Advances in Applied Probability

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

Bernt Øksendal, Agnès Sulem, and Tusheng Zhang

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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.

Article information

Adv. in Appl. Probab., Volume 43, Number 2 (2011), 572-596.

First available in Project Euclid: 21 June 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 93EXX 93E20: Optimal stochastic control 60J75: Jump processes
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H15: Stochastic partial differential equations [See also 35R60] 60H20: Stochastic integral equations 49J55: Problems involving randomness [See also 93E20] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

Optimal control stochastic delay equation Lévy process maximum principle Hamiltonian adjoint process time-advanced BSDE


Øksendal, Bernt; Sulem, Agnès; Zhang, Tusheng. Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations. Adv. in Appl. Probab. 43 (2011), no. 2, 572--596.

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