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Conjugate duality in stochastic controls with delay

Part of: Stochastic analysis Topological linear spaces and related structures Stochastic systems and control

Published online by Cambridge University Press:  17 November 2017

Zimeng Wang ,
David J. Hodge  and
Huiling Le
Zimeng Wang*
Affiliation:
University of Nottingham
David J. Hodge*
Affiliation:
University of Nottingham
Huiling Le*
Affiliation:
University of Nottingham
*
* Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
* Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
* Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
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Abstract

In this paper we use the method of conjugate duality to investigate a class of stochastic optimal control problems where state systems are described by stochastic differential equations with delay. For this, we first analyse a stochastic convex problem with delay and derive the expression for the corresponding dual problem. This enables us to obtain the relationship between the optimalities for the two problems. Then, by linking stochastic optimal control problems with delay with a particular type of stochastic convex problem, the result for the latter leads to sufficient maximum principles for the former.

Keywords

Anticipated backward stochastic differential equationconjugate convex functionstochastic delay differential equationstochastic maximum principlestochastic optimal control with delay

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 46A20: Duality theory 60H07: Stochastic calculus of variations and the Malliavin calculus
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 4 , December 2017 , pp. 1011 - 1036
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Bismut, J.-M. (1973). Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44, 384–404. Google Scholar
[2] Bismut, J.-M. (1978). An introductory approach to duality in optimal stochastic control. SIAM Rev. 20, 62–78. Google Scholar
[3] Chang, M.-H., Pang, T. and Yang, Y. (2011). A stochastic portfolio optimization model with bounded memory. Math. Operat. Res. 36, 604–619. Google Scholar
[4] Chen, L. and Wu, Z. (2010). Maximum principle for the stochastic optimal control problem with delay and application. Automatica 46, 1074–1080. CrossRefGoogle Scholar
[5] Cont, R. and Fournié, D.-A. (2013). Functional Itô calculus and stochastic integral representation of martingales. Ann. Prob. 41, 109–133. Google Scholar
[6] Dupire, B. (2009). Functional Itô calculus. Res. Rep. 2009-04, Bloomberg. CrossRefGoogle Scholar
[7] Elsanosi, I., Øksendal, B. and Sulem, A. (2000). Some solvable stochastic control problems with delay. Stoch. Stoch. Reports 71, 69–89. Google Scholar
[8] Hobson, D. G. and Rogers, L. C. G. (1998). Complete models with stochastic volatility. Math. Finance 8, 27–48. Google Scholar
[9] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York. Google Scholar
[10] Larssen, B. (2002). Dynamic programming in stochastic control of systems with delay. Stoch. Stoch. Reports 74, 651–673. CrossRefGoogle Scholar
[11] Larssen, B. and Risebro, N. H. (2003). When are HJB-equations in stochastic control of delay systems finite dimensional? Stoch. Anal. Appl. 21, 643–671. Google Scholar
[12] Meng, Q. and Shen, Y. (2016). Optimal control for stochastic delay evolution equations. Appl. Math. Optimization 74, 53–89. Google Scholar
[13] Øksendal, B. and Sulem, A. (2001). A maximum principle for optimal control of stochastic systems with delay, with applications to finance. In Optimal Control and Partial Differential Equations, IOS, Amsterdam, pp. 64–79. Google Scholar
[14] Øksendal, B., Sulem, A. and Zhang, T. (2011). Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations. Adv. Appl. Prob. 43, 572–596. Google Scholar
[15] Peng, S. and Yang, Z. (2009). Anticipated backward stochastic differential equations. Ann. Prob. 37, 877–902. Google Scholar
[16] Rockafellar, R. T. (1968). Integrals which are convex functionals. Pacific J. Math. 24, 525–539. Google Scholar
[18] Rockafellar, R. T. (1970). Conjugate convex functions in optimal control and the calculus of variations. J. Math. Anal. Appl. 32, 174–222. Google Scholar
[17] Rockafellar, R. T. (1970). Convex Analysis. Princeton University Press. CrossRefGoogle Scholar
[19] Rockafellar, R. T. (1974). Conjugate Duality and Optimization. SIAM, Philadelphia, PA. CrossRefGoogle Scholar
[20] Tsoutsinos, G. I. and Vinter, R. B. (1995). Duality theorems for convex problems with time delay. J. Optimization Theory Appl. 87, 167–195. CrossRefGoogle Scholar
[21] Yong, J. and Zhou, X. Y. (1999). Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York. CrossRefGoogle Scholar