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

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

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: oksendal@math.uio.no
∗∗ Postal address: INRIA Paris-Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex, France. Email address: agnes.sulem@inria.fr
∗∗∗ Postal address: School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK. Email address: tusheng.zhang@manchester.ac.uk
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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 

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