Advertisement

Positivity Preserving Numerical Method for Optimal Portfolio in a Power Utility Two-Dimensional Regime-Switching Model

  • Miglena N. KolevaEmail author
  • Lubin G. Vulkov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11189)

Abstract

We consider a two-dimensional regime switching model with power utility function. The problem is a system of parabolic partial differential equations with non-linear gradient terms and weakly coupled by non-linear exponential terms. We establish lower bounds for the solutions and then we construct an adequate finite difference method, preserving the qualitative properties of the exact solution. Finally, we present and discuss numerical results.

Notes

Acknowledgements

This research is supported by the Bulgarian National Science Fund under Project DN 12/4 “Advanced analytical and numerical methods for nonlinear differential equations with applications in finance and environmental pollution”, 2017.

References

  1. 1.
    Faragó, I., Horváth, R.: Discrete maximum principle and adequate discretizations of linear parabolic problems. SIAM J. Sci. Comp. 28, 2313–2336 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Gerisch, A., Griffiths, D.F., Weiner, R., Chaplain, M.A.J.: A positive splitting method for mixed hyperbolic - parabolic systems. Num. Meth. for PDEs 17(2), 152–168 (2001)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Gyulov, B., Koleva, M.N., Vulkov, L.G.: Numerical approach to optimal portfolio in a power utility regime-switching model. AIP CP 1910, 030002 (2017)Google Scholar
  4. 4.
    Gyulov, T.B., Koleva, M.N., Vulkov L.G.: Efficient finite difference method for optimal portfolio in a power utility regime-switching model. Int. J. Comp. Math., (2018).  https://doi.org/10.1080/00207160.2018.1474207
  5. 5.
    Hundsdorfer, W., Verwer, J.: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer Series in Computational Mathematics, vol. 33. Springer, Heidelberg (2003).  https://doi.org/10.1007/978-3-662-09017-6CrossRefzbMATHGoogle Scholar
  6. 6.
    Koleva, M.N. Vulkov, L.G.: Numerical method for optimal portfolio in an exponential utility regime-switching model. Int. J. Comp. Math. (2018).  https://doi.org/10.1080/00207160.2018.1440289
  7. 7.
    Kusmin, D., Turek, S.: High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter. J. Comput. Phys. 198(1), 131–158 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    van Leer, B.: Towards the ultimate conservative difference scheme II. monotonicity and conservation combined in a second order scheme. J. Comput. Phys. 14, 361–370 (1974)CrossRefGoogle Scholar
  9. 9.
    LeVeque, R.J.: Numerical Methods for Conservation Laws. Birkhäuser, Basel (1992)CrossRefGoogle Scholar
  10. 10.
    Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, NY (1992)zbMATHGoogle Scholar
  11. 11.
    Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker Inc., New York (2001)CrossRefGoogle Scholar
  12. 12.
    Valdez, A.R.L., Vargiolu, T.: Optimal portfolio in a regime-switching model. In: Dalang, R.C., Dozzi, M., Russo, F. (Eds.) Proceedings of the Ascona 2011 Seminar on Stochastic Analysis, Random Fields and Applications, pp. 435–449 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of RuseRuseBulgaria

Personalised recommendations