strategies for investment projects. J Math Anal Appl 425(2):666–694MathSciNetzbMATHGoogle Scholar
  6. Chronopoulos M, Hagspiel V, Fleten SE (2015) Stepwise investment and capacity sizing under uncertainty. OR Spectr 39(2):447–472MathSciNetzbMATHGoogle Scholar
  7. Dayanik S (2008) Optimal stopping of linear diffusions with random discounting. Math Oper Res 33(3):645–661MathSciNetzbMATHGoogle Scholar
  8. Dayanik S, Egami M (2012) Optimal stopping problems for asset management. Adv Appl Probab 44(03):655–677MathSciNetzbMATHGoogle Scholar
  9. Dayanik S, Karatzas I (2003) On the optimal stopping problem for one-dimensional diffusions. Stoch Process Appl 107(2):173–212MathSciNetzbMATHGoogle Scholar
  10. Décamps JP, Villeneuve S (2007) Optimal dividend policy and growth option. Financ Stoch 11(1):3–27MathSciNetzbMATHGoogle Scholar
  11. Dixit A (1989) Entry and exit decisions under uncertainty. J Polit Econ 97(3):620–638Google Scholar
  12. Dixit A, Pindyck R (1994) Investment under uncertainty. Princeton University Press, PrincetonGoogle Scholar
  13. Filippov AF (2013) Differential equations with discontinuous righthand sides: control systems, vol 18. Springer Science & Business Media, BerlinGoogle Scholar
  14. Guerra M, Nunes C, Oliveira C (2016) Exit option for a class of profit functions. Int J Comput Math 94(11):2178–2193MathSciNetzbMATHGoogle Scholar
  15. Hagspiel V, Huisman KJ, Kort PM, Nunes C (2016) How to escape a declining market: Capacity investment or exit? Eur J Oper Res 254(1):40–50MathSciNetzbMATHGoogle Scholar
  16. Huisman KJ, Kort PM (2002) Strategic technology investment under uncertainty. QR Spectr 24(1):79–98MathSciNetzbMATHGoogle Scholar
  17. Johnson TC (2015) The solution of some discretionary stopping problems. IMA J Math Control Inf 34(3):717-744Google Scholar
  18. Johnson TC, Zervos M (2007) The solution to a second order linear ordinary differential equation with a non-homogeneous term that is a measure. Stoch Int J Probab Stoch Process 79(3–4):363–382MathSciNetzbMATHGoogle Scholar
  19. Kensinger JW (1988) The capital investment project as a set of exchange options. Managerial Financ 14(2/3):16–27Google Scholar
  20. Knudsen TS, Meister B, Zervos M (1998) Valuation of investments in real assets with implications for the stock prices. SIAM J Control Optim 36(6):2082–2102MathSciNetzbMATHGoogle Scholar
  21. Kort PM (1998) Optimal R & D investments of the firm. Oper Res 20(3):155–164MathSciNetzbMATHGoogle Scholar
  22. Kulatilaka N, Trigeorgis L (2004) The general flexibility to switch: real options revisited. Real options and investment under uncertainty: classical readings and recent contributions, 1st edn. MIT Press, Cambridge, pp 179–198Google Scholar
  23. Lamberton D, Zervos M (2013) On the optimal stopping of a one-dimensional diffusion. Electron J Probab 18(34):1–49MathSciNetzbMATHGoogle Scholar
  24. McDonald R, Siegel D (1985) Investment and the valuation of firms when there is an option to shut down. Int Econ Rev 26:331–349zbMATHGoogle Scholar
  25. Peskir G, Shiryaev A (2006) Optimal stopping and free-boundary problems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, BaselzbMATHGoogle Scholar
  26. Revuz D, Yor M (2013) Continuous martingales and Brownian motion, vol 293. Springer Science & Business Media, BerlinzbMATHGoogle Scholar
  27. Rüschendorf L, Urusov MA (2008) On a class of optimal stopping problems for diffusions with discontinuous coefficients. Ann Appl Probab 18(3):847–878MathSciNetzbMATHGoogle Scholar
  28. Schwartz ES, Trigeorgis L (2004) Real options and investment under uncertainty: classical readings and recent contributions. MIT Press, CambridgeGoogle Scholar
  29. Stokey NL (2016) Wait-and-see: investment options under policy uncertainty. Rev Econ Dyn 21:246–265Google Scholar
  30. Trigeorgis L (1996) Real options: managerial flexibility and strategy in resource allocation. MIT Press, CambridgeGoogle Scholar
  31. Villeneuve S (2007) On threshold strategies and the smooth-fit principle for optimal stopping problems. J Appl Probab 44(1):181–198MathSciNetzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.CEMAPRE, ISEG - School of Economics and ManagementUniversidade de LisboaLisbonPortugal
  2. 2.Department of Mathematics and CEMAT, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal
  3. 3.ISEG - School of Economics and ManagementUniversidade de LisboaLisbonPortugal

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