Dynamic programming principle for stochastic recursive optimal control problem with delayed systems

Abstract

Abstract.In this paper, we study one kind of stochastic recursive optimal control problem for the systems described by stochastic differential equations with delay (SDDE). In our framework, not only the dynamics of the systems but also the recursive utility depend on the past path segment of the state process in a general form. We give the dynamic programming principle for this kind of optimal control problems and show that the value function is the viscosity solution of the corresponding infinite dimensional Hamilton-Jacobi-Bellman partial differential equation.Mathematics Subject Classification. 49L20, 60H10, 93E20.

Stochastic recursive optimal control problem with time delay and applications

“…∂t , DΦ, D 2 Φ exist and they are globally bounded and Lipschitz continuous. Then we have the following formula for the generator A, which is a slight modification of Theorem 4.2 in [7], which is also can be seen in [23].…”

“…The rest of this paper is organized as follows. In Section 2, under some suitable assumptions, we investigate that under what conditions on the coefficients, the generalized HJB equation obtained by [7] via dynamic programming can be reduced to a finite dimensional one. An the main result is a stochastic verification theorem.…”

Backward Stochastic Volterra Integral Equations With Non-Lipschitz Time Delayed Generators

“…It will permit us to study a large class of assurance and finance problems whose have been like to the works of Delong (see , has been modeled by BSVIEs with time delayed generators. On the other hand, we hope to extend, in a future paper, the stochastic control problem studied by Chen and Wu in (Chen & Wu, 2012), replacing classical BSDEs by BSVIEs with non-Lipschitz delayed generators.…”

Dynamic programming principle and associated Hamilton-Jacobi-Bellman equation for stochastic recursive control problem with non-Lipschitz aggregator

“…These settings would make much difference in the deduction of the dynamic programming principle and bring much trouble in the verification of conditions for the stability of viscosity solutions which leads to the connection between the value function and the viscosity solution of the associated HJB equation. Although there are some further results on stochastic recursive control problem from the dynamic programming principle point of view, such as Peng [25] (1997) for non-Markovian framework, Buckdahn and Li [6] (2008) for stochastic differential games, Wu and Yu [27] (2008) for the cost functional generated by reflected BSDE, Li and Peng [16] (2009) for the cost functional generated by BSDE with jumps, Chen and Wu [7] (2012) for the state equation with delay, etc., as far as we know there are no existing results in the non-Lipschitz aggregator setting. However, back to the stochastic recursive utilities, the aggregators in many situations are not Lipschitz with respect to the utilities and consumptions.…”