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Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations

Part of: Miscellaneous topics - Partial differential equations Stochastic analysis Existence theories Stochastic systems and control Markov processes

Published online by Cambridge University Press:  01 July 2016

Bernt Øksendal ,
Agnès Sulem  and
Tusheng Zhang
Bernt Øksendal*
Affiliation:
University of Oslo and Norwegian School of Economics and Business Administration
Agnès Sulem*
Affiliation:
INRIA Paris-Rocquencourt
Tusheng Zhang*
Affiliation:
University of Manchester
*
∗ Postal address: Center of Mathematics for Applications (CMA), University of Oslo, Box 1053 Blindern, N-0316 Oslo, Norway. Email address: [email protected]
∗∗ Postal address: INRIA Paris-Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex, France. Email address: [email protected]
∗∗∗ Postal address: School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK. Email address: [email protected]
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Abstract

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We study optimal control problems for (time-)delayed stochastic differential equations with jumps. We establish sufficient and necessary stochastic maximum principles for an optimal control of such systems. The associated adjoint processes are shown to satisfy a (time-)advanced backward stochastic differential equation (ABSDE). Several results on existence and uniqueness of such ABSDEs are shown. The results are illustrated by an application to optimal consumption from a cash flow with delay.

Keywords

Optimal controlstochastic delay equationLévy processmaximum principleHamiltonianadjoint processtime-advanced BSDE

MSC classification

Primary: 93E20: Optimal stochastic control 60J75: Jump processes
Secondary: 60H10: Stochastic ordinary differential equations 60H15: Stochastic partial differential equations 60H20: Stochastic integral equations 49J55: Problems involving randomness 35R60: Partial differential equations with randomness, stochastic partial differential equations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 2 , June 2011 , pp. 572 - 596
Copyright
Copyright © Applied Probability Trust 2011 

References

Chen, L. and Wu, Z. (2010). Maximum principle for the stochastic optimal control problem with delay and application. Automatica 46, 1074–1080.Google Scholar
Chojnowska-Michalik, A. (1978). Representation theorem for general stochastic delay equations. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26, 635–642.Google Scholar
David, D. (2008). Optimal control of stochastic delayed systems with Jumps. Preprint.Google Scholar
El-Karoui, N. and Hamadène, S. (2003). BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations. Stoch. Process. Appl. 107, 145–169.Google Scholar
Elsanousi, I., Øksendal, B. and Sulem, A. (2000). Some solvable stochastic control problems with delay. Stoch. Stoch. Reports 71, 69–89.Google Scholar
Federico, S. (2009). A stochastic control problem with delay arising in a pension fund model. To appear in Finance Stoch.Google Scholar
Gozzi, F. and Marinelli, C. (2004). Stochastic optimal control of delay equations arising in advertising models. In Stochastic Partial Differential Equations and Applications—VII (Lecture Notes Pure Appl. Math. 245), Chapman and Hall/CRC, Boca Raton, FL, pp. 133–148.Google Scholar
Gozzi, F., Marinelli, C. and Savin, S. (2009). On controlled linear diffusions with delay in a model of optimal advertising under uncertainty with memory effects. J. Optimization Theory Appl. 142, 291–321.Google Scholar
Kolmanovski, V. B. and Shaikhet, L. E. (1996). Control of Systems with Aftereffect. American Mathematical Society, Providence, RI.Google Scholar
Larssen, B. (2002). Dynamic programming in stochastic control of systems with delay. Stoch. Stoch. Reports 74, 651–673.Google Scholar
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
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, eds Menaldi, J. L., Rofman, E. and Sulem, A., IOS Press, Amsterdam, pp. 64–79.Google Scholar
Øksendal, B. and Sulem, A. (2007). Applied Stochastic Control of Jump Diffusions, 2nd edn. Springer, Berlin.Google Scholar
Øksendal, B. and Zhang, T. (2010). Optimal control with partial information for stochastic Volterra equations. Internat. J. Stoch. Anal. 2010, 25 pp.Google Scholar
Pardoux, É. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14, 55–61.Google Scholar
Peng, S. and Yang, Z. (2009). Anticipated backward stochastic differential equations. Ann. Prob. 37, 877–902.Google Scholar
Peszat, S. and Zabczyk, J. (2008). Stochastic Partial Differential Equations with Lévy Noise (Encyclopedia Math. Appl. 113). Cambridge University Press.Google Scholar
Situ, R. (1997). On solutions of backward stochastic differential equations with Jumps and applications. Stoch. Process. Appl. 66, 209–236.Google Scholar