Second-Order Necessary Conditions for Optimal Control with Recursive Utilities

Abstract

The necessary conditions for an optimal control of a stochastic control problem with recursive utilities is investigated. The first order condition is the the well-known Pontryagin type maximum principle. When the optimal control satisfying such first-order necessary condition is singular in some sense, certain type of the second-order necessary condition will come in naturally. The aim of this paper is to explore such kind of conditions for our optimal control problem.

Singular optimal control problems with recursive utilities of mean‐field type

“…Then it is easy to see . In this situation our second‐order adjoint system becomes the system given by Dong and Meng [17] in the classical case.…”

“…After that, many scholars contributed to this aspect, for example, Bonnans and Silva [14], Zhang and Zhang [15, 16]. In particular, recently, Dong and Meng [17] established the second‐order SMP for singular optimal controls with recursive utilities. However, we should note that due to the mean‐field feature, there are only a few works on the topic of singular optimal controls for mean‐field control systems.…”

“…Compared with the classical case [17], there is a potential obstacle met in our analysis: When dealing with the first‐order variation of , we can adapt Fubini's Theorem to write the first‐order adjoint equation as a single equation of mean‐field type. But as analysing the second‐order variation of , we need to apply Itô's formula to the square of the first‐order variational process, that is, .…”

On optimal solutions of general continuous‐singular stochastic control problem of McKean‐Vlasov type

“…Necessary conditions for optimal stochastic singular control have been investigated by many authors, see for instance previous studies . Necessary conditions for optimal singular stochastic control systems with variable delay have been studied in Aghayeva and Morali .…”

“…Necessary conditions for general optimal singular stochastic control problems have been derived by Dufour and Miller . When first‐order necessary condition is singular in some sense, second‐order necessary conditions for optimal stochastic control with recursive utilities have been studied by Dong and Meng . A second‐order maximum principle for singular optimal controls with recursive utilities of stochastic delay systems has been considered by Hao and Meng .…”