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A new class of impulse stochastic control models with non-negative control quantity

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Abstract

This paper advances and studies a new class of impulse stochastic control models. Its state structure is defined by a linear stochastic differential equation of the semi-martingale, its control cost function is a two-variable function of pre-control state quantity and control quantity, and its control quantity is kept non-negative. First this paper constructs a new type of variational equations and proves its solution exists. Using a series of stochastic analysis methods to research the solution function of this type of variational equations, this paper proves the existence of optimal control and analyzes its structure in depth. Because of the big difference between the model in this paper and stochastic control models in previous papers, the analysis method here is quite different from previous ones. It is expected that this paper will have important theoretical significance for stochastic control research, along with wide applicative value in finance control and security management.

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References

  1. Beneš V E, Shepp L A, Witsenhausen H S. Some solvable stochastic control problems. Stochastics, 1980, 4: 39–83

    MathSciNet  MATH  Google Scholar 

  2. Karatzas I. A class of singular stochastic control problems. Adv Appl Prob, 1983, 15: 225–254

    Article  MathSciNet  Google Scholar 

  3. Liu K H, Qin M D, Lu C L. A class of discounted models for singular diffusion control. Chin Sci Bull, 2000, 45: 405–408

    Article  MathSciNet  MATH  Google Scholar 

  4. Liu K H, Qin M D, Lu C L. On sufficient and necessary of existence for a class of singular optimal stochastic control. J Sys Sci Complex, 2003, 16: 424–437

    MathSciNet  MATH  Google Scholar 

  5. Liu K H, Qin M D, Lu C L. A class of stationary models of singular stochastic control. Acta Math Sci, 2004, 24: 139–150

    MathSciNet  MATH  Google Scholar 

  6. Liu K H. From stopping problems to discounted problems (in Chinese). J Sys Sci Math Scis, 1991, 11: 263–266

    MATH  Google Scholar 

  7. Liu K H, Qin M D, Lu C L. On the nonsymmetric stationary models of singular stochastic control for diffusions (in Chinese). Acta Math Sci, 2005, 25: 852–862

    MathSciNet  MATH  Google Scholar 

  8. Liu K H. A class of extended singular stochastic control problems (in Chinese). Acta Math Appl Sin, 1989, 12: 174–182

    MATH  Google Scholar 

  9. Richard S F. Optimal impulse control of a diffusion process with both fixed and proportional costs of control. SIAM J Control Optim, 1979, 1: 79–91

    MathSciNet  Google Scholar 

  10. øksendal B. Stochastic control problems where small intervention costs have big effects. Appl Math Optim, 1999, 40: 355–375

    Article  MathSciNet  Google Scholar 

  11. Cadenillas A. Optimal central bank intervention in the foreign exchange market. J Econ Theory, 1999, 87: 218–242

    Article  MathSciNet  MATH  Google Scholar 

  12. Wang H. Some control problems with random intervention times. Adv Appl Prob, 2001, 33: 404–422

    Article  MATH  Google Scholar 

  13. Ohnishi M, Tsujimura M. An impulse control of a geometric Brownian motion with quadratic costs. Europ J Oper Res, 2006, 168: 311–321

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu K H. The existence of the optimal control of Richard’s model and the structure of its cost function (in Chinese). Acta Math Appl Sin, 1986, 9: 385–408

    MATH  Google Scholar 

  15. Liu K H. A class of impulse control and the analytical expression of its optimal cost function (in Chinese). Acta Math Appl Sin, 1990, 13: 385–390

    MATH  Google Scholar 

  16. Liu K H. The problem of average expected cost of Richard’s model (in Chinese). Acta Math Sin, 1988, 31: 786–793

    MATH  Google Scholar 

  17. Liu K H, Lu C L, Qin M D. A class of stationary impulse stochastic control with state of semi-martingale (in Chinese). J Sys Sci Math Sci, 2004, 24: 249–260

    MathSciNet  MATH  Google Scholar 

  18. Liu K H. A class of extended impulse controls related to semimartingale (I) (in Chinese). J Sys Sci Math Scis, 2005, 25: 96–105

    MATH  Google Scholar 

  19. Liu K H. A class of extended impulse controls related to semimartingale (II) (in Chinese). J Sys Sci Math Scis, 2005, 25: 237–248

    Google Scholar 

  20. Liu X P, Liu K H. The generalization of a class of impulse stochastic control models of a geometric brownian motion. Sci China Ser F-Inf Sci, 2009, 52: 983–998

    Article  MATH  Google Scholar 

  21. Protter P. Stochastic Integration and Differential Equations. Beijing: World Publishing Corporation, 1992. 187–284

    Google Scholar 

  22. Klebaner F C. Introduction to Stochastic Calculus with Applications. Beijing: World Publishing Corporation, 2004. 117–137

    Google Scholar 

  23. Liu K H. A problem of symmetric variational equation. Chin Sci Bull, 1995, 40: 1502–1507

    MATH  Google Scholar 

  24. Liu K H. A new class of variational equation problems. Prog Nat Sci, 2003, 13: 149–151

    MATH  Google Scholar 

  25. Liu K H. Boundedness of a derived function of a solution about a class of diffusion variational equations. Acta Math Sin, English Ser, 2004, 20: 463–474

    Article  MATH  Google Scholar 

  26. Liu K H. The problem of a class of diffusion variational equations (in Chinese). Acta Math Sin, 2003, 46: 913–924

    MATH  Google Scholar 

  27. Liu K H. A stationary variational equation problem (in Chinese). J Eng Math, 1999, 16: 11–15

    MATH  Google Scholar 

  28. Loéve M. Probability Theory I. 4th ed. Beijing: World Publishing Corporation, 1977. 103–147

    MATH  Google Scholar 

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Authors and Affiliations

  1. College of Science, Beijing Jiaotong University, Beijing, 100044, China

    XiaoPeng Liu & KunHui Liu

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  1. XiaoPeng Liu
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  2. KunHui Liu
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Correspondence to XiaoPeng Liu.

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Liu, X., Liu, K. A new class of impulse stochastic control models with non-negative control quantity. Sci. China Inf. Sci. 54, 638–652 (2011). https://doi.org/10.1007/s11432-010-4174-7

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