Finite Difference Scheme for Stochastic Differential Games with Several Singular Control Variables and Its Environmental Application

  • Hidekazu YoshiokaEmail author
  • Yuta Yaegashi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)


Stochastic differential games have recently been key mathematical tools for resolution of environmental and ecological optimization problems. A finite difference scheme is proposed for solving a variational inequality arising in a stochastic differential game having several singular control variables: optimization of algae population management under model ambiguity. The present scheme employs fitted-exponential and upwind discretization methods to generate stable numerical solutions. Accuracy of the scheme is verified against an exact solution to a simplified problem and an asymptotic solution to a more complicated problem. The scheme is finally applied to numerical computation of the optimal algae population management policy against a range of an incurred cost. The computational results suggest that qualitatively different optimal policies are obtained depending on the magnitude of the incurred cost.



This work was supported by The River Foundation under grant The River Fund No. 285311020, The Japan Society for the Promotion Science under grant KAKENHI No. 17K15345 and No. 17J09125, and Water Resources Environment Center under grant The WEC Applied Ecology Research Grant No. 2016-02.


  1. Yaegashi, Y., Yoshioka, H., Unami, K., Fujihara, M.: Optimal policy of predator suppression for sustainable inland fishery management. In: Proceedings of 12th SDEWES Conference, pp. 309-1–309-11 (2017)Google Scholar
  2. Yoshioka, H., Yaegashi, Y.: Stochastic control model of dam discharge for algae growth management. J. Biol. Dyn. 12, 242–270 (2018a)Google Scholar
  3. Hansen, L., Sargent, T.J.: Robust control and model uncertainty. Am. Econ. Rev. 91, 60–66 (2001)CrossRefGoogle Scholar
  4. Yoshioka, H., Yaegashi, Y.: Robust stochastic control modeling of dam discharge to suppress overgrowth of downstream harmful algae. Appl. Stoch. Model. Bus. (2018b, to appear)Google Scholar
  5. Roseta-Palma, C., Xepapadeas, A.: Robust control in water management. J. Risk Uncertain. 29, 21–34 (2004)CrossRefGoogle Scholar
  6. Øksendal, B., Sulem-Bialobroda, A.: Applied Stochastic Control of Jump Diffusions. Springer, Heidelberg (2005). Scholar
  7. Al Morairi, H., Zervos, M.: Irreversible capital accumulation with economic impact. Appl. Math. Optim. 75, 525–551 (2017)MathSciNetCrossRefGoogle Scholar
  8. McAllister, T.G., Wood, S.A., Hawes, I.: The rise of toxic benthic Phormidium proliferations: a review of their taxonomy, distribution, toxin content and factors regulating prevalence and increased severity. Harmful Algae 55, 282–294 (2016)CrossRefGoogle Scholar
  9. Athanassoglou, S., Xepapadeas, A.: Pollution control with uncertain stock dynamics: when, and how, to be precautious. J. Environ. Econ. Manag. 63, 304–320 (2012)CrossRefGoogle Scholar
  10. Miao, J., Rivera, A.: Robust contracts in continuous time. Econometrica 84, 1405–1440 (2016)MathSciNetCrossRefGoogle Scholar
  11. Yoshioka, H., Unami, K., Fujihara, M.: Mathematical analysis on a conforming finite element scheme for advection-dispersion-decay equations on connected graphs. J. JSCE. Ser. A2 70, I265–I274 (2014)CrossRefGoogle Scholar
  12. De Falco, C., O’Riordan, E.: A parameter robust Petrov-Galerkin scheme for advection-diffusion-reaction equations. Numer. Algorithms 56, 107–127 (2011)MathSciNetCrossRefGoogle Scholar
  13. Oberman, A.M.: Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton-Jacobi equations and free boundary problems. SIAM J. Numer. Anal. 44, 879–895 (2006)MathSciNetCrossRefGoogle Scholar
  14. Huang, Y., Forsyth, P.A., Labahn, G.: Iterative methods for the solution of a singular control formulation of a GMWB pricing problem. Numer. Math. 122, 133–167 (2012)MathSciNetCrossRefGoogle Scholar
  15. Forsyth, P.A., Vetzal, K.R.: Numerical methods for nonlinear PDEs in finance. In: Duan, J.C., Härdle, W., Gentle, J. (eds.) Handbook of Computational Finance, pp. 503–528. Springer, Heidelberg (2012). Scholar
  16. Thomas, L.H.: Elliptic problems in linear difference equations over a network. Watson Sci. Comp. Lab. Rep. Columbia University, New York (1949)Google Scholar
  17. Cadenillas, A., Choulli, T., Taksar, M., Zhang, L.: Classical and impulse stochastic control for the optimization of the dividend and risk policies of an insurance firm. Math. Financ. 16, 181–202 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Life and Environmental ScienceShimane UniversityMatsueJapan
  2. 2.Graduate School of AgricultureKyoto UniversityKyotoJapan
  3. 3.TokyoJapan

Personalised recommendations