Stochastic Recursive Control Problem Research Articles

Extended backward stochastic Volterra integral equations and their applications to time-Inconsistent stochastic recursive control problems

<p style='text-indent:20px;'>In this paper, we study extended backward stochastic Volterra integral equations (EBSVIEs, for short). We establish the well-posedness under weaker assumptions than the literature, and prove a new kind of regularity property for the solutions. As an application, we investigate, in the open-loop framework, a time-inconsistent stochastic recursive control problem where the cost functional is defined by the solution to a backward stochastic Volterra integral equation (BSVIE, for short). We show that the corresponding adjoint equations become EBSVIEs, and provide a necessary and sufficient condition for an open-loop equilibrium control via variational methods.

Dynamic Programming Principle and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations for Stochastic Recursive Control Problem with Non-Lipschitz Generator

In this paper, we consider the stochastic recursive control problem under non-Lipschitz framework. More precisely, we assume that the generator of the backward stochastic differential equation that describes the cost functional is monotonic with respect to the first unknown variable and uniformly continuous in the second unknown variable. A dynamic programming principle is established by making use of a Girsanov transformation argument and the BSDE methods. The value function is then shown to be the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation via truncation methods, approximation techniques and the stability result of viscosity solutions.

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

In this work we study the stochastic recursive control problem, in which the aggregator (or generator) of the backward stochastic differential equation describing the running cost is continuous but not necessarily Lipschitz with respect to the first unknown variable and the control, and monotonic with respect to the first unknown variable. The dynamic programming principle and the connection between the value function and the viscosity solution of the associated Hamilton-Jacobi-Bellman equation are established in this setting by the generalized comparison theorem for backward stochastic differential equations and the stability of viscosity solutions. Finally we take the control problem of continuous-time Epstein−Zin utility with non-Lipschitz aggregator as an example to demonstrate the application of our study.