Advertisement

Dynamic Programming Principle and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations for Stochastic Recursive Control Problem with Non-Lipschitz Generator

  • Yu Zhuo
  • Yuchao Dong
  • Jiangyan Pu
Article
  • 48 Downloads

Abstract

In this paper, we consider the stochastic recursive control problem under non-Lipschitz framework. More precisely, we assume that the generator of the backward stochastic differential equation that describes the cost functional is monotonic with respect to the first unknown variable and uniformly continuous in the second unknown variable. A dynamic programming principle is established by making use of a Girsanov transformation argument and the BSDE methods. The value function is then shown to be the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation via truncation methods, approximation techniques and the stability result of viscosity solutions.

Keywords

Stochastic recursive control problem Non-Lipschitz generator Hamilton–Jacobi–Bellman equation Viscosity solution 

Notes

Acknowledgements

Y. Zhuo is supported by the National Natural Science Foundation of China (No. 11171076), and by Science and Technology Commission, Shanghai Municipality (No. 14XD1400400). Y. Dong is supported by Région Pays de la Loire throught the grant PANORisk. J. Pu is supported by the National Natural Science Foundation of China (No. 11701371), and by XuLun Scholar Project of Shanghai Lixin University of Accounting and Finance.

References

  1. 1.
    Alvarez, O., Tourin, A.: Viscosity solutions of non-linear integro-differential equations. Annales de l’Institut Henri Poincare Non Linear Analysis 13(3), 293–317 (1996)CrossRefGoogle Scholar
  2. 2.
    Barles, G., Buckdahn, R., Pardoux, E.: Backward stochastic differential equations and integral-partial differential equations. Stochastics An International Journal of Probability and Stochastic Processes 60(1), 57–83 (1997)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Briand, P., Carmona, R.: BSDEs with polynomial growth generators. Journal of Applied Mathematics and Stochastic Analysis 13(3), 207–238 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Buckdahn, R., Li, J.: Stochastic differential games and viscosity solutions of Hamilton–Jacobi–Bellman–Isaacs equations. SIAM Journal on Control and Optimization 47(1), 444–475 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, Z., Epstein, L.: Ambiguity, risk, and asset returns in continuous time. Econometrica 70(4), 1403–1443 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, L., Wu, Z.: Dynamic programming principle for stochastic recursive optimal control problem with delayed systems. ESAIM: Control, Optimisation and Calculus of Variations 18(4), 1005–1026 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bulletin of the American Mathematical Society 27(1), 1–67 (1992)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fan, S.J., Jiang, L.: Existence and uniqueness result for a backward stochastic differential equation whose generator is Lipschitz continuous in $y$ and uniformly continuous in $z$. Journal of Applied Mathematics and Computing 36(1–2), 1–10 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fan, S.J., Jiang, L.: A generalized comparison theorem for BSDEs and its applications. Journal of Theoretical Probability 25(1), 50–61 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fleming, W.H., Soner, H.M.: Controlled Markov processes and viscosity solutions. Springer, New York (2006)zbMATHGoogle Scholar
  11. 11.
    Jia, G.Y.: Some uniqueness results for one-dimensional BSDEs with uniformly continuous coefficients. Statistics & Probability Letters 79(4), 436–441 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kraft, H., Seifried, F.T., Steffensen, M.: Consumption-portfolio optimization with recursive utility in incomplete markets. Finance and Stochastics 17(1), 161–196 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lazrak, A., Quenez, M.C.: A generalized stochastic differential utility. Mathematics of Operations Research 28(1), 154–180 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lepeltier, J.P., San Martin, J.: Backward stochastic differential equations with continuous coefficient. Statistics & Probability Letters 32(4), 425–430 (1997)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Li, J., Peng, S.: Stochastic optimization theory of backward stochastic differential equations with jumps and viscosity solutions of Hamilton–Jacobi–Bellman equations. Nonlinear Analysis: Theory, Methods & Applications 70(4), 1776–1796 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Li, J., Wei, Q.M.: Optimal control problems of fully coupled FBSDEs and viscosity solutions of Hamilton–Jacobi–Bsellman equations. SIAM Journal on Control and Optimization 52(3), 1622–1662 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Liu, Z.: Control and differential game of forward-backward stochastic systems. PhD thesis, Fudan University (2016)Google Scholar
  18. 18.
    Ma, M., Fan, S.J., Song, X.: ${L}^p~ (p>1)$ solutions of backward stochastic differential equations with monotonic and uniformly continuous generators. Bulletin Des Sciences Mathématiques 137(2), 97–106 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Pardoux, E.: BSDEs, weak convergence and homogenization of semilinear PDEs. In: Clarke, F.H., Stern, R.J. (eds.) Nonlinear analysis, differential equations and control, pp. 503–549. Kluwer Academic, Dordrecht (1999)Google Scholar
  20. 20.
    Peng, S.: A generalized dynamic programming principle and Hamilton–Jacobi–Bellman equation. Stochastics: An International Journal of Probability and Stochastic Processes 38(2), 119–134 (1992)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Pu, J.Y., Zhang, Q.: Dynamic programming principle and associated Hamilton–Jacobi–Bellman equation for stochastic recursive control problem with non-Lipschitz aggregator. ESAIM: Control Optimisation and Calculus of Variations 24, 355–376 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Wu, Z., Yu, Z.: Dynamic programming principle for one kind of stochastic recursive optimal control problem and Hamilton–Jacobi–Bellman equation. SIAM Journal on Control and Optimization 47(5), 2616–2641 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Yan, J.A., Peng, S., Fang, S.Z., Wu, L.M.: Several topics in stochastic analysis. Academic Press of China, Beijing (1997)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Finance and Control Sciences, School of Mathematical SciencesFudan UniversityShanghaiChina
  2. 2.School of FinanceShanghai Lixin University of Accounting and FinanceShanghaiChina

Personalised recommendations