Computational Management Science

, Volume 14, Issue 4, pp 519–533 | Cite as

A discrete optimality system for an optimal harvesting problem

  • Hacer Öz Bakan
  • Fikriye Yılmaz
  • Gerhard-Wilhelm Weber
Original Paper


In this paper, we obtain the discrete optimality system of an optimal harvesting problem. While maximizing a combination of the total expected utility of the consumption and of the terminal size of a population, as a dynamic constraint, we assume that the density of the population is modeled by a stochastic quasi-linear heat equation. Finite-difference and symplectic partitioned Runge–Kutta (SPRK) schemes are used for space and time discretizations, respectively. It is the first time that a SPRK scheme is employed for the optimal control of stochastic partial differential equations. Monte-Carlo simulation is applied to handle expectation appearing in the cost functional. We present our results together with a numerical example. The paper ends with a conclusion and an outlook to future studies, on further research questions and applications.


Stochastic optimal control Optimal harvesting Stochastic partial differential equations Symplectic partitioned Runge–Kutta schemes 


  1. Alvarez L, Shepp L (1998) Optimal harvesting of stochastically fluctuating populations. J Math Biol 37:155–177CrossRefGoogle Scholar
  2. Barth A, Lang A (2012) Simulation of stochastic partial differential equations using finite element methods. Stoch Int J Probab Stoch Process 84(2–3):217–231CrossRefGoogle Scholar
  3. Bismut JM (1973) Conjugate convex functions in optimal stochastic control. J Math Anal Appl 44:384–404CrossRefGoogle Scholar
  4. Bismut JM (1978) An introductory approach to duality in optimal stochastic control. SIAM Rev 20(1):61–78CrossRefGoogle Scholar
  5. Bonnans JF, Laurent-Varin J (2006) Computation of order conditions for symplectic partitioned Runge–Kutta schemes with application to optimal control. Numerische Mathematik 103(1):1–10CrossRefGoogle Scholar
  6. Burrage K, Burrage PM (1999) Order conditions of stochastic Runge–Kutta methods by B-series. SIAM J Numer Anal 38:1626–1646CrossRefGoogle Scholar
  7. Burrage PM (1999) Runge–Kutta methods for stochastic differential equations. PhD Thesis, Department of Mathematics, University of Queensland, AustraliaGoogle Scholar
  8. Butcher JC (2003) Numerical methods for ordinary differential equations. Wiley, West SussexCrossRefGoogle Scholar
  9. Clark CW (1931) Mathematical bioeconomics: the optimal management of renewable resources, 2nd edn. Wiley, New YorkGoogle Scholar
  10. Debrabant K, Kværnø A (2008/2009) B-series analysis of stochastic Runge-Kutta methods that use an iterative scheme to compute their internal stage values. SIAM J Numer Anal 47(1):181–203Google Scholar
  11. Debrabant K, Rößler A (2008) A Classification of stochastic Runge-Kutta methods for the weak approximation of stochastic differential equations. Math Comput Simul 77(4):408–420CrossRefGoogle Scholar
  12. Dontchev AL, Hager WW, Veliov VM (2001) Second-order Runge–Kutta approximations in control constrained optimal control. SIAM J Numer Anal 38(1):202–226CrossRefGoogle Scholar
  13. Fadhel SF, Abdulamear AA (2011) Explicit Runge–Kutta methods for solving stochastic differential equations. J Basrah Res Sci 37(4):300–313Google Scholar
  14. Fleming WH, Rishel RW (1975) Deterministic and stochastic optimal control. Springer, New YorkCrossRefGoogle Scholar
  15. Fleming WH, Soner HM (1993) Controlled Markov processes and viscosity solutions. Springer, New YorkGoogle Scholar
  16. Glasserman P (2004) Monte-Carlo methods in financial engineering. Springer, New YorkGoogle Scholar
  17. Hager WW (2000) Runge–Kutta methods in optimal control and the transformed adjoint system. Numerische Mathematik 87(2):247–282CrossRefGoogle Scholar
  18. Itô K (1944) Stochastic integral. Proc Imperial Acad Tokyo 20:519–524CrossRefGoogle Scholar
  19. Jentzen A, Kloeden PE (2009) The numerical approximation of stochastic partial differential equations. Milan J Math 77:205–244CrossRefGoogle Scholar
  20. Kaya CY (2010) Inexact restoration for Runge–Kutta discretization of optimal control problems. SIAM J Numer Anal 48(4):1492–1517CrossRefGoogle Scholar
  21. Kloeden PE, Platen E (1999) Numerical solution of stochastic differential equations, vol 21, 2nd edn. Springer, BerlinGoogle Scholar
  22. Korn R, Korn E, Kroisandt G (2010) Monte-Carlo methods and models in finance and insurance. Chapman and Hall/CRC Financial Mathematics Series. CRC Press, Boco Raton. ISBN 9781420076196Google Scholar
  23. Kroese DP, Taimre T, Botev ZI (2011) Handbook of Monte-Carlo methods. Wiley series in probability and statistics. Wiley, New YorkCrossRefGoogle Scholar
  24. Kushner HJ, Dupuis PG (1992) Numerical methods for stochastic control problems in continuous time. Springer, New YorkCrossRefGoogle Scholar
  25. Kunita H (1990) Stochastic flows and stochastic differential equations, vol 24. Studies in advanced mathematics. Cambridge University Press, CambridgeGoogle Scholar
  26. Lungu E, Øksendal B (1997) Optimal harvesting from a population in a stochastic crowded environment. Math Biosci 145:47–75CrossRefGoogle Scholar
  27. Lungu E, Øksendal B (2001) Optimal harvesting from interacting populations in a stochastic environment. BERNOULLI 7:527–539CrossRefGoogle Scholar
  28. Øksendal B (2005) Optimal control of stochastic partial differential equations. Stoch Anal Appl 23:165–179CrossRefGoogle Scholar
  29. Øksendal B, Våge G, Zhao HZ (2000) Asymptotic properties of the solutions to stochastic KPP equations. Proc R Soc Edinb 130(6):1363–1381Google Scholar
  30. Øksendal B, Våge G, Zhao HZ (2001) Two properties of stochastic KPP equations: ergodicity and pathwise property. Nonlinearity 14:639–662CrossRefGoogle Scholar
  31. Peng S, Wu Z (1999) Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J Control Optim 37:825–843CrossRefGoogle Scholar
  32. Peng S (1990) A general stochastic maximum principle for optimal control problems. SIAM J Control Optim 28:966979CrossRefGoogle Scholar
  33. Pinheiro S (2016) Optimal harvesting for a logistic growth model with predation and a constant elasticity of variance. Ann Oper Res :1–20Google Scholar
  34. Rößler A (2006) Rooted tree analysis for order conditions of stochastic Runge–Kutta methods for the weak approximation of stochastic differential equations. Stoch Anal Appl 24(1):97–134CrossRefGoogle Scholar
  35. Rößler A (2010) Runge–Kutta methods for the strong approximation of solutions of stochastic differential equations. SIAM J Numer Anal 48(3):922–952CrossRefGoogle Scholar
  36. Sanz-Serna JM (2016) Symplectic Runge–Kutta schemes for adjoint equations, automatic differentiation, optimal control, and more. SIAM Rev 58(1):3–33CrossRefGoogle Scholar
  37. Schulstok B (1998) Optimal income by harvesting under uncertainty. Cand Scient thesis, University of OsloGoogle Scholar
  38. Yılmaz F, Öz H, Weber GW (2015) Simulation of stochastic optimal control problems with symplectic partitioned Runge–Kutta scheme. Dyn Contin Discret Impuls Syst Ser B Appl Algoritm 22:425–440Google Scholar
  39. Yong J, Zhou XY (1999) Stochastic Controls: Hamiltonian Systems and HJB Equations. Applications of mathematics, Springer, New YorkCrossRefGoogle Scholar
  40. Yoshida H (1990) Construction of higher order symplectic integrators. Phys Lett A 150:262–268CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsAtılım UniversityAnkaraTurkey
  2. 2.Department of MathematicsGazi UniversityAnkaraTurkey
  3. 3.Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey

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