Advertisement

The Stochastic Maximum Principle for a Jump-Diffusion Mean-Field Model Involving Impulse Controls and Applications in Finance

  • Cailing Li
  • Zaiming Liu
  • Jinbiao Wu
  • Xiang Huang
Article
  • 5 Downloads

Abstract

This paper establishes a stochastic maximum principle for a stochastic control of mean-field model which is governed by a Lévy process involving continuous and impulse control. The authors also show the existence and uniqueness of the solution to a jump-diffusion mean-field stochastic differential equation involving impulse control. As for its application, a mean-variance portfolio selection problem has been solved.

Keywords

Impulse control jump-diffusion Markowitz’s mean-variance model stochastic maximum principle 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors would like to thank the Editor and the anonymous referees for their constructive and insightful comments for improving the quality of this work, and Prof. Yang Shen at York University, for many helpful discussions and suggestions.

References

  1. [1]
    Oksendal B and Sulem A, Optimal consumption and portfolio with both fixed and proportional transactions costs, Society for Industrial and Applied Mathematics, 2001, 40: 1765–1790.MathSciNetzbMATHGoogle Scholar
  2. [2]
    Cadenillas A and Zapatero F, Classical and impulse stochastic control of the exchange rates and reserves, Mathematical Finance, 2000, 10: 141–156.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Wu Z and Zhang F, Stochastic maximum principle for optimal control problems of forwardbackward delay systems involving impulse controls, Journal of Systems Science & Complexity, 2017, 30(2): 280–306.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Wu J, Wang W, and Peng Y, Optimal control of fully coupled forward-backward stochastic systems with delay and noisy memory, Proceedings of the 36th Chinese Control Conference, Dalian, China, 2017.Google Scholar
  5. [5]
    Hu Y, Liu Z, and Wu J, Optimal impulse control of a mean-reverting inventory with quadratic costs, Journal of Industral & Mangagement Optimization, 2017, 13: 1–16.Google Scholar
  6. [6]
    Wu J and Liu Z, Maximum principle for mean-field zero-sum stochastic differential game with partial information and its application to finance, European Journal of Control, 2017, 37: 8–15.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Yu Z, The stochastic maximum principle for optimal control problems delay systems involving continuous and impulse controls, Automatica, 2012, 48: 2420–2432.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Korn R, Some applications of impulse control in mathematical finance, Mathematical Methods of Operations Research, 1999, 50: 493–518.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Wu J and Liu Z, Optimal control of mean-field backward doubly stochastic systems driven by It Itô-Lévy processes, International Journal of Control, 2018, DOI: 10.1080/00207179.2018.1502473.Google Scholar
  10. [10]
    Cadenillas A and Haussmann U G, The stochastic maximum principle for a singular control problem, Stochastics and Stochastic Reports, 1994, 49: 211–237.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Bahlai S and Chala A, The stochastic maximum principle in optimal control of singular diffusions with non linear coefficients, Random Operators & Stochastic Equations, 2005, 13: 1–10.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Lasry J M and Lions P L, Mean-field games, Japanese Journal of Mathematics, 2007, 2: 229–260.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Andersson D and Djehiche B, A maximum principle for SDEs of mean-field type, Applied Math & Optimization, 2011, 63: 341–356.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Buckdahn R, Djehice B, Li J, et al., Mean-field backward stochastic differential equations: a limit approach, Annals of Probability, 2009, 37: 1524–1565.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Shen Y and Siu T K, The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem, Nonliner Analysis, 2013, 86: 58–73.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Chighoub F, Sohail A, and Alia I, Near-optimality conditions in mean-field control models involving continuous and impulse controls, Nonliner Studies, 2015, 22: 719–738.MathSciNetzbMATHGoogle Scholar
  17. [17]
    Oksendal B and Sulem A, Applied Stochastic Control of Jump Diffusions, Springer, Berlin, 2009.zbMATHGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Cailing Li
    • 1
  • Zaiming Liu
    • 1
  • Jinbiao Wu
    • 1
  • Xiang Huang
    • 2
  1. 1.School of Mathematics and StatisticsCentral South UniversityChangshaChina
  2. 2.Research and Development CenterAgricultural Bank of ChinaBeijingChina

Personalised recommendations