Mathematical Control and Related Fields (MCRF)

Stochastic recursive optimal control problem with time delay and applications

Pages: 859 - 888, Volume 5, Issue 4, December 2015      doi:10.3934/mcrf.2015.5.859

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Jingtao Shi - School of Mathematics, Shandong University, Jinan 250100, China (email)
Juanjuan Xu - School of Control Science and Engineering, Shandong University, Jinan 250061, China (email)
Huanshui Zhang - School of Control Science and Engineering, Shandong University, Jinan 250061, China (email)

Abstract: This paper is concerned with a stochastic recursive optimal control problem with time delay, where the controlled system is described by a stochastic differential delayed equation (SDDE) and the cost functional is formulated as the solution to a backward SDDE (BSDDE). When there are only the pointwise and distributed time delays in the state variable, a generalized Hamilton-Jacobi-Bellman (HJB) equation for the value function in finite dimensional space is obtained, applying dynamic programming principle. This generalized HJB equation admits a smooth solution when the coefficients satisfy a particular system of first order partial differential equations (PDEs). A sufficient maximum principle is derived, where the adjoint equation is a forward-backward SDDE (FBSDDE). Under some differentiability assumptions, the relationship between the value function, the adjoint processes and the generalized Hamiltonian function is obtained. A consumption and portfolio optimization problem with recursive utility in the financial market, is discussed to show the applications of our result. Explicit solutions in a finite dimensional space derived by the two different approaches, coincide.

Keywords:  Stochastic optimal control, backward stochastic differential equation, stochastic differential delayed equation, recursive utility, generalized HJB equation, maximum principle.
Mathematics Subject Classification:  Primary: 93E20, 60H10; Secondary: 34K50, 91G80.

Received: August 2014;      Revised: March 2015;      Available Online: October 2015.