Numerische Mathematik

, Volume 132, Issue 2, pp 271–302 | Cite as

Convergence of the embedded mean-variance optimal points with discrete sampling

  • Duy-Minh Dang
  • Peter A. Forsyth
  • Yuying Li


A numerical technique based on the embedding technique proposed in (Math Finan 10:387–406, 2000), (Appl Math Optim 42:19–33, 2000) for dynamic mean-variance (MV) optimization problems may yield spurious points, i.e. points which are not on the efficient frontier. In (SIAM J Control Optim 52:1527–1546 2014), it is shown that spurious points can be eliminated by examining the left upper convex hull of the solution of the embedded problem. However, any numerical algorithm will generate only a discrete sampling of the solution set of the embedded problem. In this paper, we formally establish that, under mild assumptions, every limit point of a suitably defined sequence of upper convex hulls of the sampled solution of the embedded problem is on the original MV efficient frontier. For illustration, we discuss an MV asset-liability problem under jump diffusions, which is solved using a numerical Hamilton-Jacobi-Bellman partial differential equation approach.

Mathematics Subject Classification

65K29 91G60 93C20 


  1. 1.
    Almgren, R.: Optimal trading with stochastic liquidity and volatility. SIAM J. Finan. Math. 3, 163–181 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Andrew, A.M.: Another efficient algorithm for convex hulls in two dimensions. Inf. Process. Lett. 9, 216–219 (1979)zbMATHCrossRefGoogle Scholar
  3. 3.
    Barles, G., Souganidis, P.E.: Convergence of approximation schemes for fully nonlinear equations. Asymptot. Anal. 4, 271–283 (1991)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Basak, S., Chabakauri, G.: Dynamic mean-variance asset allocation. Rev. Finan. Stud. 23, 2970–3016 (2011)CrossRefGoogle Scholar
  5. 5.
    Bauerle, N.: Benchmark and mean-variance problems for insurers. Math. Methods Oper. Res. 62, 159–162 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bender, C., Steiner, J.: Least squares Monte Carlo for BSDEs. In: Carmona, R., Del Moral, P., Peng, H., Oudjane, N. (eds.) Numerical Methods in Finance. Springer, New York (2012)Google Scholar
  7. 7.
    Bielecki, T., Jin, H., Pliska, S., Zhou, X.Y.: Continuous time mean-variance portfolio selection with bankruptcy prohibition. Math. Finan. 15, 213–244 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Bjork, T., Murgoci, A.: A general theory of Markovian time inconsistent stochastic control problems. SSRN., (2010)
  9. 9.
    Chen, P., Yam, S.C.P.: Optimal proportional reinsurance and investment with regime switching for mean-variance insurers. Insur. Math. Econ. 53, 871–883 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Chiu, M.C., Li, D.: Asset and liability under a continuous time mean-variance optimization framework. Insur. Math. Econ. 39, 330–355 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Cui, X., Gao, J., Li, X., Li, D.: Optimal multi-period mean variance policy under no-shorting constraint. Eur. J. Oper. Res. 234, 459–468 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Czichowsky, C., Schweizer, M.: Cone-constrained continuous-time Markowitz problems. Ann. Appl. Prob. 23, 764–810 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Dang, D.M., Forsyth, P.A.: Continuous time mean-variance optimal portfolio allocation under jump diffusion: An numerical impulse control approach. Numer. Methods Partial Differ. Equations 30, 664–698 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Debrabant, K., Jakobsen, E.: Semi-Lagrangian schemes for linear and fully non-linear diffusion equations. Math. Comput. 82(283), 1433–1462 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Delong, L., Gerrard, R.: Mean-variance portfolio selection for a non-life insurance company. Math. Methods Oper. Res. 66, 339–367 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    d’Halluin, Y., Forsyth, P.A., Vetzal, K.R.: Robust numerical methods for contingent claims under jump diffusion processes. IMA J. Numer. Anal. 25, 87–112 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Forsyth, P.A.: A Hamilton Jacobi Bellman approach to optimal trade execution. Appl. Numer. Math. 61, 241–265 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Gobet, E., Lemor, J.-P., Warin, X.: A regression-based Monte Carlo method to solve backward stochastic differential equations. Ann. Appl. Prob. 15, 2172–2202 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Jose-Fombellida, R., Rincon-Zapatero, J.: Mean variance portfolio and contribution selection in stochastic pension funding. Eur. J. Oper. Res. 187, 120–137 (2008)CrossRefGoogle Scholar
  20. 20.
    Leippold, M., Trojani, F., Vanini, P.: A geometric approach to multiperiod mean variance optimization of assets and liabilities. J. Econ. Dyn. Control 28, 1079–1113 (2004)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Li, D., Ng, W.-L.: Optimal dynamic portfolio selection: multiperiod mean variance formulation. Math. Finan. 10, 387–406 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Li, X., Zhou, X.Y., Lim, E.E.B.: Dynamic mean-variance porfolio constraint selection with no-shorting constraints. SIAM J. Control Optim. 40, 1540–1555 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Lorenz, J., Almgren, R.: Mean-variance optimal adaptive execution. Appl. Math. Finan. 18, 395–422 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Merton, R.C.: Option pricing when underlying stock returns are discontinuous. J. Finan. Econ. 3, 125–144 (1976)zbMATHCrossRefGoogle Scholar
  25. 25.
    Oksendal, B., Sulem, A.: Applied Control of Jump Diffusions. Springer (2009)Google Scholar
  26. 26.
    Ruijter, M., Oosterlee, C.W.: A Fourier-cosine method for an efficient computation of solutions to BSDEs. Working paper, Centrum Wiskunde & Informatica. Amsterdam (2013)Google Scholar
  27. 27.
    Tse, S.T., Forsyth, P.A., Li, Y.: Preservation of scalarization optimal points in the embedding technique for continuous time mean variance optimization. SIAM J. Control Optim. 52, 1527–1546 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Vigna, E.: On the efficiency of mean-variance based portfolio selection in defined contibution pension schemes. Quant. Finan 14, 237–258 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Wang, J., Forsyth, P.A.: Numerical solution of the Hamilton-Jacobi-Bellman formulation for continuous time mean variance asset allocation. J. Econ. Dyn. Control 34, 207–230 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Wang, J., Forsyth, P.A.: Comparison of mean variance like strategies for optimal asset allocation problems. Int. J. Theor. Appl. Finan., 15(2) (2012). doi: 10.1142/S0219024912500148
  31. 31.
    Xie, S.X., Li, Z.F., Wang, S.Y.: Continuous time portfolio selection with liability: Mean-variance model and stochastic LQ approach. Insur. Math. Econ. 42, 943–953 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Yao, H., Lai, Y., Ma, Q., Jian, M.: Asset allocation for a DC pension fund with stochastic income and mortality risk: a multi-period mean-variance framework. Insur. Math. Econ. 54, 84–92 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Zhou, X.Y., Li, D.: Continuous-time mean-variance portfolio selection: A stochastic LQ framework. Appl. Math. Optim. 42(1), 19–33 (2000)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Mathematics and PhysicsThe University of QueenslandBrisbaneAustralia
  2. 2.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

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