Stochastic recursive optimal control problem with time delay and applications
Jingtao Shi  School of Mathematics, Shandong University, Jinan 250100, 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 HamiltonJacobiBellman (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 forwardbackward 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.
Received: August 2014; Revised: March 2015; Available Online: October 2015. 
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