Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Random Operators and Stochastic Equations

Editor-in-Chief: Girko, Vyacheslav

Managing Editor: Molchanov, S.

Editorial Board: Accardi, L. / Albeverio, Sergio / Carmona, R. / Casati, G. / Christopeit, N. / Domanski, C. / Drygas, Hilmar / Gupta, A.K. / Ibragimov, I. / Kirsch, Werner / Klein, A. / Kondratyev, Yuri / Kurotschka, V. / Leonenko, N. / Loubaton, Philippe / Orsingher, E. / Pastur, L. / Rodrigues, Waldyr A. / Shiryaev, Albert / Turbin, A.F. / Veretennikov, Alexandre

4 Issues per year

CiteScore 2017: 0.32

SCImago Journal Rank (SJR) 2017: 0.177
Source Normalized Impact per Paper (SNIP) 2017: 0.592

Mathematical Citation Quotient (MCQ) 2017: 0.12

See all formats and pricing
More options …
Volume 26, Issue 4


An HJB approach to a general continuous-time mean-variance stochastic control problem

Georgios AivaliotisORCID iD: http://orcid.org/0000-0003-4018-4820 / A. Yu. Veretennikov
  • School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom; and National Research University Higher School of Economics, Russian Federation & Institute of Information Transmission Problems, Moscow
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-11-14 | DOI: https://doi.org/10.1515/rose-2018-0020


A general continuous mean-variance problem is considered for a diffusion controlled process where the reward functional has an integral and a terminal-time component. The problem is transformed into a superposition of a static and a dynamic optimization problem. The value function of the latter can be considered as the solution to a degenerate HJB equation either in the viscosity or in the Sobolev sense (after a regularization) under suitable assumptions and with implications with regards to the optimality of strategies. There is a useful interplay between the two approaches – viscosity and Sobolev.

Keywords: Mean-variance; stochastic control; Hamilton–Jacobi–Bellman; Sobolev solutions; viscosity solutions

MSC 2010: 93E20; 60H10.


  • [1]

    G. Aivaliotis and J. Palczewski, Tutorial for viscosity solutions in optimal control of diffusions, preprint (2010), http://ssrn.com/abstract=1582548.

  • [2]

    G. Aivaliotis and J. Palczewski, Investment strategies and compensation of a mean-variance optimizing fund manager, European J. Oper. Res. 234 (2014), no. 2, 561–570. Google Scholar

  • [3]

    G. Aivaliotis and A. Yu. Veretennikov, On Bellman’s equations for mean and variance control of a Markov diffusion, Stochastics 82 (2010), no. 1–3, 41–51. Google Scholar

  • [4]

    T. R. Bielecki, H. Jin, S. R. Pliska and X. Y. Zhou, Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Math. Finance 15 (2005), no. 2, 213–244. Google Scholar

  • [5]

    T. Bielecki, S. R. Pliska and M. Sherris, Risk sensitive asset allocation, J. Econ. Dynam. Control 24 (2000), no. 8, 1145–1177. Google Scholar

  • [6]

    T. Björk, A. Murgoci and X. Y. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Math. Finance 24 (2014), no. 1, 1–24. Google Scholar

  • [7]

    F. Gozzi, A. Świȩch and X. Y. Zhou, Erratum: “A corrected proof of the stochastic verification theorem within the framework of viscosity solutions” [mr2179474], SIAM J. Control Optim. 48 (2010), no. 6, 4177–4179. Google Scholar

  • [8]

    N. V. Krylov, Controlled Diffusion Processes, Appl. Math. 14, Springer, New York, 1980. Google Scholar

  • [9]

    D. Li and W.-L. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Math. Finance 10 (2000), no. 3, 387–406. Google Scholar

  • [10]

    A. E. B. Lim, Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market, Math. Oper. Res. 29 (2004), no. 1, 132–161. Google Scholar

  • [11]

    H. M. Markowitz, Portfolio selection, J. Finance 7 (1952), no. 1, 77–91. Google Scholar

  • [12]

    J. L. Pedersen and G. Peskir, Optimal mean-variance portfolio selection, Math. Financ. Econ. 11 (2017), no. 2, 137–160. Google Scholar

  • [13]

    H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Stoch. Model. Appl. Probab. 61, Springer, Berlin, 2009. Google Scholar

  • [14]

    S. T. Tse, P. A. Forsyth and Y. Li, Preservation of scalarization optimal points in the embedding technique for continuous time mean variance optimization, SIAM J. Control Optim. 52 (2014), no. 3, 1527–1546. Google Scholar

  • [15]

    J. Wang and P. A. Forsyth, Numerical solution of the Hamilton–Jacobi–Bellman formulation for continuous time mean variance asset allocation, J. Econom. Dynam. Control 34 (2010), no. 2, 207–230. Google Scholar

  • [16]

    X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: a stochastic LQ framework, Appl. Math. Optim. 42 (2000), no. 1, 19–33. Google Scholar

About the article

Received: 2018-02-07

Revised: 2018-10-03

Accepted: 2018-10-16

Published Online: 2018-11-14

Published in Print: 2018-12-01

Funding Source: Engineering and Physical Sciences Research Council

Award identifier / Grant number: EP/N013980/1

Funding Source: Alan Turing Institute

Award identifier / Grant number: EP/N510129/1

The first author has been partially supported by EPSRC grant EP/N013980/1 and the Alan Turing Institute EP/N510129/1. For the second author this study has been funded by the Russian Academic Excellence Project ’5-100’.

Citation Information: Random Operators and Stochastic Equations, Volume 26, Issue 4, Pages 225–234, ISSN (Online) 1569-397X, ISSN (Print) 0926-6364, DOI: https://doi.org/10.1515/rose-2018-0020.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in