Advertisement

Finance and Stochastics

, Volume 23, Issue 1, pp 29–96 | Cite as

Utility maximisation in a factor model with constant and proportional transaction costs

  • Christoph BelakEmail author
  • Sören Christensen
Article

Abstract

We study the problem of maximising expected utility of terminal wealth under constant and proportional transaction costs in a multidimensional market with prices driven by a factor process. We show that the value function is the unique viscosity solution of the associated quasi-variational inequalities and construct optimal strategies. While the value function turns out to be truly discontinuous, we are able to establish a comparison principle for discontinuous viscosity solutions which is strong enough to argue that the value function is unique, globally upper semicontinuous, and continuous if restricted to either borrowing or non-borrowing portfolios.

Keywords

Portfolio optimisation Transaction costs Discontinuous viscosity solutions Comparison principle Stochastic Perron method 

Mathematics Subject Classification (2010)

93E20 49L25 91G80 

JEL Classification

G11 C61 

Notes

References

  1. 1.
    Altarovici, A., Muhle-Karbe, J., Soner, H.M.: Asymptotics for fixed transaction costs. Finance Stoch. 19, 363–414 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Altarovici, A., Reppen, M., Soner, H.M.: Optimal consumption and investment with fixed and proportional transaction costs. SIAM J. Control Optim. 55, 1673–1710 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Azimzadeh, P., Forsyth, P.A.: Weakly chained matrices, policy iteration, and impulse control. SIAM J. Numer. Anal. 54, 1341–1364 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bayraktar, E., Sîrbu, M.: Stochastic Perron’s method and verification without smoothness using viscosity comparison: the linear case. Proc. Am. Math. Soc. 140, 3645–3654 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bayraktar, E., Sîrbu, M.: Stochastic Perron’s method for Hamilton–Jacobi–Bellman equations. SIAM J. Control Optim. 51, 4274–4294 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bayraktar, E., Sîrbu, M.: Stochastic Perron’s method and verification without smoothness using viscosity comparison: Obstacle problems and Dynkin games. Proc. Am. Math. Soc. 142, 1399–1412 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Belak, C., Christensen, S., Seifried, F.T.: A general verification result for stochastic impulse control problems. SIAM J. Control Optim. 55, 627–649 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Belak, C., Sass, J.: Finite-horizon optimal investment with transaction costs: Construction of the optimal strategies. Preprint (2018). Available online at: https://ssrn.com/abstract=2636341
  9. 9.
    Bielecki, T.R., Pliska, S.R.: Risk sensitive asset management with transaction costs. Finance Stoch. 4, 1–33 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bouchard, B., Touzi, N.: Weak dynamic programming principle for viscosity solutions. SIAM J. Control Optim. 49, 948–962 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Christensen, S.: On the solution of general impulse control problems using superharmonic functions. Stoch. Process. Appl. 124, 709–729 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Eastham, J.F., Hastings, K.J.: Optimal impulse control of portfolios. Math. Oper. Res. 13, 588–605 (1988) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Feodoria, M.-R.: Optimal investment and utility indifference pricing in the presence of small fixed transaction costs. Ph.D. Thesis, Christian-Albrechts-Universität Kiel (2016). Available online at: https://macau.uni-kiel.de/receive/dissertation_diss_00019558
  15. 15.
    Ishii, K.: Viscosity solutions of nonlinear second order elliptic PDEs associated with impulse control problems. Funkc. Ekvacioj 36, 123–141 (1993) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Korn, R.: Portfolio optimisation with strictly positive transaction costs and impulse control. Finance Stoch. 2, 85–114 (1998) CrossRefzbMATHGoogle Scholar
  17. 17.
    Korn, R., Laue, S.: Portfolio optimisation with transaction costs and exponential utility. In: Buckdahn, R., et al. (eds.) Stochastic Processes and Related Topics. Proceedings of the 12th Winter School, Siegmundsburg, Germany, pp. 171–188. Taylor & Francis, London (2002) Google Scholar
  18. 18.
    Liu, H.: Optimal consumption and investment with transaction costs and multiple risky assets. J. Finance 59, 289–338 (2004) CrossRefGoogle Scholar
  19. 19.
    Øksendal, B., Sulem, A.: Optimal consumption and portfolio with both fixed and proportional transaction costs. SIAM J. Control Optim. 40, 1765–1790 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Palczewski, J., Stettner, Ł.: Impulsive control of portfolios. Appl. Math. Optim. 56, 67–103 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Peskir, G., Shiryaev, A.N.: Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel (2006) zbMATHGoogle Scholar
  22. 22.
    Schäl, M.: A selection theorem for optimization problems. Arch. Math. 25, 219–224 (1974) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Schroder, M.: Optimal portfolio selection with fixed transaction costs: numerical solutions. Preprint (1995). Available online at: https://msu.edu/~schrode7/numerical.pdf
  24. 24.
    Seydel, R.C.: Existence and uniqueness of viscosity solutions for QVI associated with impulse control of jump-diffusions. Stoch. Process. Appl. 119, 3719–3748 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Touzi, N.: Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE. Springer, New York (2013) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department IVUniversity of TrierTrierGermany
  2. 2.Department of Mathematics, SPSTUniversity of HamburgHamburgGermany

Personalised recommendations